Optimal. Leaf size=136 \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{2 c x^2}+\frac{a \left (c+d x^2\right )^{3/2} (3 a d+4 b c)}{6 c}+\frac{1}{2} a \sqrt{c+d x^2} (3 a d+4 b c)-\frac{1}{2} a \sqrt{c} (3 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d} \]
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Rubi [A] time = 0.108765, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 89, 80, 50, 63, 208} \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{2 c x^2}+\frac{a \left (c+d x^2\right )^{3/2} (3 a d+4 b c)}{6 c}+\frac{1}{2} a \sqrt{c+d x^2} (3 a d+4 b c)-\frac{1}{2} a \sqrt{c} (3 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 446
Rule 89
Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2 (c+d x)^{3/2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{a^2 \left (c+d x^2\right )^{5/2}}{2 c x^2}+\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{1}{2} a (4 b c+3 a d)+b^2 c x\right ) (c+d x)^{3/2}}{x} \, dx,x,x^2\right )}{2 c}\\ &=\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d}-\frac{a^2 \left (c+d x^2\right )^{5/2}}{2 c x^2}+\frac{(a (4 b c+3 a d)) \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2}}{x} \, dx,x,x^2\right )}{4 c}\\ &=\frac{a (4 b c+3 a d) \left (c+d x^2\right )^{3/2}}{6 c}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d}-\frac{a^2 \left (c+d x^2\right )^{5/2}}{2 c x^2}+\frac{1}{4} (a (4 b c+3 a d)) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{2} a (4 b c+3 a d) \sqrt{c+d x^2}+\frac{a (4 b c+3 a d) \left (c+d x^2\right )^{3/2}}{6 c}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d}-\frac{a^2 \left (c+d x^2\right )^{5/2}}{2 c x^2}+\frac{1}{4} (a c (4 b c+3 a d)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} a (4 b c+3 a d) \sqrt{c+d x^2}+\frac{a (4 b c+3 a d) \left (c+d x^2\right )^{3/2}}{6 c}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d}-\frac{a^2 \left (c+d x^2\right )^{5/2}}{2 c x^2}+\frac{(a c (4 b c+3 a d)) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{2 d}\\ &=\frac{1}{2} a (4 b c+3 a d) \sqrt{c+d x^2}+\frac{a (4 b c+3 a d) \left (c+d x^2\right )^{3/2}}{6 c}+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d}-\frac{a^2 \left (c+d x^2\right )^{5/2}}{2 c x^2}-\frac{1}{2} a \sqrt{c} (4 b c+3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.087984, size = 108, normalized size = 0.79 \[ \frac{\sqrt{c+d x^2} \left (-15 a^2 d \left (c-2 d x^2\right )+20 a b d x^2 \left (4 c+d x^2\right )+6 b^2 x^2 \left (c+d x^2\right )^2\right )}{30 d x^2}-\frac{1}{2} a \sqrt{c} (3 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 161, normalized size = 1.2 \begin{align*}{\frac{{b}^{2}}{5\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{2\,ab}{3} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-2\,ab\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ){c}^{3/2}+2\,ab\sqrt{d{x}^{2}+c}c-{\frac{{a}^{2}}{2\,c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}d}{2\,c} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{a}^{2}d}{2}\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ) }+{\frac{3\,{a}^{2}d}{2}\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 = 1.41225, size = 610, normalized size = 4.49 \begin{align*} \left [\frac{15 \,{\left (4 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{c} x^{2} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (6 \, b^{2} d^{2} x^{6} + 4 \,{\left (3 \, b^{2} c d + 5 \, a b d^{2}\right )} x^{4} - 15 \, a^{2} c d + 2 \,{\left (3 \, b^{2} c^{2} + 40 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{60 \, d x^{2}}, \frac{15 \,{\left (4 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) +{\left (6 \, b^{2} d^{2} x^{6} + 4 \,{\left (3 \, b^{2} c d + 5 \, a b d^{2}\right )} x^{4} - 15 \, a^{2} c d + 2 \,{\left (3 \, b^{2} c^{2} + 40 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{30 \, d x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 40.1501, size = 303, normalized size = 2.23 \begin{align*} - \frac{3 a^{2} \sqrt{c} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{2} - \frac{a^{2} c \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{2 x} + \frac{a^{2} c \sqrt{d}}{x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a^{2} d^{\frac{3}{2}} x}{\sqrt{\frac{c}{d x^{2}} + 1}} - 2 a b c^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )} + \frac{2 a b c^{2}}{\sqrt{d} x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{2 a b c \sqrt{d} x}{\sqrt{\frac{c}{d x^{2}} + 1}} + 2 a b d \left (\begin{cases} \frac{\sqrt{c} x^{2}}{2} & \text{for}\: d = 0 \\\frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) + b^{2} c \left (\begin{cases} \frac{\sqrt{c} x^{2}}{2} & \text{for}\: d = 0 \\\frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) + b^{2} d \left (\begin{cases} - \frac{2 c^{2} \sqrt{c + d x^{2}}}{15 d^{2}} + \frac{c x^{2} \sqrt{c + d x^{2}}}{15 d} + \frac{x^{4} \sqrt{c + d x^{2}}}{5} & \text{for}\: d \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13866, size = 170, normalized size = 1.25 \begin{align*} \frac{6 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} + 20 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b d + 60 \, \sqrt{d x^{2} + c} a b c d + 30 \, \sqrt{d x^{2} + c} a^{2} d^{2} - \frac{15 \, \sqrt{d x^{2} + c} a^{2} c d}{x^{2}} + \frac{15 \,{\left (4 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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