3.11.88 \(\int \frac {1}{\sqrt [3]{-b x^2+a x^3} (b+a x^4)} \, dx\)

Optimal. Leaf size=81 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^{12}-4 \text {$\#$1}^9 a+6 \text {$\#$1}^6 a^2-4 \text {$\#$1}^3 a^3+a^4+a b^3\& ,\frac {\log \left (\sqrt [3]{a x^3-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{4 b} \]

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Rubi [B]  time = 1.06, antiderivative size = 1221, normalized size of antiderivative = 15.07, number of steps used = 11, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2056, 6725, 912, 91} \begin {gather*} -\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{a-\sqrt {-\sqrt {-a}} b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a-\sqrt {-\sqrt {-a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{a+\sqrt {-\sqrt {-a}} b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a+\sqrt {-\sqrt {-a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{a-\sqrt [4]{-a} b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{a+\sqrt [4]{-a} b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [4]{b}-\sqrt {-\sqrt {-a}} x\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {-a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt {-\sqrt {-a}} x+\sqrt [4]{b}\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {-a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [4]{b}-\sqrt [4]{-a} x\right )}{8 \sqrt [3]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [4]{-a} x+\sqrt [4]{b}\right )}{8 \sqrt [3]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [3]{a-\sqrt {-\sqrt {-a}} b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {-a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [3]{a+\sqrt {-\sqrt {-a}} b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {-a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [3]{a-\sqrt [4]{-a} b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [3]{a+\sqrt [4]{-a} b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [3]{a x^3-b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((-(b*x^2) + a*x^3)^(1/3)*(b + a*x^4)),x]

[Out]

-1/4*(Sqrt[3]*x^(2/3)*(-b + a*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a*x)^(1/3))/(Sqrt[3]*(a - Sqrt[-Sqrt[-a]]*b
^(3/4))^(1/3)*x^(1/3))])/((a - Sqrt[-Sqrt[-a]]*b^(3/4))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (Sqrt[3]*x^(2/3)*(
-b + a*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a*x)^(1/3))/(Sqrt[3]*(a + Sqrt[-Sqrt[-a]]*b^(3/4))^(1/3)*x^(1/3))]
)/(4*(a + Sqrt[-Sqrt[-a]]*b^(3/4))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (Sqrt[3]*x^(2/3)*(-b + a*x)^(1/3)*ArcTa
n[1/Sqrt[3] + (2*(-b + a*x)^(1/3))/(Sqrt[3]*(a - (-a)^(1/4)*b^(3/4))^(1/3)*x^(1/3))])/(4*(a - (-a)^(1/4)*b^(3/
4))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (Sqrt[3]*x^(2/3)*(-b + a*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a*x)^(1/
3))/(Sqrt[3]*(a + (-a)^(1/4)*b^(3/4))^(1/3)*x^(1/3))])/(4*(a + (-a)^(1/4)*b^(3/4))^(1/3)*b*(-(b*x^2) + a*x^3)^
(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*Log[b^(1/4) - Sqrt[-Sqrt[-a]]*x])/(8*(a - Sqrt[-Sqrt[-a]]*b^(3/4))^(1/3)*b*
(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*Log[b^(1/4) + Sqrt[-Sqrt[-a]]*x])/(8*(a + Sqrt[-Sqrt[-a]
]*b^(3/4))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*Log[b^(1/4) - (-a)^(1/4)*x])/(8*(a -
(-a)^(1/4)*b^(3/4))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*Log[b^(1/4) + (-a)^(1/4)*x])
/(8*(a + (-a)^(1/4)*b^(3/4))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (3*x^(2/3)*(-b + a*x)^(1/3)*Log[-x^(1/3) + (-
b + a*x)^(1/3)/(a - Sqrt[-Sqrt[-a]]*b^(3/4))^(1/3)])/(8*(a - Sqrt[-Sqrt[-a]]*b^(3/4))^(1/3)*b*(-(b*x^2) + a*x^
3)^(1/3)) - (3*x^(2/3)*(-b + a*x)^(1/3)*Log[-x^(1/3) + (-b + a*x)^(1/3)/(a + Sqrt[-Sqrt[-a]]*b^(3/4))^(1/3)])/
(8*(a + Sqrt[-Sqrt[-a]]*b^(3/4))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (3*x^(2/3)*(-b + a*x)^(1/3)*Log[-x^(1/3)
+ (-b + a*x)^(1/3)/(a - (-a)^(1/4)*b^(3/4))^(1/3)])/(8*(a - (-a)^(1/4)*b^(3/4))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/
3)) - (3*x^(2/3)*(-b + a*x)^(1/3)*Log[-x^(1/3) + (-b + a*x)^(1/3)/(a + (-a)^(1/4)*b^(3/4))^(1/3)])/(8*(a + (-a
)^(1/4)*b^(3/4))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3))

Rule 91

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/
3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*
x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 912

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (b+a x^4\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (b+a x^4\right )} \, dx}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (\frac {1}{2 \sqrt {b} x^{2/3} \sqrt [3]{-b+a x} \left (\sqrt {b}-\sqrt {-a} x^2\right )}+\frac {1}{2 \sqrt {b} x^{2/3} \sqrt [3]{-b+a x} \left (\sqrt {b}+\sqrt {-a} x^2\right )}\right ) \, dx}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (\sqrt {b}-\sqrt {-a} x^2\right )} \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (\sqrt {b}+\sqrt {-a} x^2\right )} \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (\frac {1}{2 \sqrt [4]{b} x^{2/3} \left (\sqrt [4]{b}-\sqrt {-\sqrt {-a}} x\right ) \sqrt [3]{-b+a x}}+\frac {1}{2 \sqrt [4]{b} x^{2/3} \left (\sqrt [4]{b}+\sqrt {-\sqrt {-a}} x\right ) \sqrt [3]{-b+a x}}\right ) \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (\frac {1}{2 \sqrt [4]{b} x^{2/3} \left (\sqrt [4]{b}-\sqrt [4]{-a} x\right ) \sqrt [3]{-b+a x}}+\frac {1}{2 \sqrt [4]{b} x^{2/3} \left (\sqrt [4]{b}+\sqrt [4]{-a} x\right ) \sqrt [3]{-b+a x}}\right ) \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [4]{b}-\sqrt {-\sqrt {-a}} x\right ) \sqrt [3]{-b+a x}} \, dx}{4 b^{3/4} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [4]{b}+\sqrt {-\sqrt {-a}} x\right ) \sqrt [3]{-b+a x}} \, dx}{4 b^{3/4} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [4]{b}-\sqrt [4]{-a} x\right ) \sqrt [3]{-b+a x}} \, dx}{4 b^{3/4} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [4]{b}+\sqrt [4]{-a} x\right ) \sqrt [3]{-b+a x}} \, dx}{4 b^{3/4} \sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{a-\sqrt {-\sqrt {-a}} b^{3/4}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a-\sqrt {-\sqrt {-a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{a+\sqrt {-\sqrt {-a}} b^{3/4}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a+\sqrt {-\sqrt {-a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{a-\sqrt [4]{-a} b^{3/4}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{a+\sqrt [4]{-a} b^{3/4}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [4]{b}-\sqrt {-\sqrt {-a}} x\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {-a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [4]{b}+\sqrt {-\sqrt {-a}} x\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {-a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [4]{b}-\sqrt [4]{-a} x\right )}{8 \sqrt [3]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [4]{b}+\sqrt [4]{-a} x\right )}{8 \sqrt [3]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [3]{a-\sqrt {-\sqrt {-a}} b^{3/4}}}\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {-a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [3]{a+\sqrt {-\sqrt {-a}} b^{3/4}}}\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {-a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [3]{a-\sqrt [4]{-a} b^{3/4}}}\right )}{8 \sqrt [3]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [3]{a+\sqrt [4]{-a} b^{3/4}}}\right )}{8 \sqrt [3]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 167, normalized size = 2.06 \begin {gather*} \frac {3 x \left (\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a-\sqrt [4]{-a} b^{3/4}\right ) x}{a x-b}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a-i \sqrt [4]{-a} b^{3/4}\right ) x}{a x-b}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a+i \sqrt [4]{-a} b^{3/4}\right ) x}{a x-b}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a+\sqrt [4]{-a} b^{3/4}\right ) x}{a x-b}\right )\right )}{4 b \sqrt [3]{x^2 (a x-b)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((-(b*x^2) + a*x^3)^(1/3)*(b + a*x^4)),x]

[Out]

(3*x*(Hypergeometric2F1[1/3, 1, 4/3, ((a - (-a)^(1/4)*b^(3/4))*x)/(-b + a*x)] + Hypergeometric2F1[1/3, 1, 4/3,
((a - I*(-a)^(1/4)*b^(3/4))*x)/(-b + a*x)] + Hypergeometric2F1[1/3, 1, 4/3, ((a + I*(-a)^(1/4)*b^(3/4))*x)/(-
b + a*x)] + Hypergeometric2F1[1/3, 1, 4/3, ((a + (-a)^(1/4)*b^(3/4))*x)/(-b + a*x)]))/(4*b*(x^2*(-b + a*x))^(1
/3))

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IntegrateAlgebraic [A]  time = 0.00, size = 81, normalized size = 1.00 \begin {gather*} -\frac {\text {RootSum}\left [a^4+a b^3-4 a^3 \text {$\#$1}^3+6 a^2 \text {$\#$1}^6-4 a \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((-(b*x^2) + a*x^3)^(1/3)*(b + a*x^4)),x]

[Out]

-1/4*RootSum[a^4 + a*b^3 - 4*a^3*#1^3 + 6*a^2*#1^6 - 4*a*#1^9 + #1^12 & , (-Log[x] + Log[(-(b*x^2) + a*x^3)^(1
/3) - x*#1])/#1 & ]/b

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fricas [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(1/(a*x^3-b*x^2)^(1/3)/(a*x^4+b),x, algorithm="fricas")

[Out]

Timed out

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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(1/(a*x^3-b*x^2)^(1/3)/(a*x^4+b),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{3}} \left (a \,x^{4}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^3-b*x^2)^(1/3)/(a*x^4+b),x)

[Out]

int(1/(a*x^3-b*x^2)^(1/3)/(a*x^4+b),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x^{4} + b\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x^3-b*x^2)^(1/3)/(a*x^4+b),x, algorithm="maxima")

[Out]

integrate(1/((a*x^4 + b)*(a*x^3 - b*x^2)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (a\,x^4+b\right )\,{\left (a\,x^3-b\,x^2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b + a*x^4)*(a*x^3 - b*x^2)^(1/3)),x)

[Out]

int(1/((b + a*x^4)*(a*x^3 - b*x^2)^(1/3)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x^{2} \left (a x - b\right )} \left (a x^{4} + b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x**3-b*x**2)**(1/3)/(a*x**4+b),x)

[Out]

Integral(1/((x**2*(a*x - b))**(1/3)*(a*x**4 + b)), x)

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