Optimal. Leaf size=84 \[ -\frac{\left (a+b x^2\right )^{3/2} \left (2 a B d-5 b (A d+B c)-3 b B d x^2\right )}{15 b^2}+A c \sqrt{a+b x^2}-\sqrt{a} A c \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0774443, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {573, 147, 50, 63, 208} \[ -\frac{\left (a+b x^2\right )^{3/2} \left (2 a B d-5 b (A d+B c)-3 b B d x^2\right )}{15 b^2}+A c \sqrt{a+b x^2}-\sqrt{a} A c \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 573
Rule 147
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^2} \left (A+B x^2\right ) \left (c+d x^2\right )}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x} (A+B x) (c+d x)}{x} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{3/2} \left (2 a B d-5 b (B c+A d)-3 b B d x^2\right )}{15 b^2}+\frac{1}{2} (A c) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^2\right )\\ &=A c \sqrt{a+b x^2}-\frac{\left (a+b x^2\right )^{3/2} \left (2 a B d-5 b (B c+A d)-3 b B d x^2\right )}{15 b^2}+\frac{1}{2} (a A c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=A c \sqrt{a+b x^2}-\frac{\left (a+b x^2\right )^{3/2} \left (2 a B d-5 b (B c+A d)-3 b B d x^2\right )}{15 b^2}+\frac{(a A c) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{b}\\ &=A c \sqrt{a+b x^2}-\frac{\left (a+b x^2\right )^{3/2} \left (2 a B d-5 b (B c+A d)-3 b B d x^2\right )}{15 b^2}-\sqrt{a} A c \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.121499, size = 91, normalized size = 1.08 \[ \frac{\sqrt{a+b x^2} \left (5 A b \left (a d+3 b c+b d x^2\right )-B \left (a+b x^2\right ) \left (2 a d-5 b c-3 b d x^2\right )\right )}{15 b^2}-\sqrt{a} A c \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 112, normalized size = 1.3 \begin{align*}{\frac{Bd{x}^{2}}{5\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{2\,aBd}{15\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ad}{3\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bc}{3\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-A\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) c+Ac\sqrt{b{x}^{2}+a} \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 = 1.63984, size = 537, normalized size = 6.39 \begin{align*} \left [\frac{15 \, A \sqrt{a} b^{2} c \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (3 \, B b^{2} d x^{4} +{\left (5 \, B b^{2} c +{\left (B a b + 5 \, A b^{2}\right )} d\right )} x^{2} + 5 \,{\left (B a b + 3 \, A b^{2}\right )} c -{\left (2 \, B a^{2} - 5 \, A a b\right )} d\right )} \sqrt{b x^{2} + a}}{30 \, b^{2}}, \frac{15 \, A \sqrt{-a} b^{2} c \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (3 \, B b^{2} d x^{4} +{\left (5 \, B b^{2} c +{\left (B a b + 5 \, A b^{2}\right )} d\right )} x^{2} + 5 \,{\left (B a b + 3 \, A b^{2}\right )} c -{\left (2 \, B a^{2} - 5 \, A a b\right )} d\right )} \sqrt{b x^{2} + a}}{15 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 35.6397, size = 97, normalized size = 1.15 \begin{align*} \frac{A a c \operatorname{atan}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} + A c \sqrt{a + b x^{2}} + \frac{B d \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (2 A b d - 2 B a d + 2 B b c\right )}{6 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.174, size = 153, normalized size = 1.82 \begin{align*} \frac{A a c \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B b^{9} c + 15 \, \sqrt{b x^{2} + a} A b^{10} c + 3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B b^{8} d - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a b^{8} d + 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b^{9} d}{15 \, b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]