3.21 \(\int \frac{(c+d x^3)^4}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=267 \[ \frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}-\frac{(b c-a d)^3 (5 a d+b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{13/3}}+\frac{2 (b c-a d)^3 (5 a d+b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}-\frac{2 (b c-a d)^3 (5 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{13/3}}+\frac{d^3 x^4 (2 b c-a d)}{2 b^3}+\frac{x (b c-a d)^4}{3 a b^4 \left (a+b x^3\right )}+\frac{d^4 x^7}{7 b^2} \]

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (d^3*(2*b*c - a*d)*x^4)/(2*b^3) + (d^4*x^7)/(7*b^2) + ((b*c
- a*d)^4*x)/(3*a*b^4*(a + b*x^3)) - (2*(b*c - a*d)^3*(b*c + 5*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(
1/3))])/(3*Sqrt[3]*a^(5/3)*b^(13/3)) + (2*(b*c - a*d)^3*(b*c + 5*a*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(
13/3)) - ((b*c - a*d)^3*(b*c + 5*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(9*a^(5/3)*b^(13/3))

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Rubi [A]  time = 0.226458, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {390, 385, 200, 31, 634, 617, 204, 628} \[ \frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}-\frac{(b c-a d)^3 (5 a d+b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{13/3}}+\frac{2 (b c-a d)^3 (5 a d+b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}-\frac{2 (b c-a d)^3 (5 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{13/3}}+\frac{d^3 x^4 (2 b c-a d)}{2 b^3}+\frac{x (b c-a d)^4}{3 a b^4 \left (a+b x^3\right )}+\frac{d^4 x^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^4/(a + b*x^3)^2,x]

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (d^3*(2*b*c - a*d)*x^4)/(2*b^3) + (d^4*x^7)/(7*b^2) + ((b*c
- a*d)^4*x)/(3*a*b^4*(a + b*x^3)) - (2*(b*c - a*d)^3*(b*c + 5*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(
1/3))])/(3*Sqrt[3]*a^(5/3)*b^(13/3)) + (2*(b*c - a*d)^3*(b*c + 5*a*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(
13/3)) - ((b*c - a*d)^3*(b*c + 5*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(9*a^(5/3)*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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

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}{\left (a+b x^3\right )^2} \, dx &=\int \left (\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right )}{b^4}+\frac{2 d^3 (2 b c-a d) x^3}{b^3}+\frac{d^4 x^6}{b^2}+\frac{(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^3}{b^4 \left (a+b x^3\right )^2}\right ) \, dx\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{d^3 (2 b c-a d) x^4}{2 b^3}+\frac{d^4 x^7}{7 b^2}+\frac{\int \frac{(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^3}{\left (a+b x^3\right )^2} \, dx}{b^4}\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{d^3 (2 b c-a d) x^4}{2 b^3}+\frac{d^4 x^7}{7 b^2}+\frac{(b c-a d)^4 x}{3 a b^4 \left (a+b x^3\right )}+\frac{\left (2 (b c-a d)^3 (b c+5 a d)\right ) \int \frac{1}{a+b x^3} \, dx}{3 a b^4}\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{d^3 (2 b c-a d) x^4}{2 b^3}+\frac{d^4 x^7}{7 b^2}+\frac{(b c-a d)^4 x}{3 a b^4 \left (a+b x^3\right )}+\frac{\left (2 (b c-a d)^3 (b c+5 a d)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3} b^4}+\frac{\left (2 (b c-a d)^3 (b c+5 a d)\right ) \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}{9 a^{5/3} b^4}\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{d^3 (2 b c-a d) x^4}{2 b^3}+\frac{d^4 x^7}{7 b^2}+\frac{(b c-a d)^4 x}{3 a b^4 \left (a+b x^3\right )}+\frac{2 (b c-a d)^3 (b c+5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}-\frac{\left ((b c-a d)^3 (b c+5 a d)\right ) \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}{9 a^{5/3} b^{13/3}}+\frac{\left ((b c-a d)^3 (b c+5 a d)\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{4/3} b^4}\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{d^3 (2 b c-a d) x^4}{2 b^3}+\frac{d^4 x^7}{7 b^2}+\frac{(b c-a d)^4 x}{3 a b^4 \left (a+b x^3\right )}+\frac{2 (b c-a d)^3 (b c+5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}-\frac{(b c-a d)^3 (b c+5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{13/3}}+\frac{\left (2 (b c-a d)^3 (b c+5 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{5/3} b^{13/3}}\\ &=\frac{d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac{d^3 (2 b c-a d) x^4}{2 b^3}+\frac{d^4 x^7}{7 b^2}+\frac{(b c-a d)^4 x}{3 a b^4 \left (a+b x^3\right )}-\frac{2 (b c-a d)^3 (b c+5 a d) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{13/3}}+\frac{2 (b c-a d)^3 (b c+5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}-\frac{(b c-a d)^3 (b c+5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{13/3}}\\ \end{align*}

Mathematica [A]  time = 0.216735, size = 260, normalized size = 0.97 \[ \frac{126 \sqrt [3]{b} d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )+\frac{14 (a d-b c)^3 (5 a d+b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac{28 (b c-a d)^3 (5 a d+b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac{28 \sqrt{3} (b c-a d)^3 (5 a d+b c) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3}}+63 b^{4/3} d^3 x^4 (2 b c-a d)+\frac{42 \sqrt [3]{b} x (b c-a d)^4}{a \left (a+b x^3\right )}+18 b^{7/3} d^4 x^7}{126 b^{13/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(126*b^(1/3)*d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x + 63*b^(4/3)*d^3*(2*b*c - a*d)*x^4 + 18*b^(7/3)*d^4*x^7
+ (42*b^(1/3)*(b*c - a*d)^4*x)/(a*(a + b*x^3)) + (28*Sqrt[3]*(b*c - a*d)^3*(b*c + 5*a*d)*ArcTan[(-a^(1/3) + 2
*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(5/3) + (28*(b*c - a*d)^3*(b*c + 5*a*d)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) +
(14*(-(b*c) + a*d)^3*(b*c + 5*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(126*b^(13/3))

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Maple [B]  time = 0.011, size = 708, normalized size = 2.7 \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)^2,x)

[Out]

2/b^2*a*x/(b*x^3+a)*c^2*d^2-14/9/b^4*a^2/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c*d^3+4/3/b^3*a/(
1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c^2*d^2-10/9/b^5*a^3/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/
2)*(2/(1/b*a)^(1/3)*x-1))*d^4+4/9/b^2/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c^3*d+2/
9/b/a/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c^4+28/9/b^4*a^2/(1/b*a)^(2/3)*ln(x+(1/b
*a)^(1/3))*c*d^3-8/3/b^3*a/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*c^2*d^2-4/3/b^3*a^2*x/(b*x^3+a)*c*d^3+28/9/b^4*a^
2/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c*d^3-8/3/b^3*a/(1/b*a)^(2/3)*3^(1/2)*arctan
(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c^2*d^2-8*d^3/b^3*c*a*x-2/9/b^2/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*
a)^(2/3))*c^3*d-1/9/b/a/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c^4-10/9/b^5*a^3/(1/b*a)^(2/3)*ln(
x+(1/b*a)^(1/3))*d^4+4/9/b^2/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*c^3*d+2/9/b/a/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))
*c^4+5/9/b^5*a^3/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d^4+1/7*d^4*x^7/b^2+1/3/b^4*a^3*x/(b*x^3+
a)*d^4-4/3/b*x/(b*x^3+a)*c^3*d+1/3/a*x/(b*x^3+a)*c^4-1/2*d^4/b^3*x^4*a+d^3/b^2*x^4*c+3*d^4/b^4*a^2*x+6*d^2/b^2
*c^2*x

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.89248, size = 2808, normalized size = 10.52 \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)^2,x, algorithm="fricas")

[Out]

[1/126*(18*a^3*b^4*d^4*x^10 + 9*(14*a^3*b^4*c*d^3 - 5*a^4*b^3*d^4)*x^7 + 63*(12*a^3*b^4*c^2*d^2 - 14*a^4*b^3*c
*d^3 + 5*a^5*b^2*d^4)*x^4 - 42*sqrt(1/3)*(a^2*b^5*c^4 + 2*a^3*b^4*c^3*d - 12*a^4*b^3*c^2*d^2 + 14*a^5*b^2*c*d^
3 - 5*a^6*b*d^4 + (a*b^6*c^4 + 2*a^2*b^5*c^3*d - 12*a^3*b^4*c^2*d^2 + 14*a^4*b^3*c*d^3 - 5*a^5*b^2*d^4)*x^3)*s
qrt((-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)) - 14*(a*b^4*c^4 + 2*a^2*b^3*c^3*d - 12*a^3*b^2*c^2*d^
2 + 14*a^4*b*c*d^3 - 5*a^5*d^4 + (b^5*c^4 + 2*a*b^4*c^3*d - 12*a^2*b^3*c^2*d^2 + 14*a^3*b^2*c*d^3 - 5*a^4*b*d^
4)*x^3)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 28*(a*b^4*c^4 + 2*a^2*b^3*c^3*d -
12*a^3*b^2*c^2*d^2 + 14*a^4*b*c*d^3 - 5*a^5*d^4 + (b^5*c^4 + 2*a*b^4*c^3*d - 12*a^2*b^3*c^2*d^2 + 14*a^3*b^2*c
*d^3 - 5*a^4*b*d^4)*x^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) + 42*(a^2*b^5*c^4 - 4*a^3*b^4*c^3*d + 24*a
^4*b^3*c^2*d^2 - 28*a^5*b^2*c*d^3 + 10*a^6*b*d^4)*x)/(a^3*b^6*x^3 + a^4*b^5), 1/126*(18*a^3*b^4*d^4*x^10 + 9*(
14*a^3*b^4*c*d^3 - 5*a^4*b^3*d^4)*x^7 + 63*(12*a^3*b^4*c^2*d^2 - 14*a^4*b^3*c*d^3 + 5*a^5*b^2*d^4)*x^4 + 84*sq
rt(1/3)*(a^2*b^5*c^4 + 2*a^3*b^4*c^3*d - 12*a^4*b^3*c^2*d^2 + 14*a^5*b^2*c*d^3 - 5*a^6*b*d^4 + (a*b^6*c^4 + 2*
a^2*b^5*c^3*d - 12*a^3*b^4*c^2*d^2 + 14*a^4*b^3*c*d^3 - 5*a^5*b^2*d^4)*x^3)*sqrt(-(-a^2*b)^(1/3)/b)*arctan(sqr
t(1/3)*(2*(-a^2*b)^(2/3)*x + (-a^2*b)^(1/3)*a)*sqrt(-(-a^2*b)^(1/3)/b)/a^2) - 14*(a*b^4*c^4 + 2*a^2*b^3*c^3*d
- 12*a^3*b^2*c^2*d^2 + 14*a^4*b*c*d^3 - 5*a^5*d^4 + (b^5*c^4 + 2*a*b^4*c^3*d - 12*a^2*b^3*c^2*d^2 + 14*a^3*b^2
*c*d^3 - 5*a^4*b*d^4)*x^3)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 28*(a*b^4*c^4 +
2*a^2*b^3*c^3*d - 12*a^3*b^2*c^2*d^2 + 14*a^4*b*c*d^3 - 5*a^5*d^4 + (b^5*c^4 + 2*a*b^4*c^3*d - 12*a^2*b^3*c^2
*d^2 + 14*a^3*b^2*c*d^3 - 5*a^4*b*d^4)*x^3)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) + 42*(a^2*b^5*c^4 - 4*a
^3*b^4*c^3*d + 24*a^4*b^3*c^2*d^2 - 28*a^5*b^2*c*d^3 + 10*a^6*b*d^4)*x)/(a^3*b^6*x^3 + a^4*b^5)]

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Sympy [A]  time = 7.67771, size = 403, normalized size = 1.51 \begin{align*} \frac{x \left (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}\right )}{3 a^{2} b^{4} + 3 a b^{5} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{5} b^{13} + 1000 a^{12} d^{12} - 8400 a^{11} b c d^{11} + 30720 a^{10} b^{2} c^{2} d^{10} - 63472 a^{9} b^{3} c^{3} d^{9} + 79848 a^{8} b^{4} c^{4} d^{8} - 60192 a^{7} b^{5} c^{5} d^{7} + 22848 a^{6} b^{6} c^{6} d^{6} + 288 a^{5} b^{7} c^{7} d^{5} - 3528 a^{4} b^{8} c^{8} d^{4} + 752 a^{3} b^{9} c^{9} d^{3} + 192 a^{2} b^{10} c^{10} d^{2} - 48 a b^{11} c^{11} d - 8 b^{12} c^{12}, \left ( t \mapsto t \log{\left (- \frac{9 t a^{2} b^{4}}{10 a^{4} d^{4} - 28 a^{3} b c d^{3} + 24 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - 2 b^{4} c^{4}} + x \right )} \right )\right )} + \frac{d^{4} x^{7}}{7 b^{2}} - \frac{x^{4} \left (a d^{4} - 2 b c d^{3}\right )}{2 b^{3}} + \frac{x \left (3 a^{2} d^{4} - 8 a b c d^{3} + 6 b^{2} c^{2} d^{2}\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)**2,x)

[Out]

x*(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)/(3*a**2*b**4 + 3*a*b**5*
x**3) + RootSum(729*_t**3*a**5*b**13 + 1000*a**12*d**12 - 8400*a**11*b*c*d**11 + 30720*a**10*b**2*c**2*d**10 -
63472*a**9*b**3*c**3*d**9 + 79848*a**8*b**4*c**4*d**8 - 60192*a**7*b**5*c**5*d**7 + 22848*a**6*b**6*c**6*d**6
+ 288*a**5*b**7*c**7*d**5 - 3528*a**4*b**8*c**8*d**4 + 752*a**3*b**9*c**9*d**3 + 192*a**2*b**10*c**10*d**2 -
48*a*b**11*c**11*d - 8*b**12*c**12, Lambda(_t, _t*log(-9*_t*a**2*b**4/(10*a**4*d**4 - 28*a**3*b*c*d**3 + 24*a*
*2*b**2*c**2*d**2 - 4*a*b**3*c**3*d - 2*b**4*c**4) + x))) + d**4*x**7/(7*b**2) - x**4*(a*d**4 - 2*b*c*d**3)/(2
*b**3) + x*(3*a**2*d**4 - 8*a*b*c*d**3 + 6*b**2*c**2*d**2)/b**4

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Giac [B]  time = 1.14312, size = 643, normalized size = 2.41 \begin{align*} -\frac{2 \,{\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} 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 )}{9 \, a^{2} b^{4}} + \frac{2 \, \sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{4} c^{4} + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c^{3} d - 12 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} c^{2} d^{2} + 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b c d^{3} - 5 \, \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 )}{9 \, a^{2} b^{5}} + \frac{b^{4} c^{4} x - 4 \, a b^{3} c^{3} d x + 6 \, a^{2} b^{2} c^{2} d^{2} x - 4 \, a^{3} b c d^{3} x + a^{4} d^{4} x}{3 \,{\left (b x^{3} + a\right )} a b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{4} c^{4} + 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c^{3} d - 12 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} c^{2} d^{2} + 14 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b c d^{3} - 5 \, \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 )}{9 \, a^{2} b^{5}} + \frac{2 \, b^{12} d^{4} x^{7} + 14 \, b^{12} c d^{3} x^{4} - 7 \, a b^{11} d^{4} x^{4} + 84 \, b^{12} c^{2} d^{2} x - 112 \, a b^{11} c d^{3} x + 42 \, a^{2} b^{10} d^{4} x}{14 \, b^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^4/(b*x^3+a)^2,x, algorithm="giac")

[Out]

-2/9*(b^4*c^4 + 2*a*b^3*c^3*d - 12*a^2*b^2*c^2*d^2 + 14*a^3*b*c*d^3 - 5*a^4*d^4)*(-a/b)^(1/3)*log(abs(x - (-a/
b)^(1/3)))/(a^2*b^4) + 2/9*sqrt(3)*((-a*b^2)^(1/3)*b^4*c^4 + 2*(-a*b^2)^(1/3)*a*b^3*c^3*d - 12*(-a*b^2)^(1/3)*
a^2*b^2*c^2*d^2 + 14*(-a*b^2)^(1/3)*a^3*b*c*d^3 - 5*(-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^2*b^5) + 1/3*(b^4*c^4*x - 4*a*b^3*c^3*d*x + 6*a^2*b^2*c^2*d^2*x - 4*a^3*b*c*d^3*x + a^4
*d^4*x)/((b*x^3 + a)*a*b^4) + 1/9*((-a*b^2)^(1/3)*b^4*c^4 + 2*(-a*b^2)^(1/3)*a*b^3*c^3*d - 12*(-a*b^2)^(1/3)*a
^2*b^2*c^2*d^2 + 14*(-a*b^2)^(1/3)*a^3*b*c*d^3 - 5*(-a*b^2)^(1/3)*a^4*d^4)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(
2/3))/(a^2*b^5) + 1/14*(2*b^12*d^4*x^7 + 14*b^12*c*d^3*x^4 - 7*a*b^11*d^4*x^4 + 84*b^12*c^2*d^2*x - 112*a*b^11
*c*d^3*x + 42*a^2*b^10*d^4*x)/b^14