Optimal. Leaf size=188 \[ \frac{a^2 x^3 \sqrt{a+b x^2} (2 A b-a B)}{128 b^2}-\frac{3 a^3 x \sqrt{a+b x^2} (2 A b-a B)}{256 b^3}+\frac{3 a^4 (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{7/2}}+\frac{x^5 \left (a+b x^2\right )^{3/2} (2 A b-a B)}{16 b}+\frac{a x^5 \sqrt{a+b x^2} (2 A b-a B)}{32 b}+\frac{B x^5 \left (a+b x^2\right )^{5/2}}{10 b} \]
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Rubi [A] time = 0.0963296, 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} \[ \frac{a^2 x^3 \sqrt{a+b x^2} (2 A b-a B)}{128 b^2}-\frac{3 a^3 x \sqrt{a+b x^2} (2 A b-a B)}{256 b^3}+\frac{3 a^4 (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{7/2}}+\frac{x^5 \left (a+b x^2\right )^{3/2} (2 A b-a B)}{16 b}+\frac{a x^5 \sqrt{a+b x^2} (2 A b-a B)}{32 b}+\frac{B x^5 \left (a+b x^2\right )^{5/2}}{10 b} \]
Antiderivative was successfully verified.
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Rule 459
Rule 279
Rule 321
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^4 \left (a+b x^2\right )^{3/2} \left (A+B x^2\right ) \, dx &=\frac{B x^5 \left (a+b x^2\right )^{5/2}}{10 b}-\frac{(-10 A b+5 a B) \int x^4 \left (a+b x^2\right )^{3/2} \, dx}{10 b}\\ &=\frac{(2 A b-a B) x^5 \left (a+b x^2\right )^{3/2}}{16 b}+\frac{B x^5 \left (a+b x^2\right )^{5/2}}{10 b}+\frac{(3 a (2 A b-a B)) \int x^4 \sqrt{a+b x^2} \, dx}{16 b}\\ &=\frac{a (2 A b-a B) x^5 \sqrt{a+b x^2}}{32 b}+\frac{(2 A b-a B) x^5 \left (a+b x^2\right )^{3/2}}{16 b}+\frac{B x^5 \left (a+b x^2\right )^{5/2}}{10 b}+\frac{\left (a^2 (2 A b-a B)\right ) \int \frac{x^4}{\sqrt{a+b x^2}} \, dx}{32 b}\\ &=\frac{a^2 (2 A b-a B) x^3 \sqrt{a+b x^2}}{128 b^2}+\frac{a (2 A b-a B) x^5 \sqrt{a+b x^2}}{32 b}+\frac{(2 A b-a B) x^5 \left (a+b x^2\right )^{3/2}}{16 b}+\frac{B x^5 \left (a+b x^2\right )^{5/2}}{10 b}-\frac{\left (3 a^3 (2 A b-a B)\right ) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx}{128 b^2}\\ &=-\frac{3 a^3 (2 A b-a B) x \sqrt{a+b x^2}}{256 b^3}+\frac{a^2 (2 A b-a B) x^3 \sqrt{a+b x^2}}{128 b^2}+\frac{a (2 A b-a B) x^5 \sqrt{a+b x^2}}{32 b}+\frac{(2 A b-a B) x^5 \left (a+b x^2\right )^{3/2}}{16 b}+\frac{B x^5 \left (a+b x^2\right )^{5/2}}{10 b}+\frac{\left (3 a^4 (2 A b-a B)\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{256 b^3}\\ &=-\frac{3 a^3 (2 A b-a B) x \sqrt{a+b x^2}}{256 b^3}+\frac{a^2 (2 A b-a B) x^3 \sqrt{a+b x^2}}{128 b^2}+\frac{a (2 A b-a B) x^5 \sqrt{a+b x^2}}{32 b}+\frac{(2 A b-a B) x^5 \left (a+b x^2\right )^{3/2}}{16 b}+\frac{B x^5 \left (a+b x^2\right )^{5/2}}{10 b}+\frac{\left (3 a^4 (2 A b-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^3}\\ &=-\frac{3 a^3 (2 A b-a B) x \sqrt{a+b x^2}}{256 b^3}+\frac{a^2 (2 A b-a B) x^3 \sqrt{a+b x^2}}{128 b^2}+\frac{a (2 A b-a B) x^5 \sqrt{a+b x^2}}{32 b}+\frac{(2 A b-a B) x^5 \left (a+b x^2\right )^{3/2}}{16 b}+\frac{B x^5 \left (a+b x^2\right )^{5/2}}{10 b}+\frac{3 a^4 (2 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.306261, size = 150, normalized size = 0.8 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} x \left (4 a^2 b^2 x^2 \left (5 A+2 B x^2\right )-10 a^3 b \left (3 A+B x^2\right )+15 a^4 B+16 a b^3 x^4 \left (15 A+11 B x^2\right )+32 b^4 x^6 \left (5 A+4 B x^2\right )\right )-\frac{15 a^{7/2} (a B-2 A b) \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{1280 b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 219, normalized size = 1.2 \begin{align*}{\frac{B{x}^{5}}{10\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Ba{x}^{3}}{16\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}Bx}{32\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{B{a}^{3}x}{128\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,B{a}^{4}x}{256\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{3\,B{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{A{x}^{3}}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{aAx}{16\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}Ax}{64\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,A{a}^{3}x}{128\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,A{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 2.01368, size = 693, normalized size = 3.69 \begin{align*} \left [-\frac{15 \,{\left (B a^{5} - 2 \, 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 (128 \, B b^{5} x^{9} + 16 \,{\left (11 \, B a b^{4} + 10 \, A b^{5}\right )} x^{7} + 8 \,{\left (B a^{2} b^{3} + 30 \, A a b^{4}\right )} x^{5} - 10 \,{\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{3} + 15 \,{\left (B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{2560 \, b^{4}}, \frac{15 \,{\left (B a^{5} - 2 \, A a^{4} b\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (128 \, B b^{5} x^{9} + 16 \,{\left (11 \, B a b^{4} + 10 \, A b^{5}\right )} x^{7} + 8 \,{\left (B a^{2} b^{3} + 30 \, A a b^{4}\right )} x^{5} - 10 \,{\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{3} + 15 \,{\left (B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x^{2} + a}}{1280 \, b^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 34.0349, size = 345, normalized size = 1.84 \begin{align*} - \frac{3 A a^{\frac{7}{2}} x}{128 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{A a^{\frac{5}{2}} x^{3}}{128 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{13 A a^{\frac{3}{2}} x^{5}}{64 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 A \sqrt{a} b x^{7}}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{5}{2}}} + \frac{A b^{2} x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 B a^{\frac{9}{2}} x}{256 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{B a^{\frac{7}{2}} x^{3}}{256 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{\frac{5}{2}} x^{5}}{640 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{23 B a^{\frac{3}{2}} x^{7}}{160 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{19 B \sqrt{a} b 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{7}{2}}} + \frac{B b^{2} x^{11}}{10 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12732, size = 215, normalized size = 1.14 \begin{align*} \frac{1}{1280} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, B b x^{2} + \frac{11 \, B a b^{8} + 10 \, A b^{9}}{b^{8}}\right )} x^{2} + \frac{B a^{2} b^{7} + 30 \, A a b^{8}}{b^{8}}\right )} x^{2} - \frac{5 \,{\left (B a^{3} b^{6} - 2 \, A a^{2} b^{7}\right )}}{b^{8}}\right )} x^{2} + \frac{15 \,{\left (B a^{4} b^{5} - 2 \, A a^{3} b^{6}\right )}}{b^{8}}\right )} \sqrt{b x^{2} + a} x + \frac{3 \,{\left (B a^{5} - 2 \, A a^{4} b\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{256 \, b^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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