Optimal. Leaf size=34 \[ -\frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} (b c-a d)} \]
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Rubi [A] time = 0.0268717, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {444, 37} \[ -\frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 444
Rule 37
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{c+d x^2}}{(b c-a d) \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0099777, size = 33, normalized size = 0.97 \[ \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 30, normalized size = 0.9 \begin{align*}{\frac{1}{ad-bc}\sqrt{d{x}^{2}+c}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \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 = 1.8887, size = 97, normalized size = 2.85 \begin{align*} -\frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}} \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{x}{\left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18841, size = 95, normalized size = 2.79 \begin{align*} -\frac{2 \, \sqrt{b d} b}{{\left (b^{2} c - a b d -{\left (\sqrt{b x^{2} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}\right )}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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