Optimal. Leaf size=115 \[ -\frac{\left (a+b x^2\right )^{3/2} (4 a B+A b)}{8 a x^2}+\frac{3 b \sqrt{a+b x^2} (4 a B+A b)}{8 a}-\frac{3 b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{A \left (a+b x^2\right )^{5/2}}{4 a x^4} \]
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Rubi [A] time = 0.0839492, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {446, 78, 47, 50, 63, 208} \[ -\frac{\left (a+b x^2\right )^{3/2} (4 a B+A b)}{8 a x^2}+\frac{3 b \sqrt{a+b x^2} (4 a B+A b)}{8 a}-\frac{3 b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{A \left (a+b x^2\right )^{5/2}}{4 a x^4} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 47
Rule 50
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^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2} (A+B x)}{x^3} \, dx,x,x^2\right )\\ &=-\frac{A \left (a+b x^2\right )^{5/2}}{4 a x^4}+\frac{(A b+4 a B) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^2} \, dx,x,x^2\right )}{8 a}\\ &=-\frac{(A b+4 a B) \left (a+b x^2\right )^{3/2}}{8 a x^2}-\frac{A \left (a+b x^2\right )^{5/2}}{4 a x^4}+\frac{(3 b (A b+4 a B)) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^2\right )}{16 a}\\ &=\frac{3 b (A b+4 a B) \sqrt{a+b x^2}}{8 a}-\frac{(A b+4 a B) \left (a+b x^2\right )^{3/2}}{8 a x^2}-\frac{A \left (a+b x^2\right )^{5/2}}{4 a x^4}+\frac{1}{16} (3 b (A b+4 a B)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=\frac{3 b (A b+4 a B) \sqrt{a+b x^2}}{8 a}-\frac{(A b+4 a B) \left (a+b x^2\right )^{3/2}}{8 a x^2}-\frac{A \left (a+b x^2\right )^{5/2}}{4 a x^4}+\frac{1}{8} (3 (A b+4 a B)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )\\ &=\frac{3 b (A b+4 a B) \sqrt{a+b x^2}}{8 a}-\frac{(A b+4 a B) \left (a+b x^2\right )^{3/2}}{8 a x^2}-\frac{A \left (a+b x^2\right )^{5/2}}{4 a x^4}-\frac{3 b (A b+4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}\\ \end{align*}
Mathematica [C] time = 0.0275285, size = 59, normalized size = 0.51 \[ \frac{\left (a+b x^2\right )^{5/2} \left (b x^4 (4 a B+A b) \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{b x^2}{a}+1\right )-5 a^2 A\right )}{20 a^3 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 184, normalized size = 1.6 \begin{align*} -{\frac{A}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Ab}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{A{b}^{2}}{8\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,A{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\,A{b}^{2}}{8\,a}\sqrt{b{x}^{2}+a}}-{\frac{B}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bb}{2\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,Bb}{2}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{3\,Bb}{2}\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 = 1.64899, size = 436, normalized size = 3.79 \begin{align*} \left [\frac{3 \,{\left (4 \, B a b + A b^{2}\right )} \sqrt{a} x^{4} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (8 \, B a b x^{4} - 2 \, A a^{2} -{\left (4 \, B a^{2} + 5 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{16 \, a x^{4}}, \frac{3 \,{\left (4 \, B a b + A b^{2}\right )} \sqrt{-a} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (8 \, B a b x^{4} - 2 \, A a^{2} -{\left (4 \, B a^{2} + 5 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{8 \, a x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 56.6813, size = 216, normalized size = 1.88 \begin{align*} - \frac{A a^{2}}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 A a \sqrt{b}}{8 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} - \frac{A b^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 A b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 \sqrt{a}} - \frac{3 B \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2} - \frac{B a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} + \frac{B a \sqrt{b}}{x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B b^{\frac{3}{2}} x}{\sqrt{\frac{a}{b x^{2}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14449, size = 177, normalized size = 1.54 \begin{align*} \frac{8 \, \sqrt{b x^{2} + a} B b^{2} + \frac{3 \,{\left (4 \, B a b^{2} + A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{4 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x^{2} + a} B a^{2} b^{2} + 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b^{3} - 3 \, \sqrt{b x^{2} + a} A a b^{3}}{b^{2} x^{4}}}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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