3.71 \(\int \frac{(a+b x)^3}{c+d x^3} \, dx\)
Optimal. Leaf size=222 \[ \frac{\left (3 a^2 b \sqrt [3]{c} d^{2/3}+a^3 (-d)+b^3 c\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}-\frac{\left (3 a^2 b \sqrt [3]{c} d^{2/3}+a^3 (-d)+b^3 c\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac{\left (-3 a^2 b \sqrt [3]{c} d^{2/3}+a^3 (-d)+b^3 c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{4/3}}+\frac{a b^2 \log \left (c+d x^3\right )}{d}+\frac{b^3 x}{d} \]
[Out]
(b^3*x)/d + ((b^3*c - 3*a^2*b*c^(1/3)*d^(2/3) - a^3*d)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqr
t[3]*c^(2/3)*d^(4/3)) - ((b^3*c + 3*a^2*b*c^(1/3)*d^(2/3) - a^3*d)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(2/3)*d^(4/3
)) + ((b^3*c + 3*a^2*b*c^(1/3)*d^(2/3) - a^3*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(2/3)*d^(
4/3)) + (a*b^2*Log[c + d*x^3])/d
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Rubi [A] time = 0.318566, antiderivative size = 222, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used
= {1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{\left (3 a^2 b \sqrt [3]{c} d^{2/3}+a^3 (-d)+b^3 c\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}-\frac{\left (3 a^2 b \sqrt [3]{c} d^{2/3}+a^3 (-d)+b^3 c\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac{\left (-3 a^2 b \sqrt [3]{c} d^{2/3}+a^3 (-d)+b^3 c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{4/3}}+\frac{a b^2 \log \left (c+d x^3\right )}{d}+\frac{b^3 x}{d} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^3/(c + d*x^3),x]
[Out]
(b^3*x)/d + ((b^3*c - 3*a^2*b*c^(1/3)*d^(2/3) - a^3*d)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqr
t[3]*c^(2/3)*d^(4/3)) - ((b^3*c + 3*a^2*b*c^(1/3)*d^(2/3) - a^3*d)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(2/3)*d^(4/3
)) + ((b^3*c + 3*a^2*b*c^(1/3)*d^(2/3) - a^3*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(2/3)*d^(
4/3)) + (a*b^2*Log[c + d*x^3])/d
Rule 1887
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]
Rule 1871
Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] || !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]
Rule 1860
Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s
*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/
b]
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 634
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
Rule 617
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 260
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]
Rubi steps
\begin{align*} \int \frac{(a+b x)^3}{c+d x^3} \, dx &=\int \left (\frac{b^3}{d}-\frac{b^3 c-a^3 d-3 a^2 b d x-3 a b^2 d x^2}{d \left (c+d x^3\right )}\right ) \, dx\\ &=\frac{b^3 x}{d}-\frac{\int \frac{b^3 c-a^3 d-3 a^2 b d x-3 a b^2 d x^2}{c+d x^3} \, dx}{d}\\ &=\frac{b^3 x}{d}+\left (3 a b^2\right ) \int \frac{x^2}{c+d x^3} \, dx-\frac{\int \frac{b^3 c-a^3 d-3 a^2 b d x}{c+d x^3} \, dx}{d}\\ &=\frac{b^3 x}{d}+\frac{a b^2 \log \left (c+d x^3\right )}{d}-\frac{\int \frac{\sqrt [3]{c} \left (-3 a^2 b \sqrt [3]{c} d+2 \sqrt [3]{d} \left (b^3 c-a^3 d\right )\right )+\sqrt [3]{d} \left (-3 a^2 b \sqrt [3]{c} d-\sqrt [3]{d} \left (b^3 c-a^3 d\right )\right ) x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{3 c^{2/3} d^{4/3}}-\frac{\left (b^3 c+3 a^2 b \sqrt [3]{c} d^{2/3}-a^3 d\right ) \int \frac{1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{3 c^{2/3} d}\\ &=\frac{b^3 x}{d}-\frac{\left (b^3 c+3 a^2 b \sqrt [3]{c} d^{2/3}-a^3 d\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac{a b^2 \log \left (c+d x^3\right )}{d}-\frac{\left (b^3 c-3 a^2 b \sqrt [3]{c} d^{2/3}-a^3 d\right ) \int \frac{1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 \sqrt [3]{c} d}+\frac{\left (b^3 c+3 a^2 b \sqrt [3]{c} d^{2/3}-a^3 d\right ) \int \frac{-\sqrt [3]{c} \sqrt [3]{d}+2 d^{2/3} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{6 c^{2/3} d^{4/3}}\\ &=\frac{b^3 x}{d}-\frac{\left (b^3 c+3 a^2 b \sqrt [3]{c} d^{2/3}-a^3 d\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac{\left (b^3 c+3 a^2 b \sqrt [3]{c} d^{2/3}-a^3 d\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}+\frac{a b^2 \log \left (c+d x^3\right )}{d}-\frac{\left (b^3 c-3 a^2 b \sqrt [3]{c} d^{2/3}-a^3 d\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{c^{2/3} d^{4/3}}\\ &=\frac{b^3 x}{d}+\frac{\left (b^3 c-3 a^2 b \sqrt [3]{c} d^{2/3}-a^3 d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{4/3}}-\frac{\left (b^3 c+3 a^2 b \sqrt [3]{c} d^{2/3}-a^3 d\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac{\left (b^3 c+3 a^2 b \sqrt [3]{c} d^{2/3}-a^3 d\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}+\frac{a b^2 \log \left (c+d x^3\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.170185, size = 214, normalized size = 0.96 \[ \frac{\left (3 a^2 b \sqrt [3]{c} d^{2/3}+a^3 (-d)+b^3 c\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )-2 \left (3 a^2 b \sqrt [3]{c} d^{2/3}+a^3 (-d)+b^3 c\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )+2 \sqrt{3} \left (-3 a^2 b \sqrt [3]{c} d^{2/3}+a^3 (-d)+b^3 c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )+6 a b^2 c^{2/3} \sqrt [3]{d} \log \left (c+d x^3\right )+6 b^3 c^{2/3} \sqrt [3]{d} x}{6 c^{2/3} d^{4/3}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^3/(c + d*x^3),x]
[Out]
(6*b^3*c^(2/3)*d^(1/3)*x + 2*Sqrt[3]*(b^3*c - 3*a^2*b*c^(1/3)*d^(2/3) - a^3*d)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/
3))/Sqrt[3]] - 2*(b^3*c + 3*a^2*b*c^(1/3)*d^(2/3) - a^3*d)*Log[c^(1/3) + d^(1/3)*x] + (b^3*c + 3*a^2*b*c^(1/3)
*d^(2/3) - a^3*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2] + 6*a*b^2*c^(2/3)*d^(1/3)*Log[c + d*x^3])/(6*
c^(2/3)*d^(4/3))
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Maple [A] time = 0.005, size = 325, normalized size = 1.5 \begin{align*}{\frac{{b}^{3}x}{d}}+{\frac{{a}^{3}}{3\,d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{b}^{3}c}{3\,{d}^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{3}}{6\,d}\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{b}^{3}c}{6\,{d}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}{a}^{3}}{3\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}{b}^{3}c}{3\,{d}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b{a}^{2}}{d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{b{a}^{2}}{2\,d}\ln \left ({x}^{2}-\sqrt [3]{{\frac{c}{d}}}x+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{b{a}^{2}\sqrt{3}}{d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{b}^{2}a\ln \left ( d{x}^{3}+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^3/(d*x^3+c),x)
[Out]
b^3*x/d+1/3/d/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*a^3-1/3/d^2/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*b^3*c-1/6/d/(c/d)^(2/3)*
ln(x^2-(c/d)^(1/3)*x+(c/d)^(2/3))*a^3+1/6/d^2/(c/d)^(2/3)*ln(x^2-(c/d)^(1/3)*x+(c/d)^(2/3))*b^3*c+1/3/d/(c/d)^
(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*a^3-1/3/d^2/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c
/d)^(1/3)*x-1))*b^3*c-1/d*a^2*b/(c/d)^(1/3)*ln(x+(c/d)^(1/3))+1/2/d*a^2*b/(c/d)^(1/3)*ln(x^2-(c/d)^(1/3)*x+(c/
d)^(2/3))+1/d*a^2*b*3^(1/2)/(c/d)^(1/3)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))+a*b^2*ln(d*x^3+c)/d
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3/(d*x^3+c),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [C] time = 14.9484, size = 15101, normalized size = 68.02 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3/(d*x^3+c),x, algorithm="fricas")
[Out]
1/12*(12*b^3*x - 2*(6*(1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b
^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c
^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*
(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^
9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1))*
d*log(-3*a*b^8*c^3 + 15*a^4*b^5*c^2*d + 15*a^7*b^2*c*d^2 + 3/4*(6*(1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2*b^4*c +
a^5*b*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a
^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)
/(c^2*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*
c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 -
a^9*d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1))^2*a^2*b*c^2*d^3 - 1/2*(b^6*c^3*d - 20*a^3*b^3*c^2*d^2 + a^6*c*d^3)
*(6*(1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2
*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3
- 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 -
27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) -
(b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1)) - (b^9*c^3 - 3*a^3
*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)*x) + (18*a*b^2 + (6*(1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2*b^4*c + a^5*b
*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^
6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2
*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 -
3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*
d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1))*d + 3*sqrt(1/3)*d*sqrt((12*a^2*b^4*c - 48*a^5*b*d - 12*(6*(1/2)^(2/3)*(
3*a^2*b^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)
*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*
d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c +
a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3
*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1))*a*b^2*c*d - (6*(1/2)^(2/3)*(3*a^2*b
^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/
(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*
a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*
d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^
2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1))^2*c*d^2)/(c*d^2)))*log(3*a*b^8*c^3 - 15*a^
4*b^5*c^2*d - 15*a^7*b^2*c*d^2 - 3/4*(6*(1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))*(-I*sqrt
(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c
*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3) - 6*a*b^
2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*
a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)
*(I*sqrt(3) + 1))^2*a^2*b*c^2*d^3 + 1/2*(b^6*c^3*d - 20*a^3*b^3*c^2*d^2 + a^6*c*d^3)*(6*(1/2)^(2/3)*(3*a^2*b^4
/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c
*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^
6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)
*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*
d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1)) - 2*(b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*
c*d^2 - a^9*d^3)*x + 3/4*sqrt(1/3)*(2*b^6*c^3*d + 14*a^3*b^3*c^2*d^2 + 2*a^6*c*d^3 + 3*(6*(1/2)^(2/3)*(3*a^2*b
^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/
(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*
a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*
d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^
2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1))*a^2*b*c^2*d^3)*sqrt((12*a^2*b^4*c - 48*a^5
*b*d - 12*(6*(1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 -
27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) -
(b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b
^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c
^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1))*a*b^2*c*d
- (6*(1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a
^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c
^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3
- 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4)
- (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1))^2*c*d^2)/(c*d^2)
)) + (18*a*b^2 + (6*(1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6
/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2
*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(5
4*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*
d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1))*d
- 3*sqrt(1/3)*d*sqrt((12*a^2*b^4*c - 48*a^5*b*d - 12*(6*(1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(
c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2
*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4)
)^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^
3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/
(c^2*d^4))^(1/3)*(I*sqrt(3) + 1))*a*b^2*c*d - (6*(1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))
*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*
a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)
- 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c
^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^
4))^(1/3)*(I*sqrt(3) + 1))^2*c*d^2)/(c*d^2)))*log(3*a*b^8*c^3 - 15*a^4*b^5*c^2*d - 15*a^7*b^2*c*d^2 - 3/4*(6*(
1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*
c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*
a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2
*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9
*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1))^2*a^2*b*c^2*d^3 + 1/2*(
b^6*c^3*d - 20*a^3*b^3*c^2*d^2 + a^6*c*d^3)*(6*(1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))*(
-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^
6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3) -
6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2
*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4)
)^(1/3)*(I*sqrt(3) + 1)) - 2*(b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)*x - 3/4*sqrt(1/3)*(2*b^6
*c^3*d + 14*a^3*b^3*c^2*d^2 + 2*a^6*c*d^3 + 3*(6*(1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2*b^4*c + a^5*b*d)/(c*d^2))
*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*
a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)
- 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c
^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^
4))^(1/3)*(I*sqrt(3) + 1))*a^2*b*c^2*d^3)*sqrt((12*a^2*b^4*c - 48*a^5*b*d - 12*(6*(1/2)^(2/3)*(3*a^2*b^4/d^2 -
(2*a^2*b^4*c + a^5*b*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3)
- (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*
c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2
/(c*d^3) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24
*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1))*a*b^2*c*d - (6*(1/2)^(2/3)*(3*a^2*b^4/d^2 - (2*a^2
*b^4*c + a^5*b*d)/(c*d^2))*(-I*sqrt(3) + 1)/(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3) - (b^9*
c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^3*c*d^2 -
a^9*d^3)/(c^2*d^4))^(1/3) - 6*a*b^2/d + (1/2)^(1/3)*(54*a^3*b^6/d^3 - 27*(2*a^2*b^4*c + a^5*b*d)*a*b^2/(c*d^3
) - (b^9*c^3 - 3*a^3*b^6*c^2*d + 3*a^6*b^3*c*d^2 - a^9*d^3)/(c^2*d^4) - (b^9*c^3 - 3*a^3*b^6*c^2*d - 24*a^6*b^
3*c*d^2 - a^9*d^3)/(c^2*d^4))^(1/3)*(I*sqrt(3) + 1))^2*c*d^2)/(c*d^2))))/d
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Sympy [A] time = 2.64378, size = 245, normalized size = 1.1 \begin{align*} \frac{b^{3} x}{d} + \operatorname{RootSum}{\left (27 t^{3} c^{2} d^{4} - 81 t^{2} a b^{2} c^{2} d^{3} + t \left (27 a^{5} b c d^{3} + 54 a^{2} b^{4} c^{2} d^{2}\right ) - a^{9} d^{3} + 3 a^{6} b^{3} c d^{2} - 3 a^{3} b^{6} c^{2} d + b^{9} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{27 t^{2} a^{2} b c^{2} d^{3} + 3 t a^{6} c d^{3} - 60 t a^{3} b^{3} c^{2} d^{2} + 3 t b^{6} c^{3} d + 15 a^{7} b^{2} c d^{2} + 15 a^{4} b^{5} c^{2} d - 3 a b^{8} c^{3}}{a^{9} d^{3} + 24 a^{6} b^{3} c d^{2} + 3 a^{3} b^{6} c^{2} d - b^{9} c^{3}} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**3/(d*x**3+c),x)
[Out]
b**3*x/d + RootSum(27*_t**3*c**2*d**4 - 81*_t**2*a*b**2*c**2*d**3 + _t*(27*a**5*b*c*d**3 + 54*a**2*b**4*c**2*d
**2) - a**9*d**3 + 3*a**6*b**3*c*d**2 - 3*a**3*b**6*c**2*d + b**9*c**3, Lambda(_t, _t*log(x + (27*_t**2*a**2*b
*c**2*d**3 + 3*_t*a**6*c*d**3 - 60*_t*a**3*b**3*c**2*d**2 + 3*_t*b**6*c**3*d + 15*a**7*b**2*c*d**2 + 15*a**4*b
**5*c**2*d - 3*a*b**8*c**3)/(a**9*d**3 + 24*a**6*b**3*c*d**2 + 3*a**3*b**6*c**2*d - b**9*c**3))))
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Giac [A] time = 1.09219, size = 327, normalized size = 1.47 \begin{align*} \frac{b^{3} x}{d} + \frac{a b^{2} \log \left ({\left | d x^{3} + c \right |}\right )}{d} - \frac{\sqrt{3}{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-c d^{2}\right )^{\frac{1}{3}} a^{3} d + 3 \, \left (-c d^{2}\right )^{\frac{2}{3}} a^{2} b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{3 \, c d^{2}} - \frac{{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b^{3} c - \left (-c d^{2}\right )^{\frac{1}{3}} a^{3} d - 3 \, \left (-c d^{2}\right )^{\frac{2}{3}} a^{2} b\right )} \log \left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \, c d^{2}} - \frac{{\left (3 \, a^{2} b d^{3} \left (-\frac{c}{d}\right )^{\frac{1}{3}} - b^{3} c d^{2} + a^{3} d^{3}\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \, c d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3/(d*x^3+c),x, algorithm="giac")
[Out]
b^3*x/d + a*b^2*log(abs(d*x^3 + c))/d - 1/3*sqrt(3)*((-c*d^2)^(1/3)*b^3*c - (-c*d^2)^(1/3)*a^3*d + 3*(-c*d^2)^
(2/3)*a^2*b)*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(c*d^2) - 1/6*((-c*d^2)^(1/3)*b^3*c - (-c*d
^2)^(1/3)*a^3*d - 3*(-c*d^2)^(2/3)*a^2*b)*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(c*d^2) - 1/3*(3*a^2*b*d^3*
(-c/d)^(1/3) - b^3*c*d^2 + a^3*d^3)*(-c/d)^(1/3)*log(abs(x - (-c/d)^(1/3)))/(c*d^3)