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

Optimal. Leaf size=258 \[ -\frac{\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}}+\frac{\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac{\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} d^{7/3}}-\frac{x (b c-a d) (5 a d+4 b c)}{18 c^2 d^2 \left (c+d x^3\right )}-\frac{x \left (a+b x^3\right ) (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]

[Out]

-((b*c - a*d)*x*(a + b*x^3))/(6*c*d*(c + d*x^3)^2) - ((b*c - a*d)*(4*b*c + 5*a*d)*x)/(18*c^2*d^2*(c + d*x^3))
- ((2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(9*Sqrt[3]*c^(8/3)*d
^(7/3)) + ((2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(1/3) + d^(1/3)*x])/(27*c^(8/3)*d^(7/3)) - ((2*b^2*c^2 +
2*a*b*c*d + 5*a^2*d^2)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d^(7/3))

________________________________________________________________________________________

Rubi [A]  time = 0.232737, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {413, 385, 200, 31, 634, 617, 204, 628} \[ -\frac{\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}}+\frac{\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac{\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} d^{7/3}}-\frac{x (b c-a d) (5 a d+4 b c)}{18 c^2 d^2 \left (c+d x^3\right )}-\frac{x \left (a+b x^3\right ) (b c-a d)}{6 c d \left (c+d x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/(c + d*x^3)^3,x]

[Out]

-((b*c - a*d)*x*(a + b*x^3))/(6*c*d*(c + d*x^3)^2) - ((b*c - a*d)*(4*b*c + 5*a*d)*x)/(18*c^2*d^2*(c + d*x^3))
- ((2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(9*Sqrt[3]*c^(8/3)*d
^(7/3)) + ((2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(1/3) + d^(1/3)*x])/(27*c^(8/3)*d^(7/3)) - ((2*b^2*c^2 +
2*a*b*c*d + 5*a^2*d^2)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d^(7/3))

Rule 413

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

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 (a+b x^3\right )^2}{\left (c+d x^3\right )^3} \, dx &=-\frac{(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}+\frac{\int \frac{a (b c+5 a d)+2 b (2 b c+a d) x^3}{\left (c+d x^3\right )^2} \, dx}{6 c d}\\ &=-\frac{(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac{(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}+\frac{\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac{1}{c+d x^3} \, dx}{9 c^2 d^2}\\ &=-\frac{(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac{(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}+\frac{\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac{1}{\sqrt [3]{c}+\sqrt [3]{d} x} \, dx}{27 c^{8/3} d^2}+\frac{\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac{2 \sqrt [3]{c}-\sqrt [3]{d} x}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{27 c^{8/3} d^2}\\ &=-\frac{(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac{(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}+\frac{\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac{\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\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}{54 c^{8/3} d^{7/3}}+\frac{\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \int \frac{1}{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{18 c^{7/3} d^2}\\ &=-\frac{(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac{(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}+\frac{\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac{\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}}+\frac{\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{9 c^{8/3} d^{7/3}}\\ &=-\frac{(b c-a d) x \left (a+b x^3\right )}{6 c d \left (c+d x^3\right )^2}-\frac{(b c-a d) (4 b c+5 a d) x}{18 c^2 d^2 \left (c+d x^3\right )}-\frac{\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} d^{7/3}}+\frac{\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{27 c^{8/3} d^{7/3}}-\frac{\left (2 b^2 c^2+2 a b c d+5 a^2 d^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{54 c^{8/3} d^{7/3}}\\ \end{align*}

Mathematica [A]  time = 0.271457, size = 234, normalized size = 0.91 \[ \frac{-\frac{3 c^{2/3} \sqrt [3]{d} x \left (-a^2 d^2 \left (8 c+5 d x^3\right )+2 a b c d \left (2 c-d x^3\right )+b^2 c^2 \left (4 c+7 d x^3\right )\right )}{\left (c+d x^3\right )^2}-\left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )+2 \left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-2 \sqrt{3} \left (5 a^2 d^2+2 a b c d+2 b^2 c^2\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{54 c^{8/3} d^{7/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*c^(2/3)*d^(1/3)*x*(2*a*b*c*d*(2*c - d*x^3) - a^2*d^2*(8*c + 5*d*x^3) + b^2*c^2*(4*c + 7*d*x^3)))/(c + d*x
^3)^2 - 2*Sqrt[3]*(2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] + 2*(2*b^2*c
^2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(1/3) + d^(1/3)*x] - (2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*Log[c^(2/3) - c^(1/
3)*d^(1/3)*x + d^(2/3)*x^2])/(54*c^(8/3)*d^(7/3))

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 388, normalized size = 1.5 \begin{align*}{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{2}} \left ({\frac{ \left ( 5\,{a}^{2}{d}^{2}+2\,cabd-7\,{b}^{2}{c}^{2} \right ){x}^{4}}{18\,{c}^{2}d}}+{\frac{ \left ( 4\,{a}^{2}{d}^{2}-2\,cabd-2\,{b}^{2}{c}^{2} \right ) x}{9\,{d}^{2}c}} \right ) }+{\frac{5\,{a}^{2}}{27\,{c}^{2}d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,ab}{27\,{d}^{2}c}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,{b}^{2}}{27\,{d}^{3}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,{a}^{2}}{54\,{c}^{2}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{ab}{27\,{d}^{2}c}\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}^{2}}{27\,{d}^{3}}\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{5\,\sqrt{3}{a}^{2}}{27\,{c}^{2}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{2\,\sqrt{3}ab}{27\,{d}^{2}c}\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{2\,\sqrt{3}{b}^{2}}{27\,{d}^{3}}\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}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/(d*x^3+c)^3,x)

[Out]

(1/18*(5*a^2*d^2+2*a*b*c*d-7*b^2*c^2)/c^2/d*x^4+2/9*(2*a^2*d^2-a*b*c*d-b^2*c^2)/d^2/c*x)/(d*x^3+c)^2+5/27/c^2/
d/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*a^2+2/27/c/d^2/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*a*b+2/27/d^3/(c/d)^(2/3)*ln(x+(c/
d)^(1/3))*b^2-5/54/c^2/d/(c/d)^(2/3)*ln(x^2-(c/d)^(1/3)*x+(c/d)^(2/3))*a^2-1/27/c/d^2/(c/d)^(2/3)*ln(x^2-(c/d)
^(1/3)*x+(c/d)^(2/3))*a*b-1/27/d^3/(c/d)^(2/3)*ln(x^2-(c/d)^(1/3)*x+(c/d)^(2/3))*b^2+5/27/c^2/d/(c/d)^(2/3)*3^
(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*a^2+2/27/c/d^2/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(
1/3)*x-1))*a*b+2/27/d^3/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*b^2

________________________________________________________________________________________

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^3+a)^2/(d*x^3+c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.74767, size = 2296, normalized size = 8.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/54*(3*(7*b^2*c^4*d^2 - 2*a*b*c^3*d^3 - 5*a^2*c^2*d^4)*x^4 - 3*sqrt(1/3)*(2*b^2*c^5*d + 2*a*b*c^4*d^2 + 5*a
^2*c^3*d^3 + (2*b^2*c^3*d^3 + 2*a*b*c^2*d^4 + 5*a^2*c*d^5)*x^6 + 2*(2*b^2*c^4*d^2 + 2*a*b*c^3*d^3 + 5*a^2*c^2*
d^4)*x^3)*sqrt(-(c^2*d)^(1/3)/d)*log((2*c*d*x^3 - 3*(c^2*d)^(1/3)*c*x - c^2 + 3*sqrt(1/3)*(2*c*d*x^2 + (c^2*d)
^(2/3)*x - (c^2*d)^(1/3)*c)*sqrt(-(c^2*d)^(1/3)/d))/(d*x^3 + c)) + ((2*b^2*c^2*d^2 + 2*a*b*c*d^3 + 5*a^2*d^4)*
x^6 + 2*b^2*c^4 + 2*a*b*c^3*d + 5*a^2*c^2*d^2 + 2*(2*b^2*c^3*d + 2*a*b*c^2*d^2 + 5*a^2*c*d^3)*x^3)*(c^2*d)^(2/
3)*log(c*d*x^2 - (c^2*d)^(2/3)*x + (c^2*d)^(1/3)*c) - 2*((2*b^2*c^2*d^2 + 2*a*b*c*d^3 + 5*a^2*d^4)*x^6 + 2*b^2
*c^4 + 2*a*b*c^3*d + 5*a^2*c^2*d^2 + 2*(2*b^2*c^3*d + 2*a*b*c^2*d^2 + 5*a^2*c*d^3)*x^3)*(c^2*d)^(2/3)*log(c*d*
x + (c^2*d)^(2/3)) + 12*(b^2*c^5*d + a*b*c^4*d^2 - 2*a^2*c^3*d^3)*x)/(c^4*d^5*x^6 + 2*c^5*d^4*x^3 + c^6*d^3),
-1/54*(3*(7*b^2*c^4*d^2 - 2*a*b*c^3*d^3 - 5*a^2*c^2*d^4)*x^4 - 6*sqrt(1/3)*(2*b^2*c^5*d + 2*a*b*c^4*d^2 + 5*a^
2*c^3*d^3 + (2*b^2*c^3*d^3 + 2*a*b*c^2*d^4 + 5*a^2*c*d^5)*x^6 + 2*(2*b^2*c^4*d^2 + 2*a*b*c^3*d^3 + 5*a^2*c^2*d
^4)*x^3)*sqrt((c^2*d)^(1/3)/d)*arctan(sqrt(1/3)*(2*(c^2*d)^(2/3)*x - (c^2*d)^(1/3)*c)*sqrt((c^2*d)^(1/3)/d)/c^
2) + ((2*b^2*c^2*d^2 + 2*a*b*c*d^3 + 5*a^2*d^4)*x^6 + 2*b^2*c^4 + 2*a*b*c^3*d + 5*a^2*c^2*d^2 + 2*(2*b^2*c^3*d
+ 2*a*b*c^2*d^2 + 5*a^2*c*d^3)*x^3)*(c^2*d)^(2/3)*log(c*d*x^2 - (c^2*d)^(2/3)*x + (c^2*d)^(1/3)*c) - 2*((2*b^
2*c^2*d^2 + 2*a*b*c*d^3 + 5*a^2*d^4)*x^6 + 2*b^2*c^4 + 2*a*b*c^3*d + 5*a^2*c^2*d^2 + 2*(2*b^2*c^3*d + 2*a*b*c^
2*d^2 + 5*a^2*c*d^3)*x^3)*(c^2*d)^(2/3)*log(c*d*x + (c^2*d)^(2/3)) + 12*(b^2*c^5*d + a*b*c^4*d^2 - 2*a^2*c^3*d
^3)*x)/(c^4*d^5*x^6 + 2*c^5*d^4*x^3 + c^6*d^3)]

________________________________________________________________________________________

Sympy [A]  time = 2.87419, size = 233, normalized size = 0.9 \begin{align*} \frac{x^{4} \left (5 a^{2} d^{3} + 2 a b c d^{2} - 7 b^{2} c^{2} d\right ) + x \left (8 a^{2} c d^{2} - 4 a b c^{2} d - 4 b^{2} c^{3}\right )}{18 c^{4} d^{2} + 36 c^{3} d^{3} x^{3} + 18 c^{2} d^{4} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} c^{8} d^{7} - 125 a^{6} d^{6} - 150 a^{5} b c d^{5} - 210 a^{4} b^{2} c^{2} d^{4} - 128 a^{3} b^{3} c^{3} d^{3} - 84 a^{2} b^{4} c^{4} d^{2} - 24 a b^{5} c^{5} d - 8 b^{6} c^{6}, \left ( t \mapsto t \log{\left (\frac{27 t c^{3} d^{2}}{5 a^{2} d^{2} + 2 a b c d + 2 b^{2} c^{2}} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x**4*(5*a**2*d**3 + 2*a*b*c*d**2 - 7*b**2*c**2*d) + x*(8*a**2*c*d**2 - 4*a*b*c**2*d - 4*b**2*c**3))/(18*c**4*
d**2 + 36*c**3*d**3*x**3 + 18*c**2*d**4*x**6) + RootSum(19683*_t**3*c**8*d**7 - 125*a**6*d**6 - 150*a**5*b*c*d
**5 - 210*a**4*b**2*c**2*d**4 - 128*a**3*b**3*c**3*d**3 - 84*a**2*b**4*c**4*d**2 - 24*a*b**5*c**5*d - 8*b**6*c
**6, Lambda(_t, _t*log(27*_t*c**3*d**2/(5*a**2*d**2 + 2*a*b*c*d + 2*b**2*c**2) + x)))

________________________________________________________________________________________

Giac [A]  time = 1.57974, size = 400, normalized size = 1.55 \begin{align*} -\frac{{\left (2 \, b^{2} c^{2} + 2 \, a b c d + 5 \, a^{2} d^{2}\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{27 \, c^{3} d^{2}} + \frac{\sqrt{3}{\left (2 \, \left (-c d^{2}\right )^{\frac{1}{3}} b^{2} c^{2} + 2 \, \left (-c d^{2}\right )^{\frac{1}{3}} a b c d + 5 \, \left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\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 )}{27 \, c^{3} d^{3}} + \frac{{\left (2 \, \left (-c d^{2}\right )^{\frac{1}{3}} b^{2} c^{2} + 2 \, \left (-c d^{2}\right )^{\frac{1}{3}} a b c d + 5 \, \left (-c d^{2}\right )^{\frac{1}{3}} a^{2} d^{2}\right )} \log \left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{54 \, c^{3} d^{3}} - \frac{7 \, b^{2} c^{2} d x^{4} - 2 \, a b c d^{2} x^{4} - 5 \, a^{2} d^{3} x^{4} + 4 \, b^{2} c^{3} x + 4 \, a b c^{2} d x - 8 \, a^{2} c d^{2} x}{18 \,{\left (d x^{3} + c\right )}^{2} c^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*(-c/d)^(1/3)*log(abs(x - (-c/d)^(1/3)))/(c^3*d^2) + 1/27*sqrt(3)*(2*
(-c*d^2)^(1/3)*b^2*c^2 + 2*(-c*d^2)^(1/3)*a*b*c*d + 5*(-c*d^2)^(1/3)*a^2*d^2)*arctan(1/3*sqrt(3)*(2*x + (-c/d)
^(1/3))/(-c/d)^(1/3))/(c^3*d^3) + 1/54*(2*(-c*d^2)^(1/3)*b^2*c^2 + 2*(-c*d^2)^(1/3)*a*b*c*d + 5*(-c*d^2)^(1/3)
*a^2*d^2)*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(c^3*d^3) - 1/18*(7*b^2*c^2*d*x^4 - 2*a*b*c*d^2*x^4 - 5*a^2
*d^3*x^4 + 4*b^2*c^3*x + 4*a*b*c^2*d*x - 8*a^2*c*d^2*x)/((d*x^3 + c)^2*c^2*d^2)