Optimal. Leaf size=95 \[ a^2 A \sqrt{a+b x^2}-a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{1}{5} A \left (a+b x^2\right )^{5/2}+\frac{1}{3} a A \left (a+b x^2\right )^{3/2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b} \]
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Rubi [A] time = 0.0643206, antiderivative size = 95, 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 = {446, 80, 50, 63, 208} \[ a^2 A \sqrt{a+b x^2}-a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{1}{5} A \left (a+b x^2\right )^{5/2}+\frac{1}{3} a A \left (a+b x^2\right )^{3/2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b} \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2} (A+B x)}{x} \, dx,x,x^2\right )\\ &=\frac{B \left (a+b x^2\right )^{7/2}}{7 b}+\frac{1}{2} A \operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{5} A \left (a+b x^2\right )^{5/2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b}+\frac{1}{2} (a A) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{3} a A \left (a+b x^2\right )^{3/2}+\frac{1}{5} A \left (a+b x^2\right )^{5/2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b}+\frac{1}{2} \left (a^2 A\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^2\right )\\ &=a^2 A \sqrt{a+b x^2}+\frac{1}{3} a A \left (a+b x^2\right )^{3/2}+\frac{1}{5} A \left (a+b x^2\right )^{5/2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b}+\frac{1}{2} \left (a^3 A\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )\\ &=a^2 A \sqrt{a+b x^2}+\frac{1}{3} a A \left (a+b x^2\right )^{3/2}+\frac{1}{5} A \left (a+b x^2\right )^{5/2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b}+\frac{\left (a^3 A\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{b}\\ &=a^2 A \sqrt{a+b x^2}+\frac{1}{3} a A \left (a+b x^2\right )^{3/2}+\frac{1}{5} A \left (a+b x^2\right )^{5/2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b}-a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0932397, size = 88, normalized size = 0.93 \[ -a^{5/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )+\frac{1}{5} A \left (a+b x^2\right )^{5/2}+\frac{1}{3} a A \left (4 a+b x^2\right ) \sqrt{a+b x^2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 85, normalized size = 0.9 \begin{align*}{\frac{B}{7\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{A}{5} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Aa}{3} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-A{a}^{{\frac{5}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +{a}^{2}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 = 1.67919, size = 529, normalized size = 5.57 \begin{align*} \left [\frac{105 \, A a^{\frac{5}{2}} b \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (15 \, B b^{3} x^{6} + 3 \,{\left (15 \, B a b^{2} + 7 \, A b^{3}\right )} x^{4} + 15 \, B a^{3} + 161 \, A a^{2} b +{\left (45 \, B a^{2} b + 77 \, A a b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{210 \, b}, \frac{105 \, A \sqrt{-a} a^{2} b \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (15 \, B b^{3} x^{6} + 3 \,{\left (15 \, B a b^{2} + 7 \, A b^{3}\right )} x^{4} + 15 \, B a^{3} + 161 \, A a^{2} b +{\left (45 \, B a^{2} b + 77 \, A a b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 48.334, size = 88, normalized size = 0.93 \begin{align*} \frac{A a^{3} \operatorname{atan}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} + A a^{2} \sqrt{a + b x^{2}} + \frac{A a \left (a + b x^{2}\right )^{\frac{3}{2}}}{3} + \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{5} + \frac{B \left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13819, size = 131, normalized size = 1.38 \begin{align*} \frac{A a^{3} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} B b^{6} + 21 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A b^{7} + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b^{7} + 105 \, \sqrt{b x^{2} + a} A a^{2} b^{7}}{105 \, b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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