Optimal. Leaf size=185 \[ -\frac{a^2}{4 c x^4 \left (c+d x^2\right )^{3/2}}+\frac{8 b^2 c^2-5 a d (8 b c-7 a d)}{8 c^4 \sqrt{c+d x^2}}+\frac{8 b^2 c^2-5 a d (8 b c-7 a d)}{24 c^3 \left (c+d x^2\right )^{3/2}}-\frac{\left (8 b^2 c^2-5 a d (8 b c-7 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{9/2}}-\frac{a (8 b c-7 a d)}{8 c^2 x^2 \left (c+d x^2\right )^{3/2}} \]
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Rubi [A] time = 0.215523, antiderivative size = 185, 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, 78, 51, 63, 208} \[ -\frac{a^2}{4 c x^4 \left (c+d x^2\right )^{3/2}}+\frac{8 b^2 c^2-5 a d (8 b c-7 a d)}{8 c^4 \sqrt{c+d x^2}}+\frac{8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}}{24 c \left (c+d x^2\right )^{3/2}}-\frac{\left (8 b^2 c^2-5 a d (8 b c-7 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{9/2}}-\frac{a (8 b c-7 a d)}{8 c^2 x^2 \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 446
Rule 89
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^5 \left (c+d x^2\right )^{5/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^3 (c+d x)^{5/2}} \, dx,x,x^2\right )\\ &=-\frac{a^2}{4 c x^4 \left (c+d x^2\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} a (8 b c-7 a d)+2 b^2 c x}{x^2 (c+d x)^{5/2}} \, dx,x,x^2\right )}{4 c}\\ &=-\frac{a^2}{4 c x^4 \left (c+d x^2\right )^{3/2}}-\frac{a (8 b c-7 a d)}{8 c^2 x^2 \left (c+d x^2\right )^{3/2}}+\frac{1}{16} \left (8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x (c+d x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac{8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}}{24 c \left (c+d x^2\right )^{3/2}}-\frac{a^2}{4 c x^4 \left (c+d x^2\right )^{3/2}}-\frac{a (8 b c-7 a d)}{8 c^2 x^2 \left (c+d x^2\right )^{3/2}}+\frac{\left (8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x (c+d x)^{3/2}} \, dx,x,x^2\right )}{16 c}\\ &=\frac{8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}}{24 c \left (c+d x^2\right )^{3/2}}-\frac{a^2}{4 c x^4 \left (c+d x^2\right )^{3/2}}-\frac{a (8 b c-7 a d)}{8 c^2 x^2 \left (c+d x^2\right )^{3/2}}+\frac{8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}}{8 c^2 \sqrt{c+d x^2}}+\frac{\left (8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^2\right )}{16 c^2}\\ &=\frac{8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}}{24 c \left (c+d x^2\right )^{3/2}}-\frac{a^2}{4 c x^4 \left (c+d x^2\right )^{3/2}}-\frac{a (8 b c-7 a d)}{8 c^2 x^2 \left (c+d x^2\right )^{3/2}}+\frac{8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}}{8 c^2 \sqrt{c+d x^2}}+\frac{\left (8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{8 c^2 d}\\ &=\frac{8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}}{24 c \left (c+d x^2\right )^{3/2}}-\frac{a^2}{4 c x^4 \left (c+d x^2\right )^{3/2}}-\frac{a (8 b c-7 a d)}{8 c^2 x^2 \left (c+d x^2\right )^{3/2}}+\frac{8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}}{8 c^2 \sqrt{c+d x^2}}-\frac{\left (8 b^2-\frac{5 a d (8 b c-7 a d)}{c^2}\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{8 c^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0363152, size = 90, normalized size = 0.49 \[ \frac{x^4 \left (35 a^2 d^2-40 a b c d+8 b^2 c^2\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{d x^2}{c}+1\right )-3 a c \left (2 a c-7 a d x^2+8 b c x^2\right )}{24 c^3 x^4 \left (c+d x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 265, normalized size = 1.4 \begin{align*}{\frac{{b}^{2}}{3\,c} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{{b}^{2}}{{c}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{{b}^{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{5}{2}}}}-{\frac{{a}^{2}}{4\,c{x}^{4}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{7\,{a}^{2}d}{8\,{c}^{2}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,{a}^{2}{d}^{2}}{24\,{c}^{3}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}+{\frac{35\,{a}^{2}{d}^{2}}{8\,{c}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{35\,{a}^{2}{d}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{9}{2}}}}-{\frac{ab}{c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,abd}{3\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{-{\frac{3}{2}}}}-5\,{\frac{abd}{{c}^{3}\sqrt{d{x}^{2}+c}}}+5\,{\frac{abd}{{c}^{7/2}}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ) } \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 = 1.53703, size = 1150, normalized size = 6.22 \begin{align*} \left [\frac{3 \,{\left ({\left (8 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} + 2 \,{\left (8 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} +{\left (8 \, b^{2} c^{4} - 40 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4}\right )} \sqrt{c} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (3 \,{\left (8 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} - 6 \, a^{2} c^{4} + 4 \,{\left (8 \, b^{2} c^{4} - 40 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4} - 3 \,{\left (8 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{48 \,{\left (c^{5} d^{2} x^{8} + 2 \, c^{6} d x^{6} + c^{7} x^{4}\right )}}, \frac{3 \,{\left ({\left (8 \, b^{2} c^{2} d^{2} - 40 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} + 2 \,{\left (8 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} +{\left (8 \, b^{2} c^{4} - 40 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) +{\left (3 \,{\left (8 \, b^{2} c^{3} d - 40 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6} - 6 \, a^{2} c^{4} + 4 \,{\left (8 \, b^{2} c^{4} - 40 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{4} - 3 \,{\left (8 \, a b c^{4} - 7 \, a^{2} c^{3} d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{24 \,{\left (c^{5} d^{2} x^{8} + 2 \, c^{6} d x^{6} + c^{7} x^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2}\right )^{2}}{x^{5} \left (c + d x^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17544, size = 284, normalized size = 1.54 \begin{align*} \frac{{\left (8 \, b^{2} c^{2} - 40 \, a b c d + 35 \, a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{8 \, \sqrt{-c} c^{4}} + \frac{3 \,{\left (d x^{2} + c\right )} b^{2} c^{2} + b^{2} c^{3} - 12 \,{\left (d x^{2} + c\right )} a b c d - 2 \, a b c^{2} d + 9 \,{\left (d x^{2} + c\right )} a^{2} d^{2} + a^{2} c d^{2}}{3 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} c^{4}} - \frac{8 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c d - 8 \, \sqrt{d x^{2} + c} a b c^{2} d - 11 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} d^{2} + 13 \, \sqrt{d x^{2} + c} a^{2} c d^{2}}{8 \, c^{4} d^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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