3.14 \(\int \frac{(c+d x^3)^4}{a+b x^3} \, dx\)

Optimal. Leaf size=252 \[ \frac{d^2 x^4 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{4 b^3}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}-\frac{(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}}+\frac{(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{13/3}}+\frac{d^3 x^7 (4 b c-a d)}{7 b^2}+\frac{d^4 x^{10}}{10 b} \]

[Out]

(d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^4)/(4*b^3
) + (d^3*(4*b*c - a*d)*x^7)/(7*b^2) + (d^4*x^10)/(10*b) - ((b*c - a*d)^4*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[
3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*b^(13/3)) + ((b*c - a*d)^4*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(13/3)) - ((b
*c - a*d)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)*b^(13/3))

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Rubi [A]  time = 0.191462, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {390, 200, 31, 634, 617, 204, 628} \[ \frac{d^2 x^4 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{4 b^3}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}-\frac{(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}}+\frac{(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{13/3}}+\frac{d^3 x^7 (4 b c-a d)}{7 b^2}+\frac{d^4 x^{10}}{10 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^4/(a + b*x^3),x]

[Out]

(d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^4)/(4*b^3
) + (d^3*(4*b*c - a*d)*x^7)/(7*b^2) + (d^4*x^10)/(10*b) - ((b*c - a*d)^4*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[
3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*b^(13/3)) + ((b*c - a*d)^4*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(13/3)) - ((b
*c - a*d)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)*b^(13/3))

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

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]

Rubi steps

\begin{align*} \int \frac{\left (c+d x^3\right )^4}{a+b x^3} \, dx &=\int \left (\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right )}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^3}{b^3}+\frac{d^3 (4 b c-a d) x^6}{b^2}+\frac{d^4 x^9}{b}+\frac{b^4 c^4-4 a b^3 c^3 d+6 a^2 b^2 c^2 d^2-4 a^3 b c d^3+a^4 d^4}{b^4 \left (a+b x^3\right )}\right ) \, dx\\ &=\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac{d^3 (4 b c-a d) x^7}{7 b^2}+\frac{d^4 x^{10}}{10 b}+\frac{(b c-a d)^4 \int \frac{1}{a+b x^3} \, dx}{b^4}\\ &=\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac{d^3 (4 b c-a d) x^7}{7 b^2}+\frac{d^4 x^{10}}{10 b}+\frac{(b c-a d)^4 \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{2/3} b^4}+\frac{(b c-a d)^4 \int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3} b^4}\\ &=\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac{d^3 (4 b c-a d) x^7}{7 b^2}+\frac{d^4 x^{10}}{10 b}+\frac{(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac{(b c-a d)^4 \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{2/3} b^{13/3}}+\frac{(b c-a d)^4 \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{a} b^4}\\ &=\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac{d^3 (4 b c-a d) x^7}{7 b^2}+\frac{d^4 x^{10}}{10 b}+\frac{(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac{(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}}+\frac{(b c-a d)^4 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{2/3} b^{13/3}}\\ &=\frac{d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) x}{b^4}+\frac{d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac{d^3 (4 b c-a d) x^7}{7 b^2}+\frac{d^4 x^{10}}{10 b}-\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{13/3}}+\frac{(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac{(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}}\\ \end{align*}

Mathematica [A]  time = 0.115872, size = 253, normalized size = 1. \[ \frac{105 b^{4/3} d^2 x^4 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )+420 \sqrt [3]{b} d x \left (4 a^2 b c d^2-a^3 d^3-6 a b^2 c^2 d+4 b^3 c^3\right )-\frac{70 (b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{140 (b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac{140 \sqrt{3} (b c-a d)^4 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3}}+60 b^{7/3} d^3 x^7 (4 b c-a d)+42 b^{10/3} d^4 x^{10}}{420 b^{13/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3)^4/(a + b*x^3),x]

[Out]

(420*b^(1/3)*d*(4*b^3*c^3 - 6*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*x + 105*b^(4/3)*d^2*(6*b^2*c^2 - 4*a*b*c*
d + a^2*d^2)*x^4 + 60*b^(7/3)*d^3*(4*b*c - a*d)*x^7 + 42*b^(10/3)*d^4*x^10 + (140*Sqrt[3]*(b*c - a*d)^4*ArcTan
[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(2/3) + (140*(b*c - a*d)^4*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) -
(70*(b*c - a*d)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(2/3))/(420*b^(13/3))

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Maple [B]  time = 0.005, size = 661, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3+c)^4/(b*x^3+a),x)

[Out]

-4/3/b^4/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*a^3*c*d^3+2/b^3/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*a^2*c^2*d^2-4/3/b
^2/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*a*c^3*d+2/3/b^4/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*a^3*c
*d^3-1/b^3/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*a^2*c^2*d^2+2/3/b^2/(1/b*a)^(2/3)*ln(x^2-(1/b*a
)^(1/3)*x+(1/b*a)^(2/3))*a*c^3*d+1/3/b^5/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*a^4*d
^4+1/3/b/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*c^4-1/6/b/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c^4+1
/4*d^4/b^3*x^4*a^2+3/2*d^2/b*x^4*c^2-d^4/b^4*a^3*x+4*d/b*c^3*x-1/6/b^5/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1
/b*a)^(2/3))*a^4*d^4+1/3/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c^4+4*d^3/b^3*a^2*c
*x-d^3/b^2*x^4*a*c-6*d^2/b^2*a*c^2*x+1/3/b^5/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*a^4*d^4-4/3/b^4/(1/b*a)^(2/3)*3
^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*a^3*c*d^3+2/b^3/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(
1/b*a)^(1/3)*x-1))*a^2*c^2*d^2-4/3/b^2/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*a*c^3*d
+1/10*d^4*x^10/b-1/7*d^4/b^2*x^7*a+4/7*d^3/b*x^7*c

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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((d*x^3+c)^4/(b*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.72613, size = 1916, normalized size = 7.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^4/(b*x^3+a),x, algorithm="fricas")

[Out]

[1/420*(42*a^2*b^4*d^4*x^10 + 60*(4*a^2*b^4*c*d^3 - a^3*b^3*d^4)*x^7 + 105*(6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^
3 + a^4*b^2*d^4)*x^4 + 210*sqrt(1/3)*(a*b^5*c^4 - 4*a^2*b^4*c^3*d + 6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*
b*d^4)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2
/3)*x - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b))/(b*x^3 + a)) - 70*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^
2 - 4*a^3*b*c*d^3 + a^4*d^4)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 140*(b^4*c^4 - 4
*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)) + 420*(4*
a^2*b^4*c^3*d - 6*a^3*b^3*c^2*d^2 + 4*a^4*b^2*c*d^3 - a^5*b*d^4)*x)/(a^2*b^5), 1/420*(42*a^2*b^4*d^4*x^10 + 60
*(4*a^2*b^4*c*d^3 - a^3*b^3*d^4)*x^7 + 105*(6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*x^4 + 420*sqrt(
1/3)*(a*b^5*c^4 - 4*a^2*b^4*c^3*d + 6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*b*d^4)*sqrt((a^2*b)^(1/3)/b)*arc
tan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 70*(b^4*c^4 - 4*a*b^3*c^3*d +
6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) +
140*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)
^(2/3)) + 420*(4*a^2*b^4*c^3*d - 6*a^3*b^3*c^2*d^2 + 4*a^4*b^2*c*d^3 - a^5*b*d^4)*x)/(a^2*b^5)]

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Sympy [A]  time = 2.14005, size = 369, normalized size = 1.46 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{13} - a^{12} d^{12} + 12 a^{11} b c d^{11} - 66 a^{10} b^{2} c^{2} d^{10} + 220 a^{9} b^{3} c^{3} d^{9} - 495 a^{8} b^{4} c^{4} d^{8} + 792 a^{7} b^{5} c^{5} d^{7} - 924 a^{6} b^{6} c^{6} d^{6} + 792 a^{5} b^{7} c^{7} d^{5} - 495 a^{4} b^{8} c^{8} d^{4} + 220 a^{3} b^{9} c^{9} d^{3} - 66 a^{2} b^{10} c^{10} d^{2} + 12 a b^{11} c^{11} d - b^{12} c^{12}, \left ( t \mapsto t \log{\left (\frac{3 t a b^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )} \right )\right )} + \frac{d^{4} x^{10}}{10 b} - \frac{x^{7} \left (a d^{4} - 4 b c d^{3}\right )}{7 b^{2}} + \frac{x^{4} \left (a^{2} d^{4} - 4 a b c d^{3} + 6 b^{2} c^{2} d^{2}\right )}{4 b^{3}} - \frac{x \left (a^{3} d^{4} - 4 a^{2} b c d^{3} + 6 a b^{2} c^{2} d^{2} - 4 b^{3} c^{3} d\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+c)**4/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*a**2*b**13 - a**12*d**12 + 12*a**11*b*c*d**11 - 66*a**10*b**2*c**2*d**10 + 220*a**9*b**3*c**3
*d**9 - 495*a**8*b**4*c**4*d**8 + 792*a**7*b**5*c**5*d**7 - 924*a**6*b**6*c**6*d**6 + 792*a**5*b**7*c**7*d**5
- 495*a**4*b**8*c**8*d**4 + 220*a**3*b**9*c**9*d**3 - 66*a**2*b**10*c**10*d**2 + 12*a*b**11*c**11*d - b**12*c*
*12, Lambda(_t, _t*log(3*_t*a*b**4/(a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d + b*
*4*c**4) + x))) + d**4*x**10/(10*b) - x**7*(a*d**4 - 4*b*c*d**3)/(7*b**2) + x**4*(a**2*d**4 - 4*a*b*c*d**3 + 6
*b**2*c**2*d**2)/(4*b**3) - x*(a**3*d**4 - 4*a**2*b*c*d**3 + 6*a*b**2*c**2*d**2 - 4*b**3*c**3*d)/b**4

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Giac [B]  time = 1.12613, size = 622, normalized size = 2.47 \begin{align*} \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{4} c^{4} - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c^{3} d + 6 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} c^{2} d^{2} - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b c d^{3} + \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} d^{4}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{5}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{4} c^{4} - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c^{3} d + 6 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} c^{2} d^{2} - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b c d^{3} + \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} d^{4}\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{5}} - \frac{{\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{10}} + \frac{14 \, b^{9} d^{4} x^{10} + 80 \, b^{9} c d^{3} x^{7} - 20 \, a b^{8} d^{4} x^{7} + 210 \, b^{9} c^{2} d^{2} x^{4} - 140 \, a b^{8} c d^{3} x^{4} + 35 \, a^{2} b^{7} d^{4} x^{4} + 560 \, b^{9} c^{3} d x - 840 \, a b^{8} c^{2} d^{2} x + 560 \, a^{2} b^{7} c d^{3} x - 140 \, a^{3} b^{6} d^{4} x}{140 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^4/(b*x^3+a),x, algorithm="giac")

[Out]

1/3*sqrt(3)*((-a*b^2)^(1/3)*b^4*c^4 - 4*(-a*b^2)^(1/3)*a*b^3*c^3*d + 6*(-a*b^2)^(1/3)*a^2*b^2*c^2*d^2 - 4*(-a*
b^2)^(1/3)*a^3*b*c*d^3 + (-a*b^2)^(1/3)*a^4*d^4)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^5)
+ 1/6*((-a*b^2)^(1/3)*b^4*c^4 - 4*(-a*b^2)^(1/3)*a*b^3*c^3*d + 6*(-a*b^2)^(1/3)*a^2*b^2*c^2*d^2 - 4*(-a*b^2)^
(1/3)*a^3*b*c*d^3 + (-a*b^2)^(1/3)*a^4*d^4)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^5) - 1/3*(b^10*c^4 -
4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a
*b^10) + 1/140*(14*b^9*d^4*x^10 + 80*b^9*c*d^3*x^7 - 20*a*b^8*d^4*x^7 + 210*b^9*c^2*d^2*x^4 - 140*a*b^8*c*d^3*
x^4 + 35*a^2*b^7*d^4*x^4 + 560*b^9*c^3*d*x - 840*a*b^8*c^2*d^2*x + 560*a^2*b^7*c*d^3*x - 140*a^3*b^6*d^4*x)/b^
10