Optimal. Leaf size=187 \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac{\left (c+d x^2\right )^{3/2} \left (a d (12 b c-a d)+24 b^2 c^2\right )}{48 c^2 x^2}+\frac{d \sqrt{c+d x^2} \left (a d (12 b c-a d)+24 b^2 c^2\right )}{16 c^2}-\frac{d \left (a d (12 b c-a d)+24 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{3/2}}-\frac{a \left (c+d x^2\right )^{5/2} (12 b c-a d)}{24 c^2 x^4} \]
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Rubi [A] time = 0.222079, antiderivative size = 184, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {446, 89, 78, 47, 50, 63, 208} \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac{\left (c+d x^2\right )^{3/2} \left (\frac{a d (12 b c-a d)}{c^2}+24 b^2\right )}{48 x^2}+\frac{d \sqrt{c+d x^2} \left (a d (12 b c-a d)+24 b^2 c^2\right )}{16 c^2}-\frac{d \left (a d (12 b c-a d)+24 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{3/2}}-\frac{a \left (c+d x^2\right )^{5/2} (12 b c-a d)}{24 c^2 x^4} \]
Antiderivative was successfully verified.
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Rule 446
Rule 89
Rule 78
Rule 47
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^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2 (c+d x)^{3/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}+\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{1}{2} a (12 b c-a d)+3 b^2 c x\right ) (c+d x)^{3/2}}{x^3} \, dx,x,x^2\right )}{6 c}\\ &=-\frac{a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac{a (12 b c-a d) \left (c+d x^2\right )^{5/2}}{24 c^2 x^4}+\frac{1}{48} \left (24 b^2+\frac{a d (12 b c-a d)}{c^2}\right ) \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{\left (24 b^2+\frac{a d (12 b c-a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{48 x^2}-\frac{a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac{a (12 b c-a d) \left (c+d x^2\right )^{5/2}}{24 c^2 x^4}+\frac{1}{32} \left (d \left (24 b^2+\frac{a d (12 b c-a d)}{c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{16} d \left (24 b^2+\frac{a d (12 b c-a d)}{c^2}\right ) \sqrt{c+d x^2}-\frac{\left (24 b^2+\frac{a d (12 b c-a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{48 x^2}-\frac{a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac{a (12 b c-a d) \left (c+d x^2\right )^{5/2}}{24 c^2 x^4}+\frac{1}{32} \left (c d \left (24 b^2+\frac{a d (12 b c-a d)}{c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=\frac{1}{16} d \left (24 b^2+\frac{a d (12 b c-a d)}{c^2}\right ) \sqrt{c+d x^2}-\frac{\left (24 b^2+\frac{a d (12 b c-a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{48 x^2}-\frac{a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac{a (12 b c-a d) \left (c+d x^2\right )^{5/2}}{24 c^2 x^4}+\frac{1}{16} \left (c \left (24 b^2+\frac{a d (12 b c-a d)}{c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )\\ &=\frac{1}{16} d \left (24 b^2+\frac{a d (12 b c-a d)}{c^2}\right ) \sqrt{c+d x^2}-\frac{\left (24 b^2+\frac{a d (12 b c-a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}}{48 x^2}-\frac{a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac{a (12 b c-a d) \left (c+d x^2\right )^{5/2}}{24 c^2 x^4}-\frac{1}{16} \sqrt{c} d \left (24 b^2+\frac{a d (12 b c-a d)}{c^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [C] time = 0.0467302, size = 92, normalized size = 0.49 \[ -\frac{\left (c+d x^2\right )^{5/2} \left (d x^6 \left (a^2 d^2-12 a b c d-24 b^2 c^2\right ) \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{d x^2}{c}+1\right )+5 a c^2 \left (4 a c-a d x^2+12 b c x^2\right )\right )}{120 c^4 x^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 335, normalized size = 1.8 \begin{align*} -{\frac{{a}^{2}}{6\,c{x}^{6}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}d}{24\,{c}^{2}{x}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}{d}^{2}}{48\,{c}^{3}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}{d}^{3}}{48\,{c}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}{d}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}-{\frac{{a}^{2}{d}^{3}}{16\,{c}^{2}}\sqrt{d{x}^{2}+c}}-{\frac{ab}{2\,c{x}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{abd}{4\,{c}^{2}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{ab{d}^{2}}{4\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,ab{d}^{2}}{4}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}+{\frac{3\,ab{d}^{2}}{4\,c}\sqrt{d{x}^{2}+c}}-{\frac{{b}^{2}}{2\,c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}d}{2\,c} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{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\,{b}^{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 = 1.4945, size = 674, normalized size = 3.6 \begin{align*} \left [-\frac{3 \,{\left (24 \, b^{2} c^{2} d + 12 \, a b c d^{2} - a^{2} d^{3}\right )} \sqrt{c} x^{6} \log \left (-\frac{d x^{2} + 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) - 2 \,{\left (48 \, b^{2} c^{2} d x^{6} - 8 \, a^{2} c^{3} - 3 \,{\left (8 \, b^{2} c^{3} + 20 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{4} - 2 \,{\left (12 \, a b c^{3} + 7 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{96 \, c^{2} x^{6}}, \frac{3 \,{\left (24 \, b^{2} c^{2} d + 12 \, a b c d^{2} - a^{2} d^{3}\right )} \sqrt{-c} x^{6} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) +{\left (48 \, b^{2} c^{2} d x^{6} - 8 \, a^{2} c^{3} - 3 \,{\left (8 \, b^{2} c^{3} + 20 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{4} - 2 \,{\left (12 \, a b c^{3} + 7 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{48 \, c^{2} x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 118.042, size = 367, normalized size = 1.96 \begin{align*} - \frac{a^{2} c^{2}}{6 \sqrt{d} x^{7} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{11 a^{2} c \sqrt{d}}{24 x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{17 a^{2} d^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{a^{2} d^{\frac{5}{2}}}{16 c x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a^{2} d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{16 c^{\frac{3}{2}}} - \frac{a b c^{2}}{2 \sqrt{d} x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a b c \sqrt{d}}{4 x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{a b d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{x} - \frac{a b d^{\frac{3}{2}}}{4 x \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a b d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{4 \sqrt{c}} - \frac{3 b^{2} \sqrt{c} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{2} - \frac{b^{2} c \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{2 x} + \frac{b^{2} c \sqrt{d}}{x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{b^{2} d^{\frac{3}{2}} x}{\sqrt{\frac{c}{d x^{2}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16873, size = 350, normalized size = 1.87 \begin{align*} \frac{48 \, \sqrt{d x^{2} + c} b^{2} d^{2} + \frac{3 \,{\left (24 \, b^{2} c^{2} d^{2} + 12 \, a b c d^{3} - a^{2} d^{4}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c} - \frac{24 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} c^{2} d^{2} - 48 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c^{3} d^{2} + 24 \, \sqrt{d x^{2} + c} b^{2} c^{4} d^{2} + 60 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a b c d^{3} - 96 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c^{2} d^{3} + 36 \, \sqrt{d x^{2} + c} a b c^{3} d^{3} + 3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a^{2} d^{4} + 8 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} c d^{4} - 3 \, \sqrt{d x^{2} + c} a^{2} c^{2} d^{4}}{c d^{3} x^{6}}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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