Optimal. Leaf size=64 \[ \frac{2}{3} A \sqrt{a+b x^3}-\frac{2}{3} \sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0431833, antiderivative size = 64, 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, 80, 50, 63, 208} \[ \frac{2}{3} A \sqrt{a+b x^3}-\frac{2}{3} \sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^3} \left (A+B x^3\right )}{x} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x} (A+B x)}{x} \, dx,x,x^3\right )\\ &=\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b}+\frac{1}{3} A \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^3\right )\\ &=\frac{2}{3} A \sqrt{a+b x^3}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b}+\frac{1}{3} (a A) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{2}{3} A \sqrt{a+b x^3}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b}+\frac{(2 a A) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{3 b}\\ &=\frac{2}{3} A \sqrt{a+b x^3}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b}-\frac{2}{3} \sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0471751, size = 60, normalized size = 0.94 \[ \frac{2}{9} \left (\frac{\sqrt{a+b x^3} \left (B \left (a+b x^3\right )+3 A b\right )}{b}-3 \sqrt{a} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.184, size = 50, normalized size = 0.8 \begin{align*}{\frac{2\,B}{9\,b} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}}}+A \left ({\frac{2}{3}\sqrt{b{x}^{3}+a}}-{\frac{2}{3}\sqrt{a}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) } \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 = 1.77271, size = 301, normalized size = 4.7 \begin{align*} \left [\frac{3 \, A \sqrt{a} b \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) + 2 \,{\left (B b x^{3} + B a + 3 \, A b\right )} \sqrt{b x^{3} + a}}{9 \, b}, \frac{2 \,{\left (3 \, A \sqrt{-a} b \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) +{\left (B b x^{3} + B a + 3 \, A b\right )} \sqrt{b x^{3} + a}\right )}}{9 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 18.8294, size = 76, normalized size = 1.19 \begin{align*} - \frac{A \left (- \frac{2 a \operatorname{atan}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} - 2 \sqrt{a + b x^{3}}\right )}{3} - \frac{B \left (\begin{cases} - \sqrt{a} x^{3} & \text{for}\: b = 0 \\- \frac{2 \left (a + b x^{3}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right )}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09711, size = 82, normalized size = 1.28 \begin{align*} \frac{2 \, A a \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} + \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} B b^{2} + 3 \, \sqrt{b x^{3} + a} A b^{3}\right )}}{9 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]