Optimal. Leaf size=84 \[ -\frac{\left (c+d x^2\right )^{3/2} \left (-5 d (a B+A b)+2 b B c-3 b B d x^2\right )}{15 d^2}+a A \sqrt{c+d x^2}-a A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right ) \]
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Rubi [A] time = 0.0767331, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {573, 147, 50, 63, 208} \[ -\frac{\left (c+d x^2\right )^{3/2} \left (-5 d (a B+A b)+2 b B c-3 b B d x^2\right )}{15 d^2}+a A \sqrt{c+d x^2}-a A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Rule 573
Rule 147
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \left (A+B x^2\right ) \sqrt{c+d x^2}}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x) (A+B x) \sqrt{c+d x}}{x} \, dx,x,x^2\right )\\ &=-\frac{\left (c+d x^2\right )^{3/2} \left (2 b B c-5 (A b+a B) d-3 b B d x^2\right )}{15 d^2}+\frac{1}{2} (a A) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{x} \, dx,x,x^2\right )\\ &=a A \sqrt{c+d x^2}-\frac{\left (c+d x^2\right )^{3/2} \left (2 b B c-5 (A b+a B) d-3 b B d x^2\right )}{15 d^2}+\frac{1}{2} (a A c) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=a A \sqrt{c+d x^2}-\frac{\left (c+d x^2\right )^{3/2} \left (2 b B c-5 (A b+a B) d-3 b B d x^2\right )}{15 d^2}+\frac{(a A c) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{d}\\ &=a A \sqrt{c+d x^2}-\frac{\left (c+d x^2\right )^{3/2} \left (2 b B c-5 (A b+a B) d-3 b B d x^2\right )}{15 d^2}-a A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.131688, size = 91, normalized size = 1.08 \[ \frac{\sqrt{c+d x^2} \left (5 a d \left (3 A d+B \left (c+d x^2\right )\right )-b \left (c+d x^2\right ) \left (-5 A d+2 B c-3 B d x^2\right )\right )}{15 d^2}-a A \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 112, normalized size = 1.3 \begin{align*}{\frac{Bb{x}^{2}}{5\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{2\,bBc}{15\,{d}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{Ab}{3\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{Ba}{3\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-A\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ) \sqrt{c}a+aA\sqrt{d{x}^{2}+c} \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 = 1.6325, size = 518, normalized size = 6.17 \begin{align*} \left [\frac{15 \, A a \sqrt{c} d^{2} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (3 \, B b d^{2} x^{4} - 2 \, B b c^{2} + 15 \, A a d^{2} + 5 \,{\left (B a + A b\right )} c d +{\left (B b c d + 5 \,{\left (B a + A b\right )} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{30 \, d^{2}}, \frac{15 \, A a \sqrt{-c} d^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) +{\left (3 \, B b d^{2} x^{4} - 2 \, B b c^{2} + 15 \, A a d^{2} + 5 \,{\left (B a + A b\right )} c d +{\left (B b c d + 5 \,{\left (B a + A b\right )} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{15 \, d^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 35.8138, size = 97, normalized size = 1.15 \begin{align*} \frac{A a c \operatorname{atan}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{- c}} \right )}}{\sqrt{- c}} + A a \sqrt{c + d x^{2}} + \frac{B b \left (c + d x^{2}\right )^{\frac{5}{2}}}{5 d^{2}} + \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (2 A b d + 2 B a d - 2 B b c\right )}{6 d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16806, size = 153, normalized size = 1.82 \begin{align*} \frac{A a c \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}} + \frac{3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} B b d^{8} - 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} B b c d^{8} + 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} B a d^{9} + 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} A b d^{9} + 15 \, \sqrt{d x^{2} + c} A a d^{10}}{15 \, d^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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