Optimal. Leaf size=191 \[ \frac{a \sqrt{d x^2+2} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right ),1-\frac{3 d}{2 f}\right )}{\sqrt{2} \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{b x \sqrt{d x^2+2}}{d \sqrt{f x^2+3}}-\frac{\sqrt{2} b \sqrt{d x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{d \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0960592, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {531, 418, 492, 411} \[ \frac{a \sqrt{d x^2+2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{\sqrt{2} \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{b x \sqrt{d x^2+2}}{d \sqrt{f x^2+3}}-\frac{\sqrt{2} b \sqrt{d x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{d \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 531
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{a+b x^2}{\sqrt{2+d x^2} \sqrt{3+f x^2}} \, dx &=a \int \frac{1}{\sqrt{2+d x^2} \sqrt{3+f x^2}} \, dx+b \int \frac{x^2}{\sqrt{2+d x^2} \sqrt{3+f x^2}} \, dx\\ &=\frac{b x \sqrt{2+d x^2}}{d \sqrt{3+f x^2}}+\frac{a \sqrt{2+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{\sqrt{2} \sqrt{f} \sqrt{\frac{2+d x^2}{3+f x^2}} \sqrt{3+f x^2}}-\frac{(3 b) \int \frac{\sqrt{2+d x^2}}{\left (3+f x^2\right )^{3/2}} \, dx}{d}\\ &=\frac{b x \sqrt{2+d x^2}}{d \sqrt{3+f x^2}}-\frac{\sqrt{2} b \sqrt{2+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{d \sqrt{f} \sqrt{\frac{2+d x^2}{3+f x^2}} \sqrt{3+f x^2}}+\frac{a \sqrt{2+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{\sqrt{2} \sqrt{f} \sqrt{\frac{2+d x^2}{3+f x^2}} \sqrt{3+f x^2}}\\ \end{align*}
Mathematica [C] time = 0.141978, size = 81, normalized size = 0.42 \[ -\frac{i \left ((a f-3 b) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right ),\frac{2 f}{3 d}\right )+3 b E\left (i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )\right )}{\sqrt{3} \sqrt{d} f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.037, size = 105, normalized size = 0.6 \begin{align*}{\frac{\sqrt{2}}{2\,d} \left ({\it EllipticF} \left ({\frac{x\sqrt{3}}{3}\sqrt{-f}},{\frac{\sqrt{2}\sqrt{3}}{2}\sqrt{{\frac{d}{f}}}} \right ) ad-2\,{\it EllipticF} \left ( 1/3\,x\sqrt{3}\sqrt{-f},1/2\,\sqrt{2}\sqrt{3}\sqrt{{\frac{d}{f}}} \right ) b+2\,{\it EllipticE} \left ( 1/3\,x\sqrt{3}\sqrt{-f},1/2\,\sqrt{2}\sqrt{3}\sqrt{{\frac{d}{f}}} \right ) b \right ){\frac{1}{\sqrt{-f}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x^{2} + a}{\sqrt{d x^{2} + 2} \sqrt{f x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )} \sqrt{d x^{2} + 2} \sqrt{f x^{2} + 3}}{d f x^{4} +{\left (3 \, d + 2 \, f\right )} x^{2} + 6}, x\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{a + b x^{2}}{\sqrt{d x^{2} + 2} \sqrt{f x^{2} + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x^{2} + a}{\sqrt{d x^{2} + 2} \sqrt{f x^{2} + 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]