3.30.67 \(\int \frac {x^2 (b+a x^3) (-b p+3 a q+2 a p x^3)}{(q+p x^3)^{2/3} (b^3 c+d q+(3 a b^2 c+d p) x^3+3 a^2 b c x^6+a^3 c x^9)} \, dx\)
Optimal. Leaf size=367 \[ \frac {1}{3} p \text {RootSum}\left [\text {$\#$1}^9 a^3 c-3 \text {$\#$1}^6 a^3 c q+3 \text {$\#$1}^6 a^2 b c p+3 \text {$\#$1}^3 a^3 c q^2-6 \text {$\#$1}^3 a^2 b c p q+3 \text {$\#$1}^3 a b^2 c p^2+\text {$\#$1}^3 d p^3-a^3 c q^3+3 a^2 b c p q^2-3 a b^2 c p^2 q+b^3 c p^3\& ,\frac {2 \text {$\#$1}^6 a^2 \log \left (\sqrt [3]{p x^3+q}-\text {$\#$1}\right )-\text {$\#$1}^3 a^2 q \log \left (\sqrt [3]{p x^3+q}-\text {$\#$1}\right )+\text {$\#$1}^3 a b p \log \left (\sqrt [3]{p x^3+q}-\text {$\#$1}\right )-a^2 q^2 \log \left (\sqrt [3]{p x^3+q}-\text {$\#$1}\right )+2 a b p q \log \left (\sqrt [3]{p x^3+q}-\text {$\#$1}\right )-b^2 p^2 \log \left (\sqrt [3]{p x^3+q}-\text {$\#$1}\right )}{3 \text {$\#$1}^8 a^3 c-6 \text {$\#$1}^5 a^3 c q+6 \text {$\#$1}^5 a^2 b c p+3 \text {$\#$1}^2 a^3 c q^2-6 \text {$\#$1}^2 a^2 b c p q+3 \text {$\#$1}^2 a b^2 c p^2+\text {$\#$1}^2 d p^3}\& \right ] \]
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Rubi [F] time = 16.85, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \begin {gather*} \int \frac {x^2 \left (b+a x^3\right ) \left (-b p+3 a q+2 a p x^3\right )}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+\left (3 a b^2 c+d p\right ) x^3+3 a^2 b c x^6+a^3 c x^9\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(x^2*(b + a*x^3)*(-(b*p) + 3*a*q + 2*a*p*x^3))/((q + p*x^3)^(2/3)*(b^3*c + d*q + (3*a*b^2*c + d*p)*x^3 + 3
*a^2*b*c*x^6 + a^3*c*x^9)),x]
[Out]
p*(b*p - a*q)^2*Defer[Subst][Defer[Int][(-(b^3*c*p^3) - d*p^3*x^3 + 3*a*b^2*c*p^2*(q - x^3) - 3*a^2*b*c*p*(q -
x^3)^2 + a^3*c*(q - x^3)^3)^(-1), x], x, (q + p*x^3)^(1/3)] - a*p*(b*p - a*q)*Defer[Subst][Defer[Int][x^3/(-(
b^3*c*p^3) - d*p^3*x^3 - 3*a^2*b*c*p*(q - x^3)^2 + a^3*c*(q - x^3)^3 - 3*a*b^2*c*p^2*(-q + x^3)), x], x, (q +
p*x^3)^(1/3)] - 2*a^2*p*Defer[Subst][Defer[Int][x^6/(-(b^3*c*p^3) - d*p^3*x^3 - 3*a^2*b*c*p*(q - x^3)^2 + a^3*
c*(q - x^3)^3 - 3*a*b^2*c*p^2*(-q + x^3)), x], x, (q + p*x^3)^(1/3)]
Rubi steps
\begin {align*} \int \frac {x^2 \left (b+a x^3\right ) \left (-b p+3 a q+2 a p x^3\right )}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+\left (3 a b^2 c+d p\right ) x^3+3 a^2 b c x^6+a^3 c x^9\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(b+a x) (-b p+3 a q+2 a p x)}{(q+p x)^{2/3} \left (b^3 c+d q+\left (3 a b^2 c+d p\right ) x+3 a^2 b c x^2+a^3 c x^3\right )} \, dx,x,x^3\right )\\ &=-\left (p \operatorname {Subst}\left (\int \frac {\left (b p+a \left (-q+x^3\right )\right ) \left (b p-a \left (q+2 x^3\right )\right )}{b^3 c p^3+d p^3 x^3+3 a^2 b c p \left (q-x^3\right )^2-a^3 c \left (q-x^3\right )^3+3 a b^2 c p^2 \left (-q+x^3\right )} \, dx,x,\sqrt [3]{q+p x^3}\right )\right )\\ &=-\left (p \operatorname {Subst}\left (\int \frac {\left (b p-a q-2 a x^3\right ) \left (b p-a q+a x^3\right )}{b^3 c p^3+d p^3 x^3+3 a^2 b c p \left (q-x^3\right )^2-a^3 c \left (q-x^3\right )^3+3 a b^2 c p^2 \left (-q+x^3\right )} \, dx,x,\sqrt [3]{q+p x^3}\right )\right )\\ &=-\left (p \operatorname {Subst}\left (\int \left (\frac {2 a b p q \left (1-\frac {b p}{2 a q}-\frac {a q}{2 b p}\right )}{-b^3 c p^3 \left (1-\frac {a q \left (3 b^2 p^2-3 a b p q+a^2 q^2\right )}{b^3 p^3}\right )-3 a b^2 c p^2 \left (1+\frac {d p}{3 a b^2 c}+\frac {a q (-2 b p+a q)}{b^2 p^2}\right ) x^3-3 a^2 b c p \left (1-\frac {a q}{b p}\right ) x^6-a^3 c x^9}+\frac {a b p \left (1-\frac {a q}{b p}\right ) x^3}{-b^3 c p^3 \left (1-\frac {a q \left (3 b^2 p^2-3 a b p q+a^2 q^2\right )}{b^3 p^3}\right )-3 a b^2 c p^2 \left (1+\frac {d p}{3 a b^2 c}+\frac {a q (-2 b p+a q)}{b^2 p^2}\right ) x^3-3 a^2 b c p \left (1-\frac {a q}{b p}\right ) x^6-a^3 c x^9}+\frac {2 a^2 x^6}{-b^3 c p^3 \left (1-\frac {a q \left (3 b^2 p^2-3 a b p q+a^2 q^2\right )}{b^3 p^3}\right )-3 a b^2 c p^2 \left (1+\frac {d p}{3 a b^2 c}+\frac {a q (-2 b p+a q)}{b^2 p^2}\right ) x^3-3 a^2 b c p \left (1-\frac {a q}{b p}\right ) x^6-a^3 c x^9}\right ) \, dx,x,\sqrt [3]{q+p x^3}\right )\right )\\ &=-\left (\left (2 a^2 p\right ) \operatorname {Subst}\left (\int \frac {x^6}{-b^3 c p^3 \left (1-\frac {a q \left (3 b^2 p^2-3 a b p q+a^2 q^2\right )}{b^3 p^3}\right )-3 a b^2 c p^2 \left (1+\frac {d p}{3 a b^2 c}+\frac {a q (-2 b p+a q)}{b^2 p^2}\right ) x^3-3 a^2 b c p \left (1-\frac {a q}{b p}\right ) x^6-a^3 c x^9} \, dx,x,\sqrt [3]{q+p x^3}\right )\right )-(a p (b p-a q)) \operatorname {Subst}\left (\int \frac {x^3}{-b^3 c p^3 \left (1-\frac {a q \left (3 b^2 p^2-3 a b p q+a^2 q^2\right )}{b^3 p^3}\right )-3 a b^2 c p^2 \left (1+\frac {d p}{3 a b^2 c}+\frac {a q (-2 b p+a q)}{b^2 p^2}\right ) x^3-3 a^2 b c p \left (1-\frac {a q}{b p}\right ) x^6-a^3 c x^9} \, dx,x,\sqrt [3]{q+p x^3}\right )+\left (p (b p-a q)^2\right ) \operatorname {Subst}\left (\int \frac {1}{-b^3 c p^3 \left (1-\frac {a q \left (3 b^2 p^2-3 a b p q+a^2 q^2\right )}{b^3 p^3}\right )-3 a b^2 c p^2 \left (1+\frac {d p}{3 a b^2 c}+\frac {a q (-2 b p+a q)}{b^2 p^2}\right ) x^3-3 a^2 b c p \left (1-\frac {a q}{b p}\right ) x^6-a^3 c x^9} \, dx,x,\sqrt [3]{q+p x^3}\right )\\ &=-\left (\left (2 a^2 p\right ) \operatorname {Subst}\left (\int \frac {x^6}{-b^3 c p^3-d p^3 x^3-3 a^2 b c p \left (q-x^3\right )^2+a^3 c \left (q-x^3\right )^3-3 a b^2 c p^2 \left (-q+x^3\right )} \, dx,x,\sqrt [3]{q+p x^3}\right )\right )-(a p (b p-a q)) \operatorname {Subst}\left (\int \frac {x^3}{-b^3 c p^3-d p^3 x^3-3 a^2 b c p \left (q-x^3\right )^2+a^3 c \left (q-x^3\right )^3-3 a b^2 c p^2 \left (-q+x^3\right )} \, dx,x,\sqrt [3]{q+p x^3}\right )+\left (p (b p-a q)^2\right ) \operatorname {Subst}\left (\int \frac {1}{-b^3 c p^3-d p^3 x^3+3 a b^2 c p^2 \left (q-x^3\right )-3 a^2 b c p \left (q-x^3\right )^2+a^3 c \left (q-x^3\right )^3} \, dx,x,\sqrt [3]{q+p x^3}\right )\\ \end {align*}
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Mathematica [B] time = 2.19, size = 923, normalized size = 2.51 \begin {gather*} \frac {1}{6} p \left (\text {RootSum}\left [-c q^3 a^3+c \text {$\#$1}^3 a^3-3 c q \text {$\#$1}^2 a^3+3 c q^2 \text {$\#$1} a^3+3 b c p q^2 a^2+3 b c p \text {$\#$1}^2 a^2-6 b c p q \text {$\#$1} a^2-3 b^2 c p^2 q a+3 b^2 c p^2 \text {$\#$1} a+b^3 c p^3+d p^3 \text {$\#$1}\&,\frac {\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{p x^3+q}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )}{\text {$\#$1}^{2/3}}-\frac {2 \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{p x^3+q}\right )}{\text {$\#$1}^{2/3}}+\frac {\log \left (\left (p x^3+q\right )^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{p x^3+q}+\text {$\#$1}^{2/3}\right )}{\text {$\#$1}^{2/3}}}{3 c q^2 a^3+3 c \text {$\#$1}^2 a^3-6 c q \text {$\#$1} a^3-6 b c p q a^2+6 b c p \text {$\#$1} a^2+3 b^2 c p^2 a+d p^3}\&\right ] (b p-a q)^2+a \left ((a q-b p) \text {RootSum}\left [-c q^3 a^3+c \text {$\#$1}^3 a^3-3 c q \text {$\#$1}^2 a^3+3 c q^2 \text {$\#$1} a^3+3 b c p q^2 a^2+3 b c p \text {$\#$1}^2 a^2-6 b c p q \text {$\#$1} a^2-3 b^2 c p^2 q a+3 b^2 c p^2 \text {$\#$1} a+b^3 c p^3+d p^3 \text {$\#$1}\&,\frac {2 \sqrt {3} \sqrt [3]{\text {$\#$1}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{p x^3+q}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{p x^3+q}\right ) \sqrt [3]{\text {$\#$1}}+\log \left (\left (p x^3+q\right )^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{p x^3+q}+\text {$\#$1}^{2/3}\right ) \sqrt [3]{\text {$\#$1}}}{3 c q^2 a^3+3 c \text {$\#$1}^2 a^3-6 c q \text {$\#$1} a^3-6 b c p q a^2+6 b c p \text {$\#$1} a^2+3 b^2 c p^2 a+d p^3}\&\right ]-2 a \text {RootSum}\left [-c q^3 a^3+c \text {$\#$1}^3 a^3-3 c q \text {$\#$1}^2 a^3+3 c q^2 \text {$\#$1} a^3+3 b c p q^2 a^2+3 b c p \text {$\#$1}^2 a^2-6 b c p q \text {$\#$1} a^2-3 b^2 c p^2 q a+3 b^2 c p^2 \text {$\#$1} a+b^3 c p^3+d p^3 \text {$\#$1}\&,\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{p x^3+q}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right ) \text {$\#$1}^{4/3}-2 \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{p x^3+q}\right ) \text {$\#$1}^{4/3}+\log \left (\left (p x^3+q\right )^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{p x^3+q}+\text {$\#$1}^{2/3}\right ) \text {$\#$1}^{4/3}}{3 c q^2 a^3+3 c \text {$\#$1}^2 a^3-6 c q \text {$\#$1} a^3-6 b c p q a^2+6 b c p \text {$\#$1} a^2+3 b^2 c p^2 a+d p^3}\&\right ]\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x^2*(b + a*x^3)*(-(b*p) + 3*a*q + 2*a*p*x^3))/((q + p*x^3)^(2/3)*(b^3*c + d*q + (3*a*b^2*c + d*p)*x
^3 + 3*a^2*b*c*x^6 + a^3*c*x^9)),x]
[Out]
(p*((b*p - a*q)^2*RootSum[b^3*c*p^3 - 3*a*b^2*c*p^2*q + 3*a^2*b*c*p*q^2 - a^3*c*q^3 + 3*a*b^2*c*p^2*#1 + d*p^3
*#1 - 6*a^2*b*c*p*q*#1 + 3*a^3*c*q^2*#1 + 3*a^2*b*c*p*#1^2 - 3*a^3*c*q*#1^2 + a^3*c*#1^3 & , ((2*Sqrt[3]*ArcTa
n[(1 + (2*(q + p*x^3)^(1/3))/#1^(1/3))/Sqrt[3]])/#1^(2/3) - (2*Log[-(q + p*x^3)^(1/3) + #1^(1/3)])/#1^(2/3) +
Log[(q + p*x^3)^(2/3) + (q + p*x^3)^(1/3)*#1^(1/3) + #1^(2/3)]/#1^(2/3))/(3*a*b^2*c*p^2 + d*p^3 - 6*a^2*b*c*p*
q + 3*a^3*c*q^2 + 6*a^2*b*c*p*#1 - 6*a^3*c*q*#1 + 3*a^3*c*#1^2) & ] + a*((-(b*p) + a*q)*RootSum[b^3*c*p^3 - 3*
a*b^2*c*p^2*q + 3*a^2*b*c*p*q^2 - a^3*c*q^3 + 3*a*b^2*c*p^2*#1 + d*p^3*#1 - 6*a^2*b*c*p*q*#1 + 3*a^3*c*q^2*#1
+ 3*a^2*b*c*p*#1^2 - 3*a^3*c*q*#1^2 + a^3*c*#1^3 & , (2*Sqrt[3]*ArcTan[(1 + (2*(q + p*x^3)^(1/3))/#1^(1/3))/Sq
rt[3]]*#1^(1/3) - 2*Log[-(q + p*x^3)^(1/3) + #1^(1/3)]*#1^(1/3) + Log[(q + p*x^3)^(2/3) + (q + p*x^3)^(1/3)*#1
^(1/3) + #1^(2/3)]*#1^(1/3))/(3*a*b^2*c*p^2 + d*p^3 - 6*a^2*b*c*p*q + 3*a^3*c*q^2 + 6*a^2*b*c*p*#1 - 6*a^3*c*q
*#1 + 3*a^3*c*#1^2) & ] - 2*a*RootSum[b^3*c*p^3 - 3*a*b^2*c*p^2*q + 3*a^2*b*c*p*q^2 - a^3*c*q^3 + 3*a*b^2*c*p^
2*#1 + d*p^3*#1 - 6*a^2*b*c*p*q*#1 + 3*a^3*c*q^2*#1 + 3*a^2*b*c*p*#1^2 - 3*a^3*c*q*#1^2 + a^3*c*#1^3 & , (2*Sq
rt[3]*ArcTan[(1 + (2*(q + p*x^3)^(1/3))/#1^(1/3))/Sqrt[3]]*#1^(4/3) - 2*Log[-(q + p*x^3)^(1/3) + #1^(1/3)]*#1^
(4/3) + Log[(q + p*x^3)^(2/3) + (q + p*x^3)^(1/3)*#1^(1/3) + #1^(2/3)]*#1^(4/3))/(3*a*b^2*c*p^2 + d*p^3 - 6*a^
2*b*c*p*q + 3*a^3*c*q^2 + 6*a^2*b*c*p*#1 - 6*a^3*c*q*#1 + 3*a^3*c*#1^2) & ])))/6
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IntegrateAlgebraic [A] time = 0.68, size = 367, normalized size = 1.00 \begin {gather*} \frac {1}{3} p \text {RootSum}\left [b^3 c p^3-3 a b^2 c p^2 q+3 a^2 b c p q^2-a^3 c q^3+3 a b^2 c p^2 \text {$\#$1}^3+d p^3 \text {$\#$1}^3-6 a^2 b c p q \text {$\#$1}^3+3 a^3 c q^2 \text {$\#$1}^3+3 a^2 b c p \text {$\#$1}^6-3 a^3 c q \text {$\#$1}^6+a^3 c \text {$\#$1}^9\&,\frac {-b^2 p^2 \log \left (\sqrt [3]{q+p x^3}-\text {$\#$1}\right )+2 a b p q \log \left (\sqrt [3]{q+p x^3}-\text {$\#$1}\right )-a^2 q^2 \log \left (\sqrt [3]{q+p x^3}-\text {$\#$1}\right )+a b p \log \left (\sqrt [3]{q+p x^3}-\text {$\#$1}\right ) \text {$\#$1}^3-a^2 q \log \left (\sqrt [3]{q+p x^3}-\text {$\#$1}\right ) \text {$\#$1}^3+2 a^2 \log \left (\sqrt [3]{q+p x^3}-\text {$\#$1}\right ) \text {$\#$1}^6}{3 a b^2 c p^2 \text {$\#$1}^2+d p^3 \text {$\#$1}^2-6 a^2 b c p q \text {$\#$1}^2+3 a^3 c q^2 \text {$\#$1}^2+6 a^2 b c p \text {$\#$1}^5-6 a^3 c q \text {$\#$1}^5+3 a^3 c \text {$\#$1}^8}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(x^2*(b + a*x^3)*(-(b*p) + 3*a*q + 2*a*p*x^3))/((q + p*x^3)^(2/3)*(b^3*c + d*q + (3*a*b^2*c
+ d*p)*x^3 + 3*a^2*b*c*x^6 + a^3*c*x^9)),x]
[Out]
(p*RootSum[b^3*c*p^3 - 3*a*b^2*c*p^2*q + 3*a^2*b*c*p*q^2 - a^3*c*q^3 + 3*a*b^2*c*p^2*#1^3 + d*p^3*#1^3 - 6*a^2
*b*c*p*q*#1^3 + 3*a^3*c*q^2*#1^3 + 3*a^2*b*c*p*#1^6 - 3*a^3*c*q*#1^6 + a^3*c*#1^9 & , (-(b^2*p^2*Log[(q + p*x^
3)^(1/3) - #1]) + 2*a*b*p*q*Log[(q + p*x^3)^(1/3) - #1] - a^2*q^2*Log[(q + p*x^3)^(1/3) - #1] + a*b*p*Log[(q +
p*x^3)^(1/3) - #1]*#1^3 - a^2*q*Log[(q + p*x^3)^(1/3) - #1]*#1^3 + 2*a^2*Log[(q + p*x^3)^(1/3) - #1]*#1^6)/(3
*a*b^2*c*p^2*#1^2 + d*p^3*#1^2 - 6*a^2*b*c*p*q*#1^2 + 3*a^3*c*q^2*#1^2 + 6*a^2*b*c*p*#1^5 - 6*a^3*c*q*#1^5 + 3
*a^3*c*#1^8) & ])/3
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fricas [B] time = 1.28, size = 606, normalized size = 1.65 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} c d \sqrt {\frac {\left (-c^{2} d\right )^{\frac {1}{3}}}{d}} \log \left (\frac {2 \, a^{3} c^{2} x^{9} + 6 \, a^{2} b c^{2} x^{6} + 2 \, b^{3} c^{2} + {\left (6 \, a b^{2} c^{2} - c d p\right )} x^{3} - c d q - 3 \, {\left (a x^{3} + b\right )} {\left (p x^{3} + q\right )}^{\frac {2}{3}} \left (-c^{2} d\right )^{\frac {2}{3}} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (a^{2} x^{6} + 2 \, a b x^{3} + b^{2}\right )} {\left (p x^{3} + q\right )}^{\frac {1}{3}} \left (-c^{2} d\right )^{\frac {2}{3}} + {\left (a c d x^{3} + b c d\right )} {\left (p x^{3} + q\right )}^{\frac {2}{3}} + {\left (d p x^{3} + d q\right )} \left (-c^{2} d\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (-c^{2} d\right )^{\frac {1}{3}}}{d}}}{a^{3} c x^{9} + 3 \, a^{2} b c x^{6} + b^{3} c + {\left (3 \, a b^{2} c + d p\right )} x^{3} + d q}\right ) + \left (-c^{2} d\right )^{\frac {2}{3}} \log \left (a^{2} c^{2} x^{6} + 2 \, a b c^{2} x^{3} + b^{2} c^{2} + {\left (a c x^{3} + b c\right )} {\left (p x^{3} + q\right )}^{\frac {1}{3}} \left (-c^{2} d\right )^{\frac {1}{3}} + {\left (p x^{3} + q\right )}^{\frac {2}{3}} \left (-c^{2} d\right )^{\frac {2}{3}}\right ) - 2 \, \left (-c^{2} d\right )^{\frac {2}{3}} \log \left (a c x^{3} + b c - {\left (p x^{3} + q\right )}^{\frac {1}{3}} \left (-c^{2} d\right )^{\frac {1}{3}}\right )}{6 \, c^{2} d}, \frac {6 \, \sqrt {\frac {1}{3}} c d \sqrt {-\frac {\left (-c^{2} d\right )^{\frac {1}{3}}}{d}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (a c x^{3} + b c\right )} {\left (p x^{3} + q\right )}^{\frac {2}{3}} + {\left (p x^{3} + q\right )} \left (-c^{2} d\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-c^{2} d\right )^{\frac {1}{3}}}{d}}}{c p x^{3} + c q}\right ) + \left (-c^{2} d\right )^{\frac {2}{3}} \log \left (a^{2} c^{2} x^{6} + 2 \, a b c^{2} x^{3} + b^{2} c^{2} + {\left (a c x^{3} + b c\right )} {\left (p x^{3} + q\right )}^{\frac {1}{3}} \left (-c^{2} d\right )^{\frac {1}{3}} + {\left (p x^{3} + q\right )}^{\frac {2}{3}} \left (-c^{2} d\right )^{\frac {2}{3}}\right ) - 2 \, \left (-c^{2} d\right )^{\frac {2}{3}} \log \left (a c x^{3} + b c - {\left (p x^{3} + q\right )}^{\frac {1}{3}} \left (-c^{2} d\right )^{\frac {1}{3}}\right )}{6 \, c^{2} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(a*x^3+b)*(2*a*p*x^3+3*a*q-b*p)/(p*x^3+q)^(2/3)/(b^3*c+d*q+(3*a*b^2*c+d*p)*x^3+3*a^2*b*c*x^6+a^3
*c*x^9),x, algorithm="fricas")
[Out]
[1/6*(3*sqrt(1/3)*c*d*sqrt((-c^2*d)^(1/3)/d)*log((2*a^3*c^2*x^9 + 6*a^2*b*c^2*x^6 + 2*b^3*c^2 + (6*a*b^2*c^2 -
c*d*p)*x^3 - c*d*q - 3*(a*x^3 + b)*(p*x^3 + q)^(2/3)*(-c^2*d)^(2/3) + 3*sqrt(1/3)*(2*(a^2*x^6 + 2*a*b*x^3 + b
^2)*(p*x^3 + q)^(1/3)*(-c^2*d)^(2/3) + (a*c*d*x^3 + b*c*d)*(p*x^3 + q)^(2/3) + (d*p*x^3 + d*q)*(-c^2*d)^(1/3))
*sqrt((-c^2*d)^(1/3)/d))/(a^3*c*x^9 + 3*a^2*b*c*x^6 + b^3*c + (3*a*b^2*c + d*p)*x^3 + d*q)) + (-c^2*d)^(2/3)*l
og(a^2*c^2*x^6 + 2*a*b*c^2*x^3 + b^2*c^2 + (a*c*x^3 + b*c)*(p*x^3 + q)^(1/3)*(-c^2*d)^(1/3) + (p*x^3 + q)^(2/3
)*(-c^2*d)^(2/3)) - 2*(-c^2*d)^(2/3)*log(a*c*x^3 + b*c - (p*x^3 + q)^(1/3)*(-c^2*d)^(1/3)))/(c^2*d), 1/6*(6*sq
rt(1/3)*c*d*sqrt(-(-c^2*d)^(1/3)/d)*arctan(sqrt(1/3)*(2*(a*c*x^3 + b*c)*(p*x^3 + q)^(2/3) + (p*x^3 + q)*(-c^2*
d)^(1/3))*sqrt(-(-c^2*d)^(1/3)/d)/(c*p*x^3 + c*q)) + (-c^2*d)^(2/3)*log(a^2*c^2*x^6 + 2*a*b*c^2*x^3 + b^2*c^2
+ (a*c*x^3 + b*c)*(p*x^3 + q)^(1/3)*(-c^2*d)^(1/3) + (p*x^3 + q)^(2/3)*(-c^2*d)^(2/3)) - 2*(-c^2*d)^(2/3)*log(
a*c*x^3 + b*c - (p*x^3 + q)^(1/3)*(-c^2*d)^(1/3)))/(c^2*d)]
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(a*x^3+b)*(2*a*p*x^3+3*a*q-b*p)/(p*x^3+q)^(2/3)/(b^3*c+d*q+(3*a*b^2*c+d*p)*x^3+3*a^2*b*c*x^6+a^3
*c*x^9),x, algorithm="giac")
[Out]
Timed out
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a \,x^{3}+b \right ) \left (2 a p \,x^{3}+3 a q -b p \right )}{\left (p \,x^{3}+q \right )^{\frac {2}{3}} \left (b^{3} c +d q +\left (3 a \,b^{2} c +d p \right ) x^{3}+3 a^{2} b c \,x^{6}+a^{3} c \,x^{9}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(a*x^3+b)*(2*a*p*x^3+3*a*q-b*p)/(p*x^3+q)^(2/3)/(b^3*c+d*q+(3*a*b^2*c+d*p)*x^3+3*a^2*b*c*x^6+a^3*c*x^9
),x)
[Out]
int(x^2*(a*x^3+b)*(2*a*p*x^3+3*a*q-b*p)/(p*x^3+q)^(2/3)/(b^3*c+d*q+(3*a*b^2*c+d*p)*x^3+3*a^2*b*c*x^6+a^3*c*x^9
),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, a p x^{3} - b p + 3 \, a q\right )} {\left (a x^{3} + b\right )} x^{2}}{{\left (a^{3} c x^{9} + 3 \, a^{2} b c x^{6} + b^{3} c + {\left (3 \, a b^{2} c + d p\right )} x^{3} + d q\right )} {\left (p x^{3} + q\right )}^{\frac {2}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(a*x^3+b)*(2*a*p*x^3+3*a*q-b*p)/(p*x^3+q)^(2/3)/(b^3*c+d*q+(3*a*b^2*c+d*p)*x^3+3*a^2*b*c*x^6+a^3
*c*x^9),x, algorithm="maxima")
[Out]
integrate((2*a*p*x^3 - b*p + 3*a*q)*(a*x^3 + b)*x^2/((a^3*c*x^9 + 3*a^2*b*c*x^6 + b^3*c + (3*a*b^2*c + d*p)*x^
3 + d*q)*(p*x^3 + q)^(2/3)), x)
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mupad [B] time = 15.28, size = 11404, normalized size = 31.07
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2*(b + a*x^3)*(3*a*q - b*p + 2*a*p*x^3))/((q + p*x^3)^(2/3)*(d*q + b^3*c + x^3*(d*p + 3*a*b^2*c) + a^3*
c*x^9 + 3*a^2*b*c*x^6)),x)
[Out]
symsum(log(root(59049*a^2*b*c^6*d^6*h^9*p*q^2 - 59049*a*b^2*c^6*d^6*h^9*p^2*q + 19683*b^3*c^6*d^6*h^9*p^3 - 19
683*a^3*c^6*d^6*h^9*q^3 + 4374*a^2*b^4*c^5*d^4*h^6*p*q + 4374*a^2*b*c^4*d^5*h^6*p*q^2 - 4374*a*b^2*c^4*d^5*h^6
*p^2*q - 2187*a^3*b^3*c^5*d^4*h^6*q^2 - 2187*a*b^5*c^5*d^4*h^6*p^2 - 2187*a^3*c^4*d^5*h^6*q^3 + 1458*b^3*c^4*d
^5*h^6*p^3 - 567*a^2*b^4*c^3*d^3*h^3*p*q + 81*a^2*b*c^2*d^4*h^3*p*q^2 - 81*a*b^2*c^2*d^4*h^3*p^2*q + 567*a^3*b
^3*c^3*d^3*h^3*q^2 - 81*a^3*b^6*c^4*d^2*h^3*q + 162*a*b^5*c^3*d^3*h^3*p^2 + 81*a^2*b^7*c^4*d^2*h^3*p - 81*a^3*
c^2*d^4*h^3*q^3 + 27*b^3*c^2*d^4*h^3*p^3 - 3*a^3*b^3*c*d^2*q^2 - 3*a^3*b^6*c^2*d*q - a^3*d^3*q^3 - a^3*b^9*c^3
, h, k)*((q + p*x^3)^(1/3)*(19683*a^24*b^14*c^8*p^21 + 192*a^19*b^4*c^3*d^5*p^26 + 4272*a^20*b^6*c^4*d^4*p^25
+ 34596*a^21*b^8*c^5*d^3*p^24 + 119745*a^22*b^10*c^6*d^2*p^23 + 413343*a^26*b^12*c^8*p^19*q^2 - 688905*a^27*b^
11*c^8*p^18*q^3 + 688905*a^28*b^10*c^8*p^17*q^4 - 413343*a^29*b^9*c^8*p^16*q^5 + 137781*a^30*b^8*c^8*p^15*q^6
- 19683*a^31*b^7*c^8*p^14*q^7 - 192*a^20*c^2*d^6*p^25*q^2 - 3504*a^23*c^3*d^5*p^22*q^4 - 21060*a^26*c^4*d^4*p^
19*q^6 - 41553*a^29*c^5*d^3*p^16*q^8 + 150174*a^23*b^12*c^7*d*p^22 - 137781*a^25*b^13*c^8*p^20*q - 11664*a^21*
b^2*c^3*d^5*p^24*q^2 - 140292*a^22*b^4*c^4*d^4*p^23*q^2 + 264504*a^23*b^3*c^4*d^4*p^22*q^3 - 241704*a^24*b^2*c
^4*d^4*p^21*q^4 - 431751*a^23*b^6*c^5*d^3*p^22*q^2 + 1516725*a^24*b^5*c^5*d^3*p^21*q^3 - 2358207*a^25*b^4*c^5*
d^3*p^20*q^4 + 2079999*a^26*b^3*c^5*d^3*p^19*q^5 - 1088397*a^27*b^2*c^5*d^3*p^18*q^6 + 714501*a^24*b^8*c^6*d^2
*p^21*q^2 + 694305*a^25*b^7*c^6*d^2*p^20*q^3 - 3354777*a^26*b^6*c^6*d^2*p^19*q^4 + 4758183*a^27*b^5*c^6*d^2*p^
18*q^5 - 3657393*a^28*b^4*c^6*d^2*p^17*q^6 + 1629315*a^29*b^3*c^6*d^2*p^16*q^7 - 393660*a^30*b^2*c^6*d^2*p^15*
q^8 + 192*a^19*b*c^2*d^6*p^26*q - 1024245*a^24*b^11*c^7*d*p^21*q + 4080*a^20*b^3*c^3*d^5*p^25*q + 10896*a^22*b
*c^3*d^5*p^23*q^3 + 22932*a^21*b^5*c^4*d^4*p^24*q + 111348*a^25*b*c^4*d^4*p^20*q^5 - 31443*a^22*b^7*c^5*d^3*p^
23*q + 320031*a^28*b*c^5*d^3*p^17*q^7 - 549585*a^23*b^9*c^6*d^2*p^22*q + 39366*a^31*b*c^6*d^2*p^14*q^9 + 30260
79*a^25*b^10*c^7*d*p^20*q^2 - 5042493*a^26*b^9*c^7*d*p^19*q^3 + 5157675*a^27*b^8*c^7*d*p^18*q^4 - 3288519*a^28
*b^7*c^7*d*p^17*q^5 + 1257525*a^29*b^6*c^7*d*p^16*q^6 - 255879*a^30*b^5*c^7*d*p^15*q^7 + 19683*a^31*b^4*c^7*d*
p^14*q^8) + root(59049*a^2*b*c^6*d^6*h^9*p*q^2 - 59049*a*b^2*c^6*d^6*h^9*p^2*q + 19683*b^3*c^6*d^6*h^9*p^3 - 1
9683*a^3*c^6*d^6*h^9*q^3 + 4374*a^2*b^4*c^5*d^4*h^6*p*q + 4374*a^2*b*c^4*d^5*h^6*p*q^2 - 4374*a*b^2*c^4*d^5*h^
6*p^2*q - 2187*a^3*b^3*c^5*d^4*h^6*q^2 - 2187*a*b^5*c^5*d^4*h^6*p^2 - 2187*a^3*c^4*d^5*h^6*q^3 + 1458*b^3*c^4*
d^5*h^6*p^3 - 567*a^2*b^4*c^3*d^3*h^3*p*q + 81*a^2*b*c^2*d^4*h^3*p*q^2 - 81*a*b^2*c^2*d^4*h^3*p^2*q + 567*a^3*
b^3*c^3*d^3*h^3*q^2 - 81*a^3*b^6*c^4*d^2*h^3*q + 162*a*b^5*c^3*d^3*h^3*p^2 + 81*a^2*b^7*c^4*d^2*h^3*p - 81*a^3
*c^2*d^4*h^3*q^3 + 27*b^3*c^2*d^4*h^3*p^3 - 3*a^3*b^3*c*d^2*q^2 - 3*a^3*b^6*c^2*d*q - a^3*d^3*q^3 - a^3*b^9*c^
3, h, k)*(root(59049*a^2*b*c^6*d^6*h^9*p*q^2 - 59049*a*b^2*c^6*d^6*h^9*p^2*q + 19683*b^3*c^6*d^6*h^9*p^3 - 196
83*a^3*c^6*d^6*h^9*q^3 + 4374*a^2*b^4*c^5*d^4*h^6*p*q + 4374*a^2*b*c^4*d^5*h^6*p*q^2 - 4374*a*b^2*c^4*d^5*h^6*
p^2*q - 2187*a^3*b^3*c^5*d^4*h^6*q^2 - 2187*a*b^5*c^5*d^4*h^6*p^2 - 2187*a^3*c^4*d^5*h^6*q^3 + 1458*b^3*c^4*d^
5*h^6*p^3 - 567*a^2*b^4*c^3*d^3*h^3*p*q + 81*a^2*b*c^2*d^4*h^3*p*q^2 - 81*a*b^2*c^2*d^4*h^3*p^2*q + 567*a^3*b^
3*c^3*d^3*h^3*q^2 - 81*a^3*b^6*c^4*d^2*h^3*q + 162*a*b^5*c^3*d^3*h^3*p^2 + 81*a^2*b^7*c^4*d^2*h^3*p - 81*a^3*c
^2*d^4*h^3*q^3 + 27*b^3*c^2*d^4*h^3*p^3 - 3*a^3*b^3*c*d^2*q^2 - 3*a^3*b^6*c^2*d*q - a^3*d^3*q^3 - a^3*b^9*c^3,
h, k)^2*((q + p*x^3)^(1/3)*(10368*a^19*b^4*c^5*d^6*p^26 + 150336*a^20*b^6*c^6*d^5*p^25 + 589032*a^21*b^8*c^7*
d^4*p^24 + 157464*a^22*b^10*c^8*d^3*p^23 - 1062882*a^23*b^12*c^9*d^2*p^22 - 10368*a^20*c^4*d^7*p^25*q^2 - 1892
16*a^23*c^5*d^6*p^22*q^4 - 1137240*a^26*c^6*d^5*p^19*q^6 - 2243862*a^29*c^7*d^4*p^16*q^8 - 536544*a^21*b^2*c^5
*d^6*p^24*q^2 - 3431160*a^22*b^4*c^6*d^5*p^23*q^2 + 9596880*a^23*b^3*c^6*d^5*p^22*q^3 - 10999152*a^24*b^2*c^6*
d^5*p^21*q^4 + 7650126*a^23*b^6*c^7*d^4*p^22*q^2 + 7944642*a^24*b^5*c^7*d^4*p^21*q^3 - 44738730*a^25*b^4*c^7*d
^4*p^20*q^4 + 64722078*a^26*b^3*c^7*d^4*p^19*q^5 - 45703926*a^27*b^2*c^7*d^4*p^18*q^6 + 49837356*a^24*b^8*c^8*
d^3*p^21*q^2 - 139906764*a^25*b^7*c^8*d^3*p^20*q^3 + 219426084*a^26*b^6*c^8*d^3*p^19*q^4 - 204663834*a^27*b^5*
c^8*d^3*p^18*q^5 + 111405780*a^28*b^4*c^8*d^3*p^17*q^6 - 30941676*a^29*b^3*c^8*d^3*p^16*q^7 + 2125764*a^30*b^2
*c^8*d^3*p^15*q^8 - 29760696*a^25*b^10*c^9*d^2*p^20*q^2 + 59521392*a^26*b^9*c^9*d^2*p^19*q^3 - 74401740*a^27*b
^8*c^9*d^2*p^18*q^4 + 59521392*a^28*b^7*c^9*d^2*p^17*q^5 - 29760696*a^29*b^6*c^9*d^2*p^16*q^6 + 8503056*a^30*b
^5*c^9*d^2*p^15*q^7 - 1062882*a^31*b^4*c^9*d^2*p^14*q^8 + 10368*a^19*b*c^4*d^7*p^26*q + 142560*a^20*b^3*c^5*d^
6*p^25*q + 572832*a^22*b*c^5*d^6*p^23*q^3 + 52488*a^21*b^5*c^6*d^5*p^24*q + 5767848*a^25*b*c^6*d^5*p^20*q^5 -
4320054*a^22*b^7*c^7*d^4*p^23*q + 16100694*a^28*b*c^7*d^4*p^17*q^7 - 7971615*a^23*b^9*c^8*d^3*p^22*q + 531441*
a^31*b*c^8*d^3*p^14*q^9 + 8503056*a^24*b^11*c^9*d^2*p^21*q) + root(59049*a^2*b*c^6*d^6*h^9*p*q^2 - 59049*a*b^2
*c^6*d^6*h^9*p^2*q + 19683*b^3*c^6*d^6*h^9*p^3 - 19683*a^3*c^6*d^6*h^9*q^3 + 4374*a^2*b^4*c^5*d^4*h^6*p*q + 43
74*a^2*b*c^4*d^5*h^6*p*q^2 - 4374*a*b^2*c^4*d^5*h^6*p^2*q - 2187*a^3*b^3*c^5*d^4*h^6*q^2 - 2187*a*b^5*c^5*d^4*
h^6*p^2 - 2187*a^3*c^4*d^5*h^6*q^3 + 1458*b^3*c^4*d^5*h^6*p^3 - 567*a^2*b^4*c^3*d^3*h^3*p*q + 81*a^2*b*c^2*d^4
*h^3*p*q^2 - 81*a*b^2*c^2*d^4*h^3*p^2*q + 567*a^3*b^3*c^3*d^3*h^3*q^2 - 81*a^3*b^6*c^4*d^2*h^3*q + 162*a*b^5*c
^3*d^3*h^3*p^2 + 81*a^2*b^7*c^4*d^2*h^3*p - 81*a^3*c^2*d^4*h^3*q^3 + 27*b^3*c^2*d^4*h^3*p^3 - 3*a^3*b^3*c*d^2*
q^2 - 3*a^3*b^6*c^2*d*q - a^3*d^3*q^3 - a^3*b^9*c^3, h, k)*(root(59049*a^2*b*c^6*d^6*h^9*p*q^2 - 59049*a*b^2*c
^6*d^6*h^9*p^2*q + 19683*b^3*c^6*d^6*h^9*p^3 - 19683*a^3*c^6*d^6*h^9*q^3 + 4374*a^2*b^4*c^5*d^4*h^6*p*q + 4374
*a^2*b*c^4*d^5*h^6*p*q^2 - 4374*a*b^2*c^4*d^5*h^6*p^2*q - 2187*a^3*b^3*c^5*d^4*h^6*q^2 - 2187*a*b^5*c^5*d^4*h^
6*p^2 - 2187*a^3*c^4*d^5*h^6*q^3 + 1458*b^3*c^4*d^5*h^6*p^3 - 567*a^2*b^4*c^3*d^3*h^3*p*q + 81*a^2*b*c^2*d^4*h
^3*p*q^2 - 81*a*b^2*c^2*d^4*h^3*p^2*q + 567*a^3*b^3*c^3*d^3*h^3*q^2 - 81*a^3*b^6*c^4*d^2*h^3*q + 162*a*b^5*c^3
*d^3*h^3*p^2 + 81*a^2*b^7*c^4*d^2*h^3*p - 81*a^3*c^2*d^4*h^3*q^3 + 27*b^3*c^2*d^4*h^3*p^3 - 3*a^3*b^3*c*d^2*q^
2 - 3*a^3*b^6*c^2*d*q - a^3*d^3*q^3 - a^3*b^9*c^3, h, k)^2*(root(59049*a^2*b*c^6*d^6*h^9*p*q^2 - 59049*a*b^2*c
^6*d^6*h^9*p^2*q + 19683*b^3*c^6*d^6*h^9*p^3 - 19683*a^3*c^6*d^6*h^9*q^3 + 4374*a^2*b^4*c^5*d^4*h^6*p*q + 4374
*a^2*b*c^4*d^5*h^6*p*q^2 - 4374*a*b^2*c^4*d^5*h^6*p^2*q - 2187*a^3*b^3*c^5*d^4*h^6*q^2 - 2187*a*b^5*c^5*d^4*h^
6*p^2 - 2187*a^3*c^4*d^5*h^6*q^3 + 1458*b^3*c^4*d^5*h^6*p^3 - 567*a^2*b^4*c^3*d^3*h^3*p*q + 81*a^2*b*c^2*d^4*h
^3*p*q^2 - 81*a*b^2*c^2*d^4*h^3*p^2*q + 567*a^3*b^3*c^3*d^3*h^3*q^2 - 81*a^3*b^6*c^4*d^2*h^3*q + 162*a*b^5*c^3
*d^3*h^3*p^2 + 81*a^2*b^7*c^4*d^2*h^3*p - 81*a^3*c^2*d^4*h^3*q^3 + 27*b^3*c^2*d^4*h^3*p^3 - 3*a^3*b^3*c*d^2*q^
2 - 3*a^3*b^6*c^2*d*q - a^3*d^3*q^3 - a^3*b^9*c^3, h, k)*(419904*a^18*b^3*c^7*d^8*p^27 + 7243344*a^19*b^5*c^8*
d^7*p^26 + 40389516*a^20*b^7*c^9*d^6*p^25 + 71744535*a^21*b^9*c^10*d^5*p^24 - 419904*a^21*c^7*d^8*p^24*q^3 - 7
243344*a^24*c^8*d^7*p^21*q^5 - 40389516*a^27*c^9*d^6*p^18*q^7 - 71744535*a^30*c^10*d^5*p^15*q^9 + 72433440*a^2
1*b^3*c^8*d^7*p^24*q^2 - 72433440*a^22*b^2*c^8*d^7*p^23*q^3 + 848179836*a^22*b^5*c^9*d^6*p^23*q^2 - 1413633060
*a^23*b^4*c^9*d^6*p^22*q^3 + 1413633060*a^24*b^3*c^9*d^6*p^21*q^4 - 848179836*a^25*b^2*c^9*d^6*p^20*q^5 + 2582
803260*a^23*b^7*c^10*d^5*p^22*q^2 - 6026540940*a^24*b^6*c^10*d^5*p^21*q^3 + 9039811410*a^25*b^5*c^10*d^5*p^20*
q^4 - 9039811410*a^26*b^4*c^10*d^5*p^19*q^5 + 6026540940*a^27*b^3*c^10*d^5*p^18*q^6 - 2582803260*a^28*b^2*c^10
*d^5*p^17*q^7 - 1259712*a^19*b^2*c^7*d^8*p^26*q + 1259712*a^20*b*c^7*d^8*p^25*q^2 - 36216720*a^20*b^4*c^8*d^7*
p^25*q + 36216720*a^23*b*c^8*d^7*p^22*q^4 - 282726612*a^21*b^6*c^9*d^6*p^24*q + 282726612*a^26*b*c^9*d^6*p^19*
q^6 - 645700815*a^22*b^8*c^10*d^5*p^23*q + 645700815*a^29*b*c^10*d^5*p^16*q^8) + (q + p*x^3)^(1/3)*(139968*a^1
9*b^4*c^7*d^7*p^26 + 2204496*a^20*b^6*c^8*d^6*p^25 + 10628820*a^21*b^8*c^9*d^5*p^24 + 14348907*a^22*b^10*c^10*
d^4*p^23 - 139968*a^20*c^6*d^8*p^25*q^2 - 2554416*a^23*c^7*d^7*p^22*q^4 - 15352740*a^26*c^8*d^6*p^19*q^6 - 302
92137*a^29*c^9*d^5*p^16*q^8 - 7243344*a^21*b^2*c^7*d^7*p^24*q^2 - 54718740*a^22*b^4*c^8*d^6*p^23*q^2 + 1314824
40*a^23*b^3*c^8*d^6*p^22*q^3 - 142504920*a^24*b^2*c^8*d^6*p^21*q^4 + 11160261*a^23*b^6*c^9*d^5*p^22*q^2 + 2641
26177*a^24*b^5*c^9*d^5*p^21*q^3 - 688216095*a^25*b^4*c^9*d^5*p^20*q^4 + 837019575*a^26*b^3*c^9*d^5*p^19*q^5 -
561733137*a^27*b^2*c^9*d^5*p^18*q^6 + 516560652*a^24*b^8*c^10*d^4*p^21*q^2 - 1205308188*a^25*b^7*c^10*d^4*p^20
*q^3 + 1807962282*a^26*b^6*c^10*d^4*p^19*q^4 - 1807962282*a^27*b^5*c^10*d^4*p^18*q^5 + 1205308188*a^28*b^4*c^1
0*d^4*p^17*q^6 - 516560652*a^29*b^3*c^10*d^4*p^16*q^7 + 129140163*a^30*b^2*c^10*d^4*p^15*q^8 + 139968*a^19*b*c
^6*d^8*p^26*q + 2134512*a^20*b^3*c^7*d^7*p^25*q + 7523280*a^22*b*c^7*d^7*p^23*q^3 + 4330260*a^21*b^5*c^8*d^6*p
^24*q + 74559204*a^25*b*c^8*d^6*p^20*q^5 - 44109603*a^22*b^7*c^9*d^5*p^23*q + 201416139*a^28*b*c^9*d^5*p^17*q^
7 - 129140163*a^23*b^9*c^10*d^4*p^22*q - 14348907*a^31*b*c^10*d^4*p^14*q^9)) + 31104*a^18*b^3*c^5*d^7*p^27 + 4
89888*a^19*b^5*c^6*d^6*p^26 + 2175336*a^20*b^7*c^7*d^5*p^25 + 669222*a^21*b^9*c^8*d^4*p^24 - 8503056*a^22*b^11
*c^9*d^3*p^23 - 31104*a^21*c^5*d^7*p^24*q^3 - 536544*a^24*c^6*d^6*p^21*q^5 - 2991816*a^27*c^7*d^5*p^18*q^7 - 5
314410*a^30*c^8*d^4*p^15*q^9 + 5365440*a^21*b^3*c^6*d^6*p^24*q^2 - 5458752*a^22*b^2*c^6*d^6*p^23*q^3 + 5813920
8*a^22*b^5*c^7*d^5*p^23*q^2 - 103337208*a^23*b^4*c^7*d^5*p^22*q^3 + 107256312*a^24*b^3*c^7*d^5*p^21*q^4 - 6524
2584*a^25*b^2*c^7*d^5*p^20*q^5 + 115263648*a^23*b^7*c^8*d^4*p^22*q^2 - 345003624*a^24*b^6*c^8*d^4*p^21*q^3 + 6
03480780*a^25*b^5*c^8*d^4*p^20*q^4 - 663002172*a^26*b^4*c^8*d^4*p^19*q^5 + 465148656*a^27*b^3*c^8*d^4*p^18*q^6
- 202656168*a^28*b^2*c^8*d^4*p^17*q^7 - 276349320*a^24*b^9*c^9*d^3*p^21*q^2 + 610094268*a^25*b^8*c^9*d^3*p^20
*q^3 - 863060184*a^26*b^7*c^9*d^3*p^19*q^4 + 810978966*a^27*b^6*c^9*d^3*p^18*q^5 - 505931832*a^28*b^5*c^9*d^3*
p^17*q^6 + 201947580*a^29*b^4*c^9*d^3*p^16*q^7 - 46766808*a^30*b^3*c^9*d^3*p^15*q^8 + 4782969*a^31*b^2*c^9*d^3
*p^14*q^9 - 93312*a^19*b^2*c^5*d^7*p^26*q + 93312*a^20*b*c^5*d^7*p^25*q^2 - 2589408*a^20*b^4*c^6*d^6*p^25*q +
2729376*a^23*b*c^6*d^6*p^22*q^4 - 17571816*a^21*b^6*c^7*d^5*p^24*q + 21572568*a^26*b*c^7*d^5*p^19*q^6 - 185413
86*a^22*b^8*c^8*d^4*p^23*q + 49955454*a^29*b*c^8*d^4*p^16*q^8 + 72807417*a^23*b^10*c^9*d^3*p^22*q)) + 576*a^18
*b^3*c^3*d^6*p^27 + 13392*a^19*b^5*c^4*d^5*p^26 + 115452*a^20*b^7*c^5*d^4*p^25 + 441423*a^21*b^9*c^6*d^3*p^24
+ 674325*a^22*b^11*c^7*d^2*p^23 - 576*a^21*c^3*d^6*p^24*q^3 - 9936*a^24*c^4*d^5*p^21*q^5 - 55404*a^27*c^5*d^4*
p^18*q^7 - 98415*a^30*c^6*d^3*p^15*q^9 + 216513*a^23*b^13*c^8*d*p^22 + 114912*a^21*b^3*c^4*d^5*p^24*q^2 - 1080
00*a^22*b^2*c^4*d^5*p^23*q^3 + 1855548*a^22*b^5*c^5*d^4*p^23*q^2 - 2736612*a^23*b^4*c^5*d^4*p^22*q^3 + 2449332
*a^24*b^3*c^5*d^4*p^21*q^4 - 1334556*a^25*b^2*c^5*d^4*p^20*q^5 + 10710144*a^23*b^7*c^6*d^3*p^22*q^2 - 20521728
*a^24*b^6*c^6*d^3*p^21*q^3 + 25365798*a^25*b^5*c^6*d^3*p^20*q^4 - 21063726*a^26*b^4*c^6*d^3*p^19*q^5 + 1182146
4*a^27*b^3*c^6*d^3*p^18*q^6 - 4356504*a^28*b^2*c^6*d^3*p^17*q^7 + 20547594*a^24*b^9*c^7*d^2*p^21*q^2 - 4442963
4*a^25*b^8*c^7*d^2*p^20*q^3 + 62020404*a^26*b^7*c^7*d^2*p^19*q^4 - 57919050*a^27*b^6*c^7*d^2*p^18*q^5 + 361511
10*a^28*b^5*c^7*d^2*p^17*q^6 - 14526054*a^29*b^4*c^7*d^2*p^16*q^7 + 3405159*a^30*b^3*c^7*d^2*p^15*q^8 - 354294
*a^31*b^2*c^7*d^2*p^14*q^9 - 1692738*a^24*b^12*c^8*d*p^21*q - 1728*a^19*b^2*c^3*d^6*p^26*q + 1728*a^20*b*c^3*d
^6*p^25*q^2 - 61776*a^20*b^4*c^4*d^5*p^25*q + 51408*a^23*b*c^4*d^5*p^22*q^4 - 704916*a^21*b^6*c^5*d^4*p^24*q +
411156*a^26*b*c^5*d^4*p^19*q^6 - 3262923*a^22*b^8*c^6*d^3*p^23*q + 964467*a^29*b*c^6*d^3*p^16*q^8 - 5569560*a
^23*b^10*c^7*d^2*p^22*q + 5786802*a^25*b^11*c^8*d*p^20*q^2 - 11298042*a^26*b^10*c^8*d*p^19*q^3 + 13778100*a^27
*b^9*c^8*d*p^18*q^4 - 10746918*a^28*b^8*c^8*d*p^17*q^5 + 5235678*a^29*b^7*c^8*d*p^16*q^6 - 1456542*a^30*b^6*c^
8*d*p^15*q^7 + 177147*a^31*b^5*c^8*d*p^14*q^8)))*root(59049*a^2*b*c^6*d^6*h^9*p*q^2 - 59049*a*b^2*c^6*d^6*h^9*
p^2*q + 19683*b^3*c^6*d^6*h^9*p^3 - 19683*a^3*c^6*d^6*h^9*q^3 + 4374*a^2*b^4*c^5*d^4*h^6*p*q + 4374*a^2*b*c^4*
d^5*h^6*p*q^2 - 4374*a*b^2*c^4*d^5*h^6*p^2*q - 2187*a^3*b^3*c^5*d^4*h^6*q^2 - 2187*a*b^5*c^5*d^4*h^6*p^2 - 218
7*a^3*c^4*d^5*h^6*q^3 + 1458*b^3*c^4*d^5*h^6*p^3 - 567*a^2*b^4*c^3*d^3*h^3*p*q + 81*a^2*b*c^2*d^4*h^3*p*q^2 -
81*a*b^2*c^2*d^4*h^3*p^2*q + 567*a^3*b^3*c^3*d^3*h^3*q^2 - 81*a^3*b^6*c^4*d^2*h^3*q + 162*a*b^5*c^3*d^3*h^3*p^
2 + 81*a^2*b^7*c^4*d^2*h^3*p - 81*a^3*c^2*d^4*h^3*q^3 + 27*b^3*c^2*d^4*h^3*p^3 - 3*a^3*b^3*c*d^2*q^2 - 3*a^3*b
^6*c^2*d*q - a^3*d^3*q^3 - a^3*b^9*c^3, h, k), k, 1, 9) + symsum(log(-root(59049*a^2*b*c^3*d^6*h^9*p*q^2 - 590
49*a*b^2*c^3*d^6*h^9*p^2*q + 19683*b^3*c^3*d^6*h^9*p^3 - 19683*a^3*c^3*d^6*h^9*q^3 - 4374*a^2*b^4*c^2*d^4*h^6*
p*q + 2187*a^3*b^3*c^2*d^4*h^6*q^2 + 2187*a*b^5*c^2*d^4*h^6*p^2 + 729*b^3*c*d^5*h^6*p^3 - 81*a^3*b^6*c*d^2*h^3
*q + 81*a^2*b^7*c*d^2*h^3*p + a^3*b^9, h, k)*((q + p*x^3)^(1/3)*(19683*a^24*b^14*c^8*p^21 + 192*a^21*b^8*c^5*d
^3*p^24 + 3024*a^22*b^10*c^6*d^2*p^23 + 413343*a^26*b^12*c^8*p^19*q^2 - 688905*a^27*b^11*c^8*p^18*q^3 + 688905
*a^28*b^10*c^8*p^17*q^4 - 413343*a^29*b^9*c^8*p^16*q^5 + 137781*a^30*b^8*c^8*p^15*q^6 - 19683*a^31*b^7*c^8*p^1
4*q^7 + 14580*a^23*b^12*c^7*d*p^22 - 137781*a^25*b^13*c^8*p^20*q + 9072*a^24*b^8*c^6*d^2*p^21*q^2 - 3024*a^25*
b^7*c^6*d^2*p^20*q^3 - 72900*a^24*b^11*c^7*d*p^21*q - 192*a^22*b^7*c^5*d^3*p^23*q - 9072*a^23*b^9*c^6*d^2*p^22
*q + 145800*a^25*b^10*c^7*d*p^20*q^2 - 145800*a^26*b^9*c^7*d*p^19*q^3 + 72900*a^27*b^8*c^7*d*p^18*q^4 - 14580*
a^28*b^7*c^7*d*p^17*q^5) + root(59049*a^2*b*c^3*d^6*h^9*p*q^2 - 59049*a*b^2*c^3*d^6*h^9*p^2*q + 19683*b^3*c^3*
d^6*h^9*p^3 - 19683*a^3*c^3*d^6*h^9*q^3 - 4374*a^2*b^4*c^2*d^4*h^6*p*q + 2187*a^3*b^3*c^2*d^4*h^6*q^2 + 2187*a
*b^5*c^2*d^4*h^6*p^2 + 729*b^3*c*d^5*h^6*p^3 - 81*a^3*b^6*c*d^2*h^3*q + 81*a^2*b^7*c*d^2*h^3*p + a^3*b^9, h, k
)*(root(59049*a^2*b*c^3*d^6*h^9*p*q^2 - 59049*a*b^2*c^3*d^6*h^9*p^2*q + 19683*b^3*c^3*d^6*h^9*p^3 - 19683*a^3*
c^3*d^6*h^9*q^3 - 4374*a^2*b^4*c^2*d^4*h^6*p*q + 2187*a^3*b^3*c^2*d^4*h^6*q^2 + 2187*a*b^5*c^2*d^4*h^6*p^2 + 7
29*b^3*c*d^5*h^6*p^3 - 81*a^3*b^6*c*d^2*h^3*q + 81*a^2*b^7*c*d^2*h^3*p + a^3*b^9, h, k)^2*((q + p*x^3)^(1/3)*(
15552*a^20*b^6*c^6*d^5*p^25 + 233280*a^21*b^8*c^7*d^4*p^24 + 1023516*a^22*b^10*c^8*d^3*p^23 + 1062882*a^23*b^1
2*c^9*d^2*p^22 + 15552*a^22*b^4*c^6*d^5*p^23*q^2 + 1399680*a^23*b^6*c^7*d^4*p^22*q^2 - 933120*a^24*b^5*c^7*d^4
*p^21*q^3 + 233280*a^25*b^4*c^7*d^4*p^20*q^4 + 15352740*a^24*b^8*c^8*d^3*p^21*q^2 - 20470320*a^25*b^7*c^8*d^3*
p^20*q^3 + 15352740*a^26*b^6*c^8*d^3*p^19*q^4 - 6141096*a^27*b^5*c^8*d^3*p^18*q^5 + 1023516*a^28*b^4*c^8*d^3*p
^17*q^6 + 29760696*a^25*b^10*c^9*d^2*p^20*q^2 - 59521392*a^26*b^9*c^9*d^2*p^19*q^3 + 74401740*a^27*b^8*c^9*d^2
*p^18*q^4 - 59521392*a^28*b^7*c^9*d^2*p^17*q^5 + 29760696*a^29*b^6*c^9*d^2*p^16*q^6 - 8503056*a^30*b^5*c^9*d^2
*p^15*q^7 + 1062882*a^31*b^4*c^9*d^2*p^14*q^8 - 31104*a^21*b^5*c^6*d^5*p^24*q - 933120*a^22*b^7*c^7*d^4*p^23*q
- 6141096*a^23*b^9*c^8*d^3*p^22*q - 8503056*a^24*b^11*c^9*d^2*p^21*q) - root(59049*a^2*b*c^3*d^6*h^9*p*q^2 -
59049*a*b^2*c^3*d^6*h^9*p^2*q + 19683*b^3*c^3*d^6*h^9*p^3 - 19683*a^3*c^3*d^6*h^9*q^3 - 4374*a^2*b^4*c^2*d^4*h
^6*p*q + 2187*a^3*b^3*c^2*d^4*h^6*q^2 + 2187*a*b^5*c^2*d^4*h^6*p^2 + 729*b^3*c*d^5*h^6*p^3 - 81*a^3*b^6*c*d^2*
h^3*q + 81*a^2*b^7*c*d^2*h^3*p + a^3*b^9, h, k)*(root(59049*a^2*b*c^3*d^6*h^9*p*q^2 - 59049*a*b^2*c^3*d^6*h^9*
p^2*q + 19683*b^3*c^3*d^6*h^9*p^3 - 19683*a^3*c^3*d^6*h^9*q^3 - 4374*a^2*b^4*c^2*d^4*h^6*p*q + 2187*a^3*b^3*c^
2*d^4*h^6*q^2 + 2187*a*b^5*c^2*d^4*h^6*p^2 + 729*b^3*c*d^5*h^6*p^3 - 81*a^3*b^6*c*d^2*h^3*q + 81*a^2*b^7*c*d^2
*h^3*p + a^3*b^9, h, k)^2*(root(59049*a^2*b*c^3*d^6*h^9*p*q^2 - 59049*a*b^2*c^3*d^6*h^9*p^2*q + 19683*b^3*c^3*
d^6*h^9*p^3 - 19683*a^3*c^3*d^6*h^9*q^3 - 4374*a^2*b^4*c^2*d^4*h^6*p*q + 2187*a^3*b^3*c^2*d^4*h^6*q^2 + 2187*a
*b^5*c^2*d^4*h^6*p^2 + 729*b^3*c*d^5*h^6*p^3 - 81*a^3*b^6*c*d^2*h^3*q + 81*a^2*b^7*c*d^2*h^3*p + a^3*b^9, h, k
)*(419904*a^18*b^3*c^7*d^8*p^27 + 7243344*a^19*b^5*c^8*d^7*p^26 + 40389516*a^20*b^7*c^9*d^6*p^25 + 71744535*a^
21*b^9*c^10*d^5*p^24 - 419904*a^21*c^7*d^8*p^24*q^3 - 7243344*a^24*c^8*d^7*p^21*q^5 - 40389516*a^27*c^9*d^6*p^
18*q^7 - 71744535*a^30*c^10*d^5*p^15*q^9 + 72433440*a^21*b^3*c^8*d^7*p^24*q^2 - 72433440*a^22*b^2*c^8*d^7*p^23
*q^3 + 848179836*a^22*b^5*c^9*d^6*p^23*q^2 - 1413633060*a^23*b^4*c^9*d^6*p^22*q^3 + 1413633060*a^24*b^3*c^9*d^
6*p^21*q^4 - 848179836*a^25*b^2*c^9*d^6*p^20*q^5 + 2582803260*a^23*b^7*c^10*d^5*p^22*q^2 - 6026540940*a^24*b^6
*c^10*d^5*p^21*q^3 + 9039811410*a^25*b^5*c^10*d^5*p^20*q^4 - 9039811410*a^26*b^4*c^10*d^5*p^19*q^5 + 602654094
0*a^27*b^3*c^10*d^5*p^18*q^6 - 2582803260*a^28*b^2*c^10*d^5*p^17*q^7 - 1259712*a^19*b^2*c^7*d^8*p^26*q + 12597
12*a^20*b*c^7*d^8*p^25*q^2 - 36216720*a^20*b^4*c^8*d^7*p^25*q + 36216720*a^23*b*c^8*d^7*p^22*q^4 - 282726612*a
^21*b^6*c^9*d^6*p^24*q + 282726612*a^26*b*c^9*d^6*p^19*q^6 - 645700815*a^22*b^8*c^10*d^5*p^23*q + 645700815*a^
29*b*c^10*d^5*p^16*q^8) - (q + p*x^3)^(1/3)*(139968*a^18*b^2*c^6*d^8*p^27 + 2694384*a^19*b^4*c^7*d^7*p^26 + 17
557236*a^20*b^6*c^8*d^6*p^25 + 40920957*a^21*b^8*c^9*d^5*p^24 + 14348907*a^22*b^10*c^10*d^4*p^23 + 8083152*a^2
1*b^2*c^7*d^7*p^24*q^2 + 175572360*a^22*b^4*c^8*d^6*p^23*q^2 - 175572360*a^23*b^3*c^8*d^6*p^22*q^3 + 87786180*
a^24*b^2*c^8*d^6*p^21*q^4 + 859340097*a^23*b^6*c^9*d^5*p^22*q^2 - 1432233495*a^24*b^5*c^9*d^5*p^21*q^3 + 14322
33495*a^25*b^4*c^9*d^5*p^20*q^4 - 859340097*a^26*b^3*c^9*d^5*p^19*q^5 + 286446699*a^27*b^2*c^9*d^5*p^18*q^6 +
516560652*a^24*b^8*c^10*d^4*p^21*q^2 - 1205308188*a^25*b^7*c^10*d^4*p^20*q^3 + 1807962282*a^26*b^6*c^10*d^4*p^
19*q^4 - 1807962282*a^27*b^5*c^10*d^4*p^18*q^5 + 1205308188*a^28*b^4*c^10*d^4*p^17*q^6 - 516560652*a^29*b^3*c^
10*d^4*p^16*q^7 + 129140163*a^30*b^2*c^10*d^4*p^15*q^8 - 139968*a^19*b*c^6*d^8*p^26*q - 8083152*a^20*b^3*c^7*d
^7*p^25*q - 2694384*a^22*b*c^7*d^7*p^23*q^3 - 87786180*a^21*b^5*c^8*d^6*p^24*q - 17557236*a^25*b*c^8*d^6*p^20*
q^5 - 286446699*a^22*b^7*c^9*d^5*p^23*q - 40920957*a^28*b*c^9*d^5*p^17*q^7 - 129140163*a^23*b^9*c^10*d^4*p^22*
q - 14348907*a^31*b*c^10*d^4*p^14*q^9)) + 81648*a^20*b^7*c^7*d^5*p^25 + 1102248*a^21*b^9*c^8*d^4*p^24 + 372008
7*a^22*b^11*c^9*d^3*p^23 + 489888*a^22*b^5*c^7*d^5*p^23*q^2 - 326592*a^23*b^4*c^7*d^5*p^22*q^3 + 81648*a^24*b^
3*c^7*d^5*p^21*q^4 + 16533720*a^23*b^7*c^8*d^4*p^22*q^2 - 22044960*a^24*b^6*c^8*d^4*p^21*q^3 + 16533720*a^25*b
^5*c^8*d^4*p^20*q^4 - 6613488*a^26*b^4*c^8*d^4*p^19*q^5 + 1102248*a^27*b^3*c^8*d^4*p^18*q^6 + 104162436*a^24*b
^9*c^9*d^3*p^21*q^2 - 208324872*a^25*b^8*c^9*d^3*p^20*q^3 + 260406090*a^26*b^7*c^9*d^3*p^19*q^4 - 208324872*a^
27*b^6*c^9*d^3*p^18*q^5 + 104162436*a^28*b^5*c^9*d^3*p^17*q^6 - 29760696*a^29*b^4*c^9*d^3*p^16*q^7 + 3720087*a
^30*b^3*c^9*d^3*p^15*q^8 - 326592*a^21*b^6*c^7*d^5*p^24*q - 6613488*a^22*b^8*c^8*d^4*p^23*q - 29760696*a^23*b^
10*c^9*d^3*p^22*q)) - 864*a^21*b^9*c^6*d^3*p^24 - 11664*a^22*b^11*c^7*d^2*p^23 - 39366*a^23*b^13*c^8*d*p^22 -
2592*a^23*b^7*c^6*d^3*p^22*q^2 + 864*a^24*b^6*c^6*d^3*p^21*q^3 - 116640*a^24*b^9*c^7*d^2*p^21*q^2 + 116640*a^2
5*b^8*c^7*d^2*p^20*q^3 - 58320*a^26*b^7*c^7*d^2*p^19*q^4 + 11664*a^27*b^6*c^7*d^2*p^18*q^5 + 275562*a^24*b^12*
c^8*d*p^21*q + 2592*a^22*b^8*c^6*d^3*p^23*q + 58320*a^23*b^10*c^7*d^2*p^22*q - 826686*a^25*b^11*c^8*d*p^20*q^2
+ 1377810*a^26*b^10*c^8*d*p^19*q^3 - 1377810*a^27*b^9*c^8*d*p^18*q^4 + 826686*a^28*b^8*c^8*d*p^17*q^5 - 27556
2*a^29*b^7*c^8*d*p^16*q^6 + 39366*a^30*b^6*c^8*d*p^15*q^7)))*root(59049*a^2*b*c^3*d^6*h^9*p*q^2 - 59049*a*b^2*
c^3*d^6*h^9*p^2*q + 19683*b^3*c^3*d^6*h^9*p^3 - 19683*a^3*c^3*d^6*h^9*q^3 - 4374*a^2*b^4*c^2*d^4*h^6*p*q + 218
7*a^3*b^3*c^2*d^4*h^6*q^2 + 2187*a*b^5*c^2*d^4*h^6*p^2 + 729*b^3*c*d^5*h^6*p^3 - 81*a^3*b^6*c*d^2*h^3*q + 81*a
^2*b^7*c*d^2*h^3*p + a^3*b^9, h, k), k, 1, 9)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*(a*x**3+b)*(2*a*p*x**3+3*a*q-b*p)/(p*x**3+q)**(2/3)/(b**3*c+d*q+(3*a*b**2*c+d*p)*x**3+3*a**2*b*
c*x**6+a**3*c*x**9),x)
[Out]
Timed out
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