Optimal. Leaf size=173 \[ \frac{3 a^4 B x \sqrt{a+b x^2}}{256 b^2}+\frac{a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac{a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac{3 a^5 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}-\frac{a \left (a+b x^2\right )^{7/2} (160 A+189 B x)}{5040 b^2}+\frac{A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b} \]
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Rubi [A] time = 0.104559, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {833, 780, 195, 217, 206} \[ \frac{3 a^4 B x \sqrt{a+b x^2}}{256 b^2}+\frac{a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac{a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac{3 a^5 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}-\frac{a \left (a+b x^2\right )^{7/2} (160 A+189 B x)}{5040 b^2}+\frac{A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b} \]
Antiderivative was successfully verified.
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Rule 833
Rule 780
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^3 (A+B x) \left (a+b x^2\right )^{5/2} \, dx &=\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}+\frac{\int x^2 (-3 a B+10 A b x) \left (a+b x^2\right )^{5/2} \, dx}{10 b}\\ &=\frac{A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}+\frac{\int x (-20 a A b-27 a b B x) \left (a+b x^2\right )^{5/2} \, dx}{90 b^2}\\ &=\frac{A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac{a (160 A+189 B x) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac{\left (3 a^2 B\right ) \int \left (a+b x^2\right )^{5/2} \, dx}{80 b^2}\\ &=\frac{a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac{A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac{a (160 A+189 B x) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac{\left (a^3 B\right ) \int \left (a+b x^2\right )^{3/2} \, dx}{32 b^2}\\ &=\frac{a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac{a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac{A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac{a (160 A+189 B x) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac{\left (3 a^4 B\right ) \int \sqrt{a+b x^2} \, dx}{128 b^2}\\ &=\frac{3 a^4 B x \sqrt{a+b x^2}}{256 b^2}+\frac{a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac{a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac{A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac{a (160 A+189 B x) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac{\left (3 a^5 B\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{256 b^2}\\ &=\frac{3 a^4 B x \sqrt{a+b x^2}}{256 b^2}+\frac{a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac{a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac{A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac{a (160 A+189 B x) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac{\left (3 a^5 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{3 a^4 B x \sqrt{a+b x^2}}{256 b^2}+\frac{a^3 B x \left (a+b x^2\right )^{3/2}}{128 b^2}+\frac{a^2 B x \left (a+b x^2\right )^{5/2}}{160 b^2}+\frac{A x^2 \left (a+b x^2\right )^{7/2}}{9 b}+\frac{B x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac{a (160 A+189 B x) \left (a+b x^2\right )^{7/2}}{5040 b^2}+\frac{3 a^5 B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{256 b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.276433, size = 145, normalized size = 0.84 \[ \frac{\sqrt{a+b x^2} \left (\sqrt{b} \left (24 a^2 b^2 x^4 (800 A+651 B x)+10 a^3 b x^2 (128 A+63 B x)-5 a^4 (512 A+189 B x)+16 a b^3 x^6 (1520 A+1323 B x)+896 b^4 x^8 (10 A+9 B x)\right )+\frac{945 a^{9/2} B \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{\frac{b x^2}{a}+1}}\right )}{80640 b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 153, normalized size = 0.9 \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{{a}^{3}Bx}{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{A{x}^{2}}{9\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,Aa}{63\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{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 = 1.7001, size = 775, normalized size = 4.48 \begin{align*} \left [\frac{945 \, B a^{5} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (8064 \, B b^{5} x^{9} + 8960 \, A b^{5} x^{8} + 21168 \, B a b^{4} x^{7} + 24320 \, A a b^{4} x^{6} + 15624 \, B a^{2} b^{3} x^{5} + 19200 \, A a^{2} b^{3} x^{4} + 630 \, B a^{3} b^{2} x^{3} + 1280 \, A a^{3} b^{2} x^{2} - 945 \, B a^{4} b x - 2560 \, A a^{4} b\right )} \sqrt{b x^{2} + a}}{161280 \, b^{3}}, -\frac{945 \, B a^{5} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (8064 \, B b^{5} x^{9} + 8960 \, A b^{5} x^{8} + 21168 \, B a b^{4} x^{7} + 24320 \, A a b^{4} x^{6} + 15624 \, B a^{2} b^{3} x^{5} + 19200 \, A a^{2} b^{3} x^{4} + 630 \, B a^{3} b^{2} x^{3} + 1280 \, A a^{3} b^{2} x^{2} - 945 \, B a^{4} b x - 2560 \, A a^{4} b\right )} \sqrt{b x^{2} + a}}{80640 \, b^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 30.4327, size = 469, normalized size = 2.71 \begin{align*} A a^{2} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + 2 A a b \left (\begin{cases} \frac{8 a^{3} \sqrt{a + b x^{2}}}{105 b^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{2}}}{35 b} + \frac{x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + A b^{2} \left (\begin{cases} - \frac{16 a^{4} \sqrt{a + b x^{2}}}{315 b^{4}} + \frac{8 a^{3} x^{2} \sqrt{a + b x^{2}}}{315 b^{3}} - \frac{2 a^{2} x^{4} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{a x^{6} \sqrt{a + b x^{2}}}{63 b} + \frac{x^{8} \sqrt{a + b x^{2}}}{9} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases}\right ) - \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.
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Giac [A] time = 1.23683, size = 189, normalized size = 1.09 \begin{align*} -\frac{3 \, B a^{5} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{256 \, b^{\frac{5}{2}}} - \frac{1}{80640} \,{\left (\frac{2560 \, A a^{4}}{b^{2}} +{\left (\frac{945 \, B a^{4}}{b^{2}} - 2 \,{\left (\frac{640 \, A a^{3}}{b} +{\left (\frac{315 \, B a^{3}}{b} + 4 \,{\left (2400 \, A a^{2} +{\left (1953 \, B a^{2} + 2 \,{\left (1520 \, A a b + 7 \,{\left (189 \, B a b + 8 \,{\left (9 \, B b^{2} x + 10 \, A b^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{b x^{2} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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