Optimal. Leaf size=48 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-b x^2}}{\sqrt{b} \sqrt{c-d x^2}}\right )}{\sqrt{b} \sqrt{d}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0580089, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {444, 63, 217, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-b x^2}}{\sqrt{b} \sqrt{c-d x^2}}\right )}{\sqrt{b} \sqrt{d}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 444
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{a-b x^2} \sqrt{c-d x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b x} \sqrt{c-d x}} \, dx,x,x^2\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a-b x^2}\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a-b x^2}}{\sqrt{c-d x^2}}\right )}{b}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-b x^2}}{\sqrt{b} \sqrt{c-d x^2}}\right )}{\sqrt{b} \sqrt{d}}\\ \end{align*}
Mathematica [B] time = 0.0897496, size = 109, normalized size = 2.27 \[ \frac{\sqrt{-b} \sqrt{a d-b c} \sqrt{\frac{b \left (c-d x^2\right )}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{-b} \sqrt{d} \sqrt{a-b x^2}}{\sqrt{b} \sqrt{a d-b c}}\right )}{b^{3/2} \sqrt{d} \sqrt{c-d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.04, size = 111, normalized size = 2.3 \begin{align*}{\frac{1}{2}\ln \left ({\frac{1}{2} \left ( 2\,d{x}^{2}b+2\,\sqrt{bd{x}^{4}-ad{x}^{2}-bc{x}^{2}+ac}\sqrt{bd}-ad-bc \right ){\frac{1}{\sqrt{bd}}}} \right ) \sqrt{-b{x}^{2}+a}\sqrt{-d{x}^{2}+c}{\frac{1}{\sqrt{bd{x}^{4}-ad{x}^{2}-bc{x}^{2}+ac}}}{\frac{1}{\sqrt{bd}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.89428, size = 446, normalized size = 9.29 \begin{align*} \left [\frac{\sqrt{b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \,{\left (2 \, b d x^{2} - b c - a d\right )} \sqrt{-b x^{2} + a} \sqrt{-d x^{2} + c} \sqrt{b d}\right )}{4 \, b d}, -\frac{\sqrt{-b d} \arctan \left (\frac{{\left (2 \, b d x^{2} - b c - a d\right )} \sqrt{-b x^{2} + a} \sqrt{-d x^{2} + c} \sqrt{-b d}}{2 \,{\left (b^{2} d^{2} x^{4} + a b c d -{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right )}{2 \, b d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{a - b x^{2}} \sqrt{c - d x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20376, size = 77, normalized size = 1.6 \begin{align*} \frac{b \log \left ({\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 |}\right )}{\sqrt{b d}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]