Optimal. Leaf size=120 \[ \frac{b^2 (A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{3/2}}+\frac{\left (a+b x^2\right )^{3/2} (A b-6 a B)}{24 a x^4}+\frac{b \sqrt{a+b x^2} (A b-6 a B)}{16 a x^2}-\frac{A \left (a+b x^2\right )^{5/2}}{6 a x^6} \]
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Rubi [A] time = 0.0961466, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {446, 78, 47, 63, 208} \[ \frac{b^2 (A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{3/2}}+\frac{\left (a+b x^2\right )^{3/2} (A b-6 a B)}{24 a x^4}+\frac{b \sqrt{a+b x^2} (A b-6 a B)}{16 a x^2}-\frac{A \left (a+b x^2\right )^{5/2}}{6 a x^6} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2} (A+B x)}{x^4} \, dx,x,x^2\right )\\ &=-\frac{A \left (a+b x^2\right )^{5/2}}{6 a x^6}+\frac{\left (-\frac{A b}{2}+3 a B\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^3} \, dx,x,x^2\right )}{6 a}\\ &=\frac{(A b-6 a B) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac{A \left (a+b x^2\right )^{5/2}}{6 a x^6}-\frac{(b (A b-6 a B)) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,x^2\right )}{16 a}\\ &=\frac{b (A b-6 a B) \sqrt{a+b x^2}}{16 a x^2}+\frac{(A b-6 a B) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac{A \left (a+b x^2\right )^{5/2}}{6 a x^6}-\frac{\left (b^2 (A b-6 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{32 a}\\ &=\frac{b (A b-6 a B) \sqrt{a+b x^2}}{16 a x^2}+\frac{(A b-6 a B) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac{A \left (a+b x^2\right )^{5/2}}{6 a x^6}-\frac{(b (A b-6 a B)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{16 a}\\ &=\frac{b (A b-6 a B) \sqrt{a+b x^2}}{16 a x^2}+\frac{(A b-6 a B) \left (a+b x^2\right )^{3/2}}{24 a x^4}-\frac{A \left (a+b x^2\right )^{5/2}}{6 a x^6}+\frac{b^2 (A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0804227, size = 119, normalized size = 0.99 \[ \frac{-\left (a+b x^2\right ) \left (4 a^2 \left (2 A+3 B x^2\right )+2 a b x^2 \left (7 A+15 B x^2\right )+3 A b^2 x^4\right )-3 b^2 x^6 \sqrt{\frac{b x^2}{a}+1} (6 a B-A b) \tanh ^{-1}\left (\sqrt{\frac{b x^2}{a}+1}\right )}{48 a x^6 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 233, normalized size = 1.9 \begin{align*} -{\frac{A}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ab}{24\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{A{b}^{2}}{48\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{A{b}^{3}}{48\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{A{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{A{b}^{3}}{16\,{a}^{2}}\sqrt{b{x}^{2}+a}}-{\frac{B}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Bb}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{B{b}^{2}}{8\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,B{b}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{3\,B{b}^{2}}{8\,a}\sqrt{b{x}^{2}+a}} \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 = 1.70586, size = 506, normalized size = 4.22 \begin{align*} \left [-\frac{3 \,{\left (6 \, B a b^{2} - A b^{3}\right )} \sqrt{a} x^{6} \log \left (-\frac{b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (3 \,{\left (10 \, B a^{2} b + A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 2 \,{\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{96 \, a^{2} x^{6}}, \frac{3 \,{\left (6 \, B a b^{2} - A b^{3}\right )} \sqrt{-a} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (3 \,{\left (10 \, B a^{2} b + A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 2 \,{\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{48 \, a^{2} x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 82.4467, size = 253, normalized size = 2.11 \begin{align*} - \frac{A a^{2}}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{11 A a \sqrt{b}}{24 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{17 A b^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{A b^{\frac{5}{2}}}{16 a x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 a^{\frac{3}{2}}} - \frac{B a^{2}}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 B a \sqrt{b}}{8 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{B b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} - \frac{B b^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 B b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12376, size = 215, normalized size = 1.79 \begin{align*} \frac{\frac{3 \,{\left (6 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{30 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a b^{3} - 48 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{2} b^{3} + 18 \, \sqrt{b x^{2} + a} B a^{3} b^{3} + 3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A b^{4} + 8 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b^{4} - 3 \, \sqrt{b x^{2} + a} A a^{2} b^{4}}{a b^{3} x^{6}}}{48 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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