Optimal. Leaf size=77 \[ \frac{2 A}{3 a^2 \sqrt{a+b x^3}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{5/2}}+\frac{2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0504637, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{2 A}{3 a^2 \sqrt{a+b x^3}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{5/2}}+\frac{2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}} \]
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 \left (a+b x^3\right )^{5/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{x (a+b x)^{5/2}} \, dx,x,x^3\right )\\ &=\frac{2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}}+\frac{A \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,x^3\right )}{3 a}\\ &=\frac{2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}}+\frac{2 A}{3 a^2 \sqrt{a+b x^3}}+\frac{A \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{3 a^2}\\ &=\frac{2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}}+\frac{2 A}{3 a^2 \sqrt{a+b x^3}}+\frac{(2 A) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{3 a^2 b}\\ &=\frac{2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}}+\frac{2 A}{3 a^2 \sqrt{a+b x^3}}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0221555, size = 62, normalized size = 0.81 \[ \frac{2 a (A b-a B)+6 A b \left (a+b x^3\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x^3}{a}+1\right )}{9 a^2 b \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.027, size = 85, normalized size = 1.1 \begin{align*} -{\frac{2\,B}{9\,b} \left ( b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}+A \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 = 1.82575, size = 522, normalized size = 6.78 \begin{align*} \left [\frac{3 \,{\left (A b^{3} x^{6} + 2 \, A a b^{2} x^{3} + A a^{2} b\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 \, A a b^{2} x^{3} - B a^{3} + 4 \, A a^{2} b\right )} \sqrt{b x^{3} + a}}{9 \,{\left (a^{3} b^{3} x^{6} + 2 \, a^{4} b^{2} x^{3} + a^{5} b\right )}}, \frac{2 \,{\left (3 \,{\left (A b^{3} x^{6} + 2 \, A a b^{2} x^{3} + A a^{2} b\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) +{\left (3 \, A a b^{2} x^{3} - B a^{3} + 4 \, A a^{2} b\right )} \sqrt{b x^{3} + a}\right )}}{9 \,{\left (a^{3} b^{3} x^{6} + 2 \, a^{4} b^{2} x^{3} + a^{5} b\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 17.1417, size = 76, normalized size = 0.99 \begin{align*} \frac{2 A}{3 a^{2} \sqrt{a + b x^{3}}} + \frac{2 A \operatorname{atan}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{- a}} \right )}}{3 a^{2} \sqrt{- a}} - \frac{2 \left (- A b + B a\right )}{9 a b \left (a + b x^{3}\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1286, size = 90, normalized size = 1.17 \begin{align*} \frac{2 \, A \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a} a^{2}} - \frac{2 \,{\left (B a^{2} - 3 \,{\left (b x^{3} + a\right )} A b - A a b\right )}}{9 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]