Optimal. Leaf size=113 \[ -\frac{5 A b-2 a B}{3 a^3 \sqrt{a+b x^3}}-\frac{5 A b-2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}+\frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{7/2}}-\frac{A}{3 a x^3 \left (a+b x^3\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0869774, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {446, 78, 51, 63, 208} \[ -\frac{5 A b-2 a B}{3 a^3 \sqrt{a+b x^3}}-\frac{5 A b-2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}+\frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{7/2}}-\frac{A}{3 a x^3 \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^4 \left (a+b x^3\right )^{5/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{x^2 (a+b x)^{5/2}} \, dx,x,x^3\right )\\ &=-\frac{A}{3 a x^3 \left (a+b x^3\right )^{3/2}}+\frac{\left (-\frac{5 A b}{2}+a B\right ) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{5/2}} \, dx,x,x^3\right )}{3 a}\\ &=-\frac{5 A b-2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac{A}{3 a x^3 \left (a+b x^3\right )^{3/2}}-\frac{(5 A b-2 a B) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,x^3\right )}{6 a^2}\\ &=-\frac{5 A b-2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac{A}{3 a x^3 \left (a+b x^3\right )^{3/2}}-\frac{5 A b-2 a B}{3 a^3 \sqrt{a+b x^3}}-\frac{(5 A b-2 a B) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{6 a^3}\\ &=-\frac{5 A b-2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac{A}{3 a x^3 \left (a+b x^3\right )^{3/2}}-\frac{5 A b-2 a B}{3 a^3 \sqrt{a+b x^3}}-\frac{(5 A b-2 a B) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{3 a^3 b}\\ &=-\frac{5 A b-2 a B}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac{A}{3 a x^3 \left (a+b x^3\right )^{3/2}}-\frac{5 A b-2 a B}{3 a^3 \sqrt{a+b x^3}}+\frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0203189, size = 57, normalized size = 0.5 \[ \frac{x^3 (2 a B-5 A b) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b x^3}{a}+1\right )-3 a A}{9 a^2 x^3 \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.028, size = 157, normalized size = 1.4 \begin{align*} A \left ( -{\frac{2}{9\,{a}^{2}b}\sqrt{b{x}^{3}+a} \left ({x}^{3}+{\frac{a}{b}} \right ) ^{-2}}-{\frac{4\,b}{3\,{a}^{3}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}-{\frac{1}{3\,{a}^{3}{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{5\,b}{3}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{7}{2}}}} \right ) +B \left ({\frac{2}{9\,a{b}^{2}}\sqrt{b{x}^{3}+a} \left ({x}^{3}+{\frac{a}{b}} \right ) ^{-2}}+{\frac{2}{3\,{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}-{\frac{2}{3}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.77859, size = 757, normalized size = 6.7 \begin{align*} \left [-\frac{3 \,{\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{9} + 2 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} +{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3}\right )} \sqrt{a} \log \left (\frac{b x^{3} + 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) - 2 \,{\left (3 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} - 3 \, A a^{3} + 4 \,{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3}\right )} \sqrt{b x^{3} + a}}{18 \,{\left (a^{4} b^{2} x^{9} + 2 \, a^{5} b x^{6} + a^{6} x^{3}\right )}}, \frac{3 \,{\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{9} + 2 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} +{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) +{\left (3 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} - 3 \, A a^{3} + 4 \,{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3}\right )} \sqrt{b x^{3} + a}}{9 \,{\left (a^{4} b^{2} x^{9} + 2 \, a^{5} b x^{6} + a^{6} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17357, size = 136, normalized size = 1.2 \begin{align*} \frac{{\left (2 \, B a - 5 \, A b\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a} a^{3}} + \frac{2 \,{\left (3 \,{\left (b x^{3} + a\right )} B a + B a^{2} - 6 \,{\left (b x^{3} + a\right )} A b - A a b\right )}}{9 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{3}} - \frac{\sqrt{b x^{3} + a} A}{3 \, a^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]