Optimal. Leaf size=113 \[ -\frac{3 d}{2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{1}{2 \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}+\frac{3 \sqrt{b} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 (b c-a d)^{5/2}} \]
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Rubi [A] time = 0.0852439, antiderivative size = 113, 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} \[ -\frac{3 d}{2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{1}{2 \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}+\frac{3 \sqrt{b} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 (b c-a d)^{5/2}} \]
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 )^2 \left (c+d x^2\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(a+b x)^2 (c+d x)^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{1}{2 (b c-a d) \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{(3 d) \operatorname{Subst}\left (\int \frac{1}{(a+b x) (c+d x)^{3/2}} \, dx,x,x^2\right )}{4 (b c-a d)}\\ &=-\frac{3 d}{2 (b c-a d)^2 \sqrt{c+d x^2}}-\frac{1}{2 (b c-a d) \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{(3 b d) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^2\right )}{4 (b c-a d)^2}\\ &=-\frac{3 d}{2 (b c-a d)^2 \sqrt{c+d x^2}}-\frac{1}{2 (b c-a d) \left (a+b x^2\right ) \sqrt{c+d x^2}}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^2}\right )}{2 (b c-a d)^2}\\ &=-\frac{3 d}{2 (b c-a d)^2 \sqrt{c+d x^2}}-\frac{1}{2 (b c-a d) \left (a+b x^2\right ) \sqrt{c+d x^2}}+\frac{3 \sqrt{b} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 (b c-a d)^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0160044, size = 52, normalized size = 0.46 \[ -\frac{d \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};-\frac{b \left (d x^2+c\right )}{a d-b c}\right )}{\sqrt{c+d x^2} (a d-b c)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 989, normalized size = 8.8 \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 = 2.04625, size = 1110, normalized size = 9.82 \begin{align*} \left [\frac{3 \,{\left (b d^{2} x^{4} + a c d +{\left (b c d + a d^{2}\right )} x^{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} + b c + 2 \, a d\right )} \sqrt{d x^{2} + c}}{8 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}}, -\frac{3 \,{\left (b d^{2} x^{4} + a c d +{\left (b c d + a d^{2}\right )} x^{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} + b c + 2 \, a d\right )} \sqrt{d x^{2} + c}}{4 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A] time = 1.12848, size = 203, normalized size = 1.8 \begin{align*} -\frac{1}{2} \, d{\left (\frac{3 \, b \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 - 2 \, b c + 2 \, a d}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left ({\left (d x^{2} + c\right )}^{\frac{3}{2}} b - \sqrt{d x^{2} + c} b c + \sqrt{d x^{2} + c} a d\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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