Optimal. Leaf size=98 \[ -\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}}+\frac{b}{\sqrt{c+d x^2} (b c-a d)^2}+\frac{1}{3 \left (c+d x^2\right )^{3/2} (b c-a d)} \]
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Rubi [A] time = 0.0848251, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {444, 51, 63, 208} \[ -\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}}+\frac{b}{\sqrt{c+d x^2} (b c-a d)^2}+\frac{1}{3 \left (c+d x^2\right )^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 444
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(a+b x) (c+d x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac{1}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{(a+b x) (c+d x)^{3/2}} \, dx,x,x^2\right )}{2 (b c-a d)}\\ &=\frac{1}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac{b}{(b c-a d)^2 \sqrt{c+d x^2}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^2\right )}{2 (b c-a d)^2}\\ &=\frac{1}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac{b}{(b c-a d)^2 \sqrt{c+d x^2}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{d (b c-a d)^2}\\ &=\frac{1}{3 (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac{b}{(b c-a d)^2 \sqrt{c+d x^2}}-\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.021596, size = 52, normalized size = 0.53 \[ \frac{\, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b \left (d x^2+c\right )}{b c-a d}\right )}{3 \left (c+d x^2\right )^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 1086, normalized size = 11.1 \begin{align*} \text{result too large to display} \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 [B] time = 1.86772, size = 1061, normalized size = 10.83 \begin{align*} \left [\frac{3 \,{\left (b d^{2} x^{4} + 2 \, b c d x^{2} + b c^{2}\right )} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \,{\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \,{\left (3 \, b d x^{2} + 4 \, b c - a d\right )} \sqrt{d x^{2} + c}}{12 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{4} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}, \frac{3 \,{\left (b d^{2} x^{4} + 2 \, b c d x^{2} + b c^{2}\right )} \sqrt{-\frac{b}{b c - a d}} \arctan \left (\frac{{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{b}{b c - a d}}}{2 \,{\left (b d x^{2} + b c\right )}}\right ) + 2 \,{\left (3 \, b d x^{2} + 4 \, b c - a d\right )} \sqrt{d x^{2} + c}}{6 \,{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{4} + 2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.8985, size = 85, normalized size = 0.87 \begin{align*} \frac{b}{\sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} + \frac{b \operatorname{atan}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{\frac{a d - b c}{b}}} \right )}}{\sqrt{\frac{a d - b c}{b}} \left (a d - b c\right )^{2}} - \frac{1}{3 \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10842, size = 159, normalized size = 1.62 \begin{align*} \frac{b^{2} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{3 \,{\left (d x^{2} + c\right )} b + b c - a d}{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left (d x^{2} + c\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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