Optimal. Leaf size=188 \[ \frac{a^3 x \sqrt{a+b x^2} (10 A b-3 a B)}{256 b^2}-\frac{a^4 (10 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}+\frac{a^2 x^3 \sqrt{a+b x^2} (10 A b-3 a B)}{128 b}+\frac{a x^3 \left (a+b x^2\right )^{3/2} (10 A b-3 a B)}{96 b}+\frac{x^3 \left (a+b x^2\right )^{5/2} (10 A b-3 a B)}{80 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08813, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {459, 279, 321, 217, 206} \[ \frac{a^3 x \sqrt{a+b x^2} (10 A b-3 a B)}{256 b^2}-\frac{a^4 (10 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}+\frac{a^2 x^3 \sqrt{a+b x^2} (10 A b-3 a B)}{128 b}+\frac{a x^3 \left (a+b x^2\right )^{3/2} (10 A b-3 a B)}{96 b}+\frac{x^3 \left (a+b x^2\right )^{5/2} (10 A b-3 a B)}{80 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 459
Rule 279
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^2 \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx &=\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac{(-10 A b+3 a B) \int x^2 \left (a+b x^2\right )^{5/2} \, dx}{10 b}\\ &=\frac{(10 A b-3 a B) x^3 \left (a+b x^2\right )^{5/2}}{80 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}+\frac{(a (10 A b-3 a B)) \int x^2 \left (a+b x^2\right )^{3/2} \, dx}{16 b}\\ &=\frac{a (10 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{96 b}+\frac{(10 A b-3 a B) x^3 \left (a+b x^2\right )^{5/2}}{80 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}+\frac{\left (a^2 (10 A b-3 a B)\right ) \int x^2 \sqrt{a+b x^2} \, dx}{32 b}\\ &=\frac{a^2 (10 A b-3 a B) x^3 \sqrt{a+b x^2}}{128 b}+\frac{a (10 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{96 b}+\frac{(10 A b-3 a B) x^3 \left (a+b x^2\right )^{5/2}}{80 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}+\frac{\left (a^3 (10 A b-3 a B)\right ) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx}{128 b}\\ &=\frac{a^3 (10 A b-3 a B) x \sqrt{a+b x^2}}{256 b^2}+\frac{a^2 (10 A b-3 a B) x^3 \sqrt{a+b x^2}}{128 b}+\frac{a (10 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{96 b}+\frac{(10 A b-3 a B) x^3 \left (a+b x^2\right )^{5/2}}{80 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac{\left (a^4 (10 A b-3 a B)\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{256 b^2}\\ &=\frac{a^3 (10 A b-3 a B) x \sqrt{a+b x^2}}{256 b^2}+\frac{a^2 (10 A b-3 a B) x^3 \sqrt{a+b x^2}}{128 b}+\frac{a (10 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{96 b}+\frac{(10 A b-3 a B) x^3 \left (a+b x^2\right )^{5/2}}{80 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac{\left (a^4 (10 A b-3 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{256 b^2}\\ &=\frac{a^3 (10 A b-3 a B) x \sqrt{a+b x^2}}{256 b^2}+\frac{a^2 (10 A b-3 a B) x^3 \sqrt{a+b x^2}}{128 b}+\frac{a (10 A b-3 a B) x^3 \left (a+b x^2\right )^{3/2}}{96 b}+\frac{(10 A b-3 a B) x^3 \left (a+b x^2\right )^{5/2}}{80 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac{a^4 (10 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.295342, size = 151, normalized size = 0.8 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} x \left (4 a^2 b^2 x^2 \left (295 A+186 B x^2\right )+30 a^3 b \left (5 A+B x^2\right )-45 a^4 B+16 a b^3 x^4 \left (85 A+63 B x^2\right )+96 b^4 x^6 \left (5 A+4 B x^2\right )\right )+\frac{15 a^{7/2} (3 a B-10 A b) \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{3840 b^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 215, normalized size = 1.1 \begin{align*}{\frac{B{x}^{3}}{10\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,Bax}{80\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}Bx}{160\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{B{a}^{3}x}{128\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,B{a}^{4}x}{256\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,B{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{Ax}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{aAx}{48\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{a}^{2}Ax}{192\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,A{a}^{3}x}{128\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,A{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \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 = 1.99864, size = 732, normalized size = 3.89 \begin{align*} \left [-\frac{15 \,{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (384 \, B b^{5} x^{9} + 48 \,{\left (21 \, B a b^{4} + 10 \, A b^{5}\right )} x^{7} + 8 \,{\left (93 \, B a^{2} b^{3} + 170 \, A a b^{4}\right )} x^{5} + 10 \,{\left (3 \, B a^{3} b^{2} + 118 \, A a^{2} b^{3}\right )} x^{3} - 15 \,{\left (3 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{7680 \, b^{3}}, -\frac{15 \,{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (384 \, B b^{5} x^{9} + 48 \,{\left (21 \, B a b^{4} + 10 \, A b^{5}\right )} x^{7} + 8 \,{\left (93 \, B a^{2} b^{3} + 170 \, A a b^{4}\right )} x^{5} + 10 \,{\left (3 \, B a^{3} b^{2} + 118 \, A a^{2} b^{3}\right )} x^{3} - 15 \,{\left (3 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{3840 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 40.913, size = 348, normalized size = 1.85 \begin{align*} \frac{5 A a^{\frac{7}{2}} x}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{133 A a^{\frac{5}{2}} x^{3}}{384 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{127 A a^{\frac{3}{2}} b x^{5}}{192 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 A \sqrt{a} b^{2} x^{7}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 A a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{3}{2}}} + \frac{A b^{3} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 B a^{\frac{9}{2}} x}{256 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{7}{2}} x^{3}}{256 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{129 B a^{\frac{5}{2}} x^{5}}{640 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{73 B a^{\frac{3}{2}} b x^{7}}{160 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{29 B \sqrt{a} b^{2} x^{9}}{80 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B a^{5} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{256 b^{\frac{5}{2}}} + \frac{B b^{3} x^{11}}{10 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15397, size = 223, normalized size = 1.19 \begin{align*} \frac{1}{3840} \,{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, B b^{2} x^{2} + \frac{21 \, B a b^{9} + 10 \, A b^{10}}{b^{8}}\right )} x^{2} + \frac{93 \, B a^{2} b^{8} + 170 \, A a b^{9}}{b^{8}}\right )} x^{2} + \frac{5 \,{\left (3 \, B a^{3} b^{7} + 118 \, A a^{2} b^{8}\right )}}{b^{8}}\right )} x^{2} - \frac{15 \,{\left (3 \, B a^{4} b^{6} - 10 \, A a^{3} b^{7}\right )}}{b^{8}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{256 \, b^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]