3.20 \(\int \frac{(c+d x^3)^5}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=320 \[ \frac{d^3 x^4 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )}{4 b^4}+\frac{d^2 x \left (15 a^2 b c d^2-4 a^3 d^3-20 a b^2 c^2 d+10 b^3 c^3\right )}{b^5}-\frac{(b c-a d)^4 (13 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{16/3}}+\frac{(b c-a d)^4 (13 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{16/3}}-\frac{(b c-a d)^4 (13 a d+2 b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{16/3}}+\frac{d^4 x^7 (5 b c-2 a d)}{7 b^3}+\frac{x (b c-a d)^5}{3 a b^5 \left (a+b x^3\right )}+\frac{d^5 x^{10}}{10 b^2} \]

[Out]

(d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x)/b^5 + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2
*d^2)*x^4)/(4*b^4) + (d^4*(5*b*c - 2*a*d)*x^7)/(7*b^3) + (d^5*x^10)/(10*b^2) + ((b*c - a*d)^5*x)/(3*a*b^5*(a +
b*x^3)) - ((b*c - a*d)^4*(2*b*c + 13*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/
3)*b^(16/3)) + ((b*c - a*d)^4*(2*b*c + 13*a*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(16/3)) - ((b*c - a*d)^4
*(2*b*c + 13*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(5/3)*b^(16/3))

________________________________________________________________________________________

Rubi [A]  time = 0.298205, antiderivative size = 320, 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} \[ \frac{d^3 x^4 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )}{4 b^4}+\frac{d^2 x \left (15 a^2 b c d^2-4 a^3 d^3-20 a b^2 c^2 d+10 b^3 c^3\right )}{b^5}-\frac{(b c-a d)^4 (13 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{16/3}}+\frac{(b c-a d)^4 (13 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{16/3}}-\frac{(b c-a d)^4 (13 a d+2 b c) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} b^{16/3}}+\frac{d^4 x^7 (5 b c-2 a d)}{7 b^3}+\frac{x (b c-a d)^5}{3 a b^5 \left (a+b x^3\right )}+\frac{d^5 x^{10}}{10 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3)^5/(a + b*x^3)^2,x]

[Out]

(d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x)/b^5 + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2
*d^2)*x^4)/(4*b^4) + (d^4*(5*b*c - 2*a*d)*x^7)/(7*b^3) + (d^5*x^10)/(10*b^2) + ((b*c - a*d)^5*x)/(3*a*b^5*(a +
b*x^3)) - ((b*c - a*d)^4*(2*b*c + 13*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/
3)*b^(16/3)) + ((b*c - a*d)^4*(2*b*c + 13*a*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*b^(16/3)) - ((b*c - a*d)^4
*(2*b*c + 13*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(5/3)*b^(16/3))

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 )^5}{\left (a+b x^3\right )^2} \, dx &=\int \left (\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right )}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^3}{b^4}+\frac{d^4 (5 b c-2 a d) x^6}{b^3}+\frac{d^5 x^9}{b^2}+\frac{(b c-a d)^4 (b c+4 a d)+5 b d (b c-a d)^4 x^3}{b^5 \left (a+b x^3\right )^2}\right ) \, dx\\ &=\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4}{4 b^4}+\frac{d^4 (5 b c-2 a d) x^7}{7 b^3}+\frac{d^5 x^{10}}{10 b^2}+\frac{\int \frac{(b c-a d)^4 (b c+4 a d)+5 b d (b c-a d)^4 x^3}{\left (a+b x^3\right )^2} \, dx}{b^5}\\ &=\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4}{4 b^4}+\frac{d^4 (5 b c-2 a d) x^7}{7 b^3}+\frac{d^5 x^{10}}{10 b^2}+\frac{(b c-a d)^5 x}{3 a b^5 \left (a+b x^3\right )}+\frac{\left ((b c-a d)^4 (2 b c+13 a d)\right ) \int \frac{1}{a+b x^3} \, dx}{3 a b^5}\\ &=\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4}{4 b^4}+\frac{d^4 (5 b c-2 a d) x^7}{7 b^3}+\frac{d^5 x^{10}}{10 b^2}+\frac{(b c-a d)^5 x}{3 a b^5 \left (a+b x^3\right )}+\frac{\left ((b c-a d)^4 (2 b c+13 a d)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3} b^5}+\frac{\left ((b c-a d)^4 (2 b c+13 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^5}\\ &=\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4}{4 b^4}+\frac{d^4 (5 b c-2 a d) x^7}{7 b^3}+\frac{d^5 x^{10}}{10 b^2}+\frac{(b c-a d)^5 x}{3 a b^5 \left (a+b x^3\right )}+\frac{(b c-a d)^4 (2 b c+13 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{16/3}}-\frac{\left ((b c-a d)^4 (2 b c+13 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}{18 a^{5/3} b^{16/3}}+\frac{\left ((b c-a d)^4 (2 b c+13 a d)\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{4/3} b^5}\\ &=\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4}{4 b^4}+\frac{d^4 (5 b c-2 a d) x^7}{7 b^3}+\frac{d^5 x^{10}}{10 b^2}+\frac{(b c-a d)^5 x}{3 a b^5 \left (a+b x^3\right )}+\frac{(b c-a d)^4 (2 b c+13 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{16/3}}-\frac{(b c-a d)^4 (2 b c+13 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{16/3}}+\frac{\left ((b c-a d)^4 (2 b c+13 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^{16/3}}\\ &=\frac{d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac{d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^4}{4 b^4}+\frac{d^4 (5 b c-2 a d) x^7}{7 b^3}+\frac{d^5 x^{10}}{10 b^2}+\frac{(b c-a d)^5 x}{3 a b^5 \left (a+b x^3\right )}-\frac{(b c-a d)^4 (2 b c+13 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^{16/3}}+\frac{(b c-a d)^4 (2 b c+13 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{16/3}}-\frac{(b c-a d)^4 (2 b c+13 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{16/3}}\\ \end{align*}

Mathematica [A]  time = 0.252526, size = 313, normalized size = 0.98 \[ \frac{315 b^{4/3} d^3 x^4 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )+1260 \sqrt [3]{b} d^2 x \left (15 a^2 b c d^2-4 a^3 d^3-20 a b^2 c^2 d+10 b^3 c^3\right )-\frac{70 (b c-a d)^4 (13 a d+2 b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac{140 (b c-a d)^4 (13 a d+2 b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac{140 \sqrt{3} (b c-a d)^4 (13 a d+2 b c) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{5/3}}+180 b^{7/3} d^4 x^7 (5 b c-2 a d)+\frac{420 \sqrt [3]{b} x (b c-a d)^5}{a \left (a+b x^3\right )}+126 b^{10/3} d^5 x^{10}}{1260 b^{16/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3)^5/(a + b*x^3)^2,x]

[Out]

(1260*b^(1/3)*d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x + 315*b^(4/3)*d^3*(10*b^2*c^2 -
10*a*b*c*d + 3*a^2*d^2)*x^4 + 180*b^(7/3)*d^4*(5*b*c - 2*a*d)*x^7 + 126*b^(10/3)*d^5*x^10 + (420*b^(1/3)*(b*c
- a*d)^5*x)/(a*(a + b*x^3)) + (140*Sqrt[3]*(b*c - a*d)^4*(2*b*c + 13*a*d)*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sq
rt[3]*a^(1/3))])/a^(5/3) + (140*(b*c - a*d)^4*(2*b*c + 13*a*d)*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - (70*(b*c -
a*d)^4*(2*b*c + 13*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3))/(1260*b^(16/3))

________________________________________________________________________________________

Maple [B]  time = 0.013, size = 905, normalized size = 2.8 \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)^5/(b*x^3+a)^2,x)

[Out]

5/9/b^2/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*c^4*d-13/18/b^6*a^4/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/
3))*d^5-20*d^3/b^3*a*c^2*x+13/9/b^6*a^4/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*d^5-5/3/b*x/(b*x^3+a)*c^4*d-50/9/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))*c*d^4+70/9/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^2*d^3-40/9/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^3*d^2-50/9/b^5*a^3/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*c*d^4+70/9/b^4*a^2/(1/b*a)^(2/3)*ln(x+(1
/b*a)^(1/3))*c^2*d^3-40/9/b^3*a/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*c^3*d^2+25/9/b^5*a^3/(1/b*a)^(2/3)*ln(x^2-(1
/b*a)^(1/3)*x+(1/b*a)^(2/3))*c*d^4-35/9/b^4*a^2/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c^2*d^3+20
/9/b^3*a/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c^3*d^2+13/9/b^6*a^4/(1/b*a)^(2/3)*3^(1/2)*arctan
(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*d^5+5/9/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^4*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^5+5/3/b^4*a^3*x/(b*x^3+a)*c*
d^4-10/3/b^3*a^2*x/(b*x^3+a)*c^2*d^3+10/3/b^2*a*x/(b*x^3+a)*c^3*d^2+1/10*d^5*x^10/b^2-5/18/b^2/(1/b*a)^(2/3)*l
n(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c^4*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^5+1/3
/a*x/(b*x^3+a)*c^5+5/2*d^3/b^2*x^4*c^2-4*d^5/b^5*a^3*x+10*d^2/b^2*c^3*x-2/7*d^5/b^3*x^7*a+5/7*d^4/b^2*x^7*c+3/
4*d^5/b^4*x^4*a^2-5/2*d^4/b^3*x^4*a*c+15*d^4/b^4*a^2*c*x-1/3/b^5*a^4*x/(b*x^3+a)*d^5+2/9/b/a/(1/b*a)^(2/3)*ln(
x+(1/b*a)^(1/3))*c^5

________________________________________________________________________________________

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)^5/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.81118, size = 3490, normalized size = 10.91 \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)^5/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

[1/1260*(126*a^3*b^5*d^5*x^13 + 18*(50*a^3*b^5*c*d^4 - 13*a^4*b^4*d^5)*x^10 + 45*(70*a^3*b^5*c^2*d^3 - 50*a^4*
b^4*c*d^4 + 13*a^5*b^3*d^5)*x^7 + 315*(40*a^3*b^5*c^3*d^2 - 70*a^4*b^4*c^2*d^3 + 50*a^5*b^3*c*d^4 - 13*a^6*b^2
*d^5)*x^4 + 210*sqrt(1/3)*(2*a^2*b^6*c^5 + 5*a^3*b^5*c^4*d - 40*a^4*b^4*c^3*d^2 + 70*a^5*b^3*c^2*d^3 - 50*a^6*
b^2*c*d^4 + 13*a^7*b*d^5 + (2*a*b^7*c^5 + 5*a^2*b^6*c^4*d - 40*a^3*b^5*c^3*d^2 + 70*a^4*b^4*c^2*d^3 - 50*a^5*b
^3*c*d^4 + 13*a^6*b^2*d^5)*x^3)*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*(2*a*b^5*c^5 + 5*a
^2*b^4*c^4*d - 40*a^3*b^3*c^3*d^2 + 70*a^4*b^2*c^2*d^3 - 50*a^5*b*c*d^4 + 13*a^6*d^5 + (2*b^6*c^5 + 5*a*b^5*c^
4*d - 40*a^2*b^4*c^3*d^2 + 70*a^3*b^3*c^2*d^3 - 50*a^4*b^2*c*d^4 + 13*a^5*b*d^5)*x^3)*(a^2*b)^(2/3)*log(a*b*x^
2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 140*(2*a*b^5*c^5 + 5*a^2*b^4*c^4*d - 40*a^3*b^3*c^3*d^2 + 70*a^4*b^2*
c^2*d^3 - 50*a^5*b*c*d^4 + 13*a^6*d^5 + (2*b^6*c^5 + 5*a*b^5*c^4*d - 40*a^2*b^4*c^3*d^2 + 70*a^3*b^3*c^2*d^3 -
50*a^4*b^2*c*d^4 + 13*a^5*b*d^5)*x^3)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)) + 420*(a^2*b^6*c^5 - 5*a^3*b^5
*c^4*d + 40*a^4*b^4*c^3*d^2 - 70*a^5*b^3*c^2*d^3 + 50*a^6*b^2*c*d^4 - 13*a^7*b*d^5)*x)/(a^3*b^7*x^3 + a^4*b^6)
, 1/1260*(126*a^3*b^5*d^5*x^13 + 18*(50*a^3*b^5*c*d^4 - 13*a^4*b^4*d^5)*x^10 + 45*(70*a^3*b^5*c^2*d^3 - 50*a^4
*b^4*c*d^4 + 13*a^5*b^3*d^5)*x^7 + 315*(40*a^3*b^5*c^3*d^2 - 70*a^4*b^4*c^2*d^3 + 50*a^5*b^3*c*d^4 - 13*a^6*b^
2*d^5)*x^4 + 420*sqrt(1/3)*(2*a^2*b^6*c^5 + 5*a^3*b^5*c^4*d - 40*a^4*b^4*c^3*d^2 + 70*a^5*b^3*c^2*d^3 - 50*a^6
*b^2*c*d^4 + 13*a^7*b*d^5 + (2*a*b^7*c^5 + 5*a^2*b^6*c^4*d - 40*a^3*b^5*c^3*d^2 + 70*a^4*b^4*c^2*d^3 - 50*a^5*
b^3*c*d^4 + 13*a^6*b^2*d^5)*x^3)*sqrt((a^2*b)^(1/3)/b)*arctan(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*(2*a*b^5*c^5 + 5*a^2*b^4*c^4*d - 40*a^3*b^3*c^3*d^2 + 70*a^4*b^2*c^2*d^3 - 50*
a^5*b*c*d^4 + 13*a^6*d^5 + (2*b^6*c^5 + 5*a*b^5*c^4*d - 40*a^2*b^4*c^3*d^2 + 70*a^3*b^3*c^2*d^3 - 50*a^4*b^2*c
*d^4 + 13*a^5*b*d^5)*x^3)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 140*(2*a*b^5*c^5 +
5*a^2*b^4*c^4*d - 40*a^3*b^3*c^3*d^2 + 70*a^4*b^2*c^2*d^3 - 50*a^5*b*c*d^4 + 13*a^6*d^5 + (2*b^6*c^5 + 5*a*b^5
*c^4*d - 40*a^2*b^4*c^3*d^2 + 70*a^3*b^3*c^2*d^3 - 50*a^4*b^2*c*d^4 + 13*a^5*b*d^5)*x^3)*(a^2*b)^(2/3)*log(a*b
*x + (a^2*b)^(2/3)) + 420*(a^2*b^6*c^5 - 5*a^3*b^5*c^4*d + 40*a^4*b^4*c^3*d^2 - 70*a^5*b^3*c^2*d^3 + 50*a^6*b^
2*c*d^4 - 13*a^7*b*d^5)*x)/(a^3*b^7*x^3 + a^4*b^6)]

________________________________________________________________________________________

Sympy [A]  time = 15.5391, size = 536, normalized size = 1.68 \begin{align*} - \frac{x \left (a^{5} d^{5} - 5 a^{4} b c d^{4} + 10 a^{3} b^{2} c^{2} d^{3} - 10 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d - b^{5} c^{5}\right )}{3 a^{2} b^{5} + 3 a b^{6} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{5} b^{16} - 2197 a^{15} d^{15} + 25350 a^{14} b c d^{14} - 132990 a^{13} b^{2} c^{2} d^{13} + 418280 a^{12} b^{3} c^{3} d^{12} - 874635 a^{11} b^{4} c^{4} d^{11} + 1271886 a^{10} b^{5} c^{5} d^{10} - 1302400 a^{9} b^{6} c^{6} d^{9} + 922680 a^{8} b^{7} c^{7} d^{8} - 422235 a^{7} b^{8} c^{8} d^{7} + 97570 a^{6} b^{9} c^{9} d^{6} + 7194 a^{5} b^{10} c^{10} d^{5} - 10200 a^{4} b^{11} c^{11} d^{4} + 1435 a^{3} b^{12} c^{12} d^{3} + 330 a^{2} b^{13} c^{13} d^{2} - 60 a b^{14} c^{14} d - 8 b^{15} c^{15}, \left ( t \mapsto t \log{\left (\frac{9 t a^{2} b^{5}}{13 a^{5} d^{5} - 50 a^{4} b c d^{4} + 70 a^{3} b^{2} c^{2} d^{3} - 40 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d + 2 b^{5} c^{5}} + x \right )} \right )\right )} + \frac{d^{5} x^{10}}{10 b^{2}} - \frac{x^{7} \left (2 a d^{5} - 5 b c d^{4}\right )}{7 b^{3}} + \frac{x^{4} \left (3 a^{2} d^{5} - 10 a b c d^{4} + 10 b^{2} c^{2} d^{3}\right )}{4 b^{4}} - \frac{x \left (4 a^{3} d^{5} - 15 a^{2} b c d^{4} + 20 a b^{2} c^{2} d^{3} - 10 b^{3} c^{3} d^{2}\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**3+c)**5/(b*x**3+a)**2,x)

[Out]

-x*(a**5*d**5 - 5*a**4*b*c*d**4 + 10*a**3*b**2*c**2*d**3 - 10*a**2*b**3*c**3*d**2 + 5*a*b**4*c**4*d - b**5*c**
5)/(3*a**2*b**5 + 3*a*b**6*x**3) + RootSum(729*_t**3*a**5*b**16 - 2197*a**15*d**15 + 25350*a**14*b*c*d**14 - 1
32990*a**13*b**2*c**2*d**13 + 418280*a**12*b**3*c**3*d**12 - 874635*a**11*b**4*c**4*d**11 + 1271886*a**10*b**5
*c**5*d**10 - 1302400*a**9*b**6*c**6*d**9 + 922680*a**8*b**7*c**7*d**8 - 422235*a**7*b**8*c**8*d**7 + 97570*a*
*6*b**9*c**9*d**6 + 7194*a**5*b**10*c**10*d**5 - 10200*a**4*b**11*c**11*d**4 + 1435*a**3*b**12*c**12*d**3 + 33
0*a**2*b**13*c**13*d**2 - 60*a*b**14*c**14*d - 8*b**15*c**15, Lambda(_t, _t*log(9*_t*a**2*b**5/(13*a**5*d**5 -
50*a**4*b*c*d**4 + 70*a**3*b**2*c**2*d**3 - 40*a**2*b**3*c**3*d**2 + 5*a*b**4*c**4*d + 2*b**5*c**5) + x))) +
d**5*x**10/(10*b**2) - x**7*(2*a*d**5 - 5*b*c*d**4)/(7*b**3) + x**4*(3*a**2*d**5 - 10*a*b*c*d**4 + 10*b**2*c**
2*d**3)/(4*b**4) - x*(4*a**3*d**5 - 15*a**2*b*c*d**4 + 20*a*b**2*c**2*d**3 - 10*b**3*c**3*d**2)/b**5

________________________________________________________________________________________

Giac [B]  time = 1.11113, size = 822, normalized size = 2.57 \begin{align*} -\frac{{\left (2 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d - 40 \, a^{2} b^{3} c^{3} d^{2} + 70 \, a^{3} b^{2} c^{2} d^{3} - 50 \, a^{4} b c d^{4} + 13 \, a^{5} d^{5}\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^{5}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{5} c^{5} + 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{4} c^{4} d - 40 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{3} c^{3} d^{2} + 70 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b^{2} c^{2} d^{3} - 50 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} b c d^{4} + 13 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{5} d^{5}\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^{6}} + \frac{b^{5} c^{5} x - 5 \, a b^{4} c^{4} d x + 10 \, a^{2} b^{3} c^{3} d^{2} x - 10 \, a^{3} b^{2} c^{2} d^{3} x + 5 \, a^{4} b c d^{4} x - a^{5} d^{5} x}{3 \,{\left (b x^{3} + a\right )} a b^{5}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{5} c^{5} + 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{4} c^{4} d - 40 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{3} c^{3} d^{2} + 70 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b^{2} c^{2} d^{3} - 50 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} b c d^{4} + 13 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{5} d^{5}\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{6}} + \frac{14 \, b^{18} d^{5} x^{10} + 100 \, b^{18} c d^{4} x^{7} - 40 \, a b^{17} d^{5} x^{7} + 350 \, b^{18} c^{2} d^{3} x^{4} - 350 \, a b^{17} c d^{4} x^{4} + 105 \, a^{2} b^{16} d^{5} x^{4} + 1400 \, b^{18} c^{3} d^{2} x - 2800 \, a b^{17} c^{2} d^{3} x + 2100 \, a^{2} b^{16} c d^{4} x - 560 \, a^{3} b^{15} d^{5} x}{140 \, b^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^3+c)^5/(b*x^3+a)^2,x, algorithm="giac")

[Out]

-1/9*(2*b^5*c^5 + 5*a*b^4*c^4*d - 40*a^2*b^3*c^3*d^2 + 70*a^3*b^2*c^2*d^3 - 50*a^4*b*c*d^4 + 13*a^5*d^5)*(-a/b
)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^5) + 1/9*sqrt(3)*(2*(-a*b^2)^(1/3)*b^5*c^5 + 5*(-a*b^2)^(1/3)*a*b^4*
c^4*d - 40*(-a*b^2)^(1/3)*a^2*b^3*c^3*d^2 + 70*(-a*b^2)^(1/3)*a^3*b^2*c^2*d^3 - 50*(-a*b^2)^(1/3)*a^4*b*c*d^4
+ 13*(-a*b^2)^(1/3)*a^5*d^5)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b^6) + 1/3*(b^5*c^5*x
- 5*a*b^4*c^4*d*x + 10*a^2*b^3*c^3*d^2*x - 10*a^3*b^2*c^2*d^3*x + 5*a^4*b*c*d^4*x - a^5*d^5*x)/((b*x^3 + a)*a*
b^5) + 1/18*(2*(-a*b^2)^(1/3)*b^5*c^5 + 5*(-a*b^2)^(1/3)*a*b^4*c^4*d - 40*(-a*b^2)^(1/3)*a^2*b^3*c^3*d^2 + 70*
(-a*b^2)^(1/3)*a^3*b^2*c^2*d^3 - 50*(-a*b^2)^(1/3)*a^4*b*c*d^4 + 13*(-a*b^2)^(1/3)*a^5*d^5)*log(x^2 + x*(-a/b)
^(1/3) + (-a/b)^(2/3))/(a^2*b^6) + 1/140*(14*b^18*d^5*x^10 + 100*b^18*c*d^4*x^7 - 40*a*b^17*d^5*x^7 + 350*b^18
*c^2*d^3*x^4 - 350*a*b^17*c*d^4*x^4 + 105*a^2*b^16*d^5*x^4 + 1400*b^18*c^3*d^2*x - 2800*a*b^17*c^2*d^3*x + 210
0*a^2*b^16*c*d^4*x - 560*a^3*b^15*d^5*x)/b^20