Optimal. Leaf size=190 \[ -\frac{(A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{7/3}}+\frac{(A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}-\frac{(A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{7/3}}-\frac{x (A b-4 a B)}{3 a b^2}+\frac{x^4 (A b-a B)}{3 a b \left (a+b x^3\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.30186, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{(A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{7/3}}+\frac{(A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}-\frac{(A b-4 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3} b^{7/3}}-\frac{x (A b-4 a B)}{3 a b^2}+\frac{x^4 (A b-a B)}{3 a b \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(A + B*x^3))/(a + b*x^3)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 43.0382, size = 173, normalized size = 0.91 \[ \frac{x^{4} \left (A b - B a\right )}{3 a b \left (a + b x^{3}\right )} - \frac{x \left (A b - 4 B a\right )}{3 a b^{2}} + \frac{\left (A b - 4 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{2}{3}} b^{\frac{7}{3}}} - \frac{\left (A b - 4 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{2}{3}} b^{\frac{7}{3}}} - \frac{\sqrt{3} \left (A b - 4 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{2}{3}} b^{\frac{7}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(B*x**3+A)/(b*x**3+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.240734, size = 160, normalized size = 0.84 \[ \frac{\frac{(4 a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{2 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac{2 \sqrt{3} (4 a B-A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{2/3}}-\frac{6 \sqrt [3]{b} x (A b-a B)}{a+b x^3}+18 \sqrt [3]{b} B x}{18 b^{7/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(A + B*x^3))/(a + b*x^3)^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.011, size = 228, normalized size = 1.2 \[{\frac{Bx}{{b}^{2}}}-{\frac{xA}{3\,b \left ( b{x}^{3}+a \right ) }}+{\frac{Bxa}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{4\,Ba}{9\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,Ba}{9\,{b}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,Ba\sqrt{3}}{9\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{A}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{A}{18\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{A\sqrt{3}}{9\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(B*x^3+A)/(b*x^3+a)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^3/(b*x^3 + a)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.237483, size = 296, normalized size = 1.56 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (4 \, B a b - A b^{2}\right )} x^{3} + 4 \, B a^{2} - A a b\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left ({\left (4 \, B a b - A b^{2}\right )} x^{3} + 4 \, B a^{2} - A a b\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 6 \,{\left ({\left (4 \, B a b - A b^{2}\right )} x^{3} + 4 \, B a^{2} - A a b\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) + 6 \, \sqrt{3}{\left (3 \, B b x^{4} +{\left (4 \, B a - A b\right )} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{54 \,{\left (b^{3} x^{3} + a b^{2}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^3/(b*x^3 + a)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 3.59246, size = 102, normalized size = 0.54 \[ \frac{B x}{b^{2}} + \frac{x \left (- A b + B a\right )}{3 a b^{2} + 3 b^{3} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{2} b^{7} - A^{3} b^{3} + 12 A^{2} B a b^{2} - 48 A B^{2} a^{2} b + 64 B^{3} a^{3}, \left ( t \mapsto t \log{\left (- \frac{9 t a b^{2}}{- A b + 4 B a} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(B*x**3+A)/(b*x**3+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.219189, size = 254, normalized size = 1.34 \[ \frac{B x}{b^{2}} + \frac{{\left (4 \, B a - A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{2}} - \frac{\sqrt{3}{\left (4 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{3}} + \frac{B a x - A b x}{3 \,{\left (b x^{3} + a\right )} b^{2}} - \frac{{\left (4 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*x^3/(b*x^3 + a)^2,x, algorithm="giac")
[Out]