Optimal. Leaf size=86 \[ -\frac{2 \sqrt{\frac{f^2 (c+d x)}{c f^2+d}} \Pi \left (\frac{2 b}{a f^2+b};\sin ^{-1}\left (\frac{\sqrt{1-f^2 x}}{\sqrt{2}}\right )|\frac{2 d}{c f^2+d}\right )}{\left (a f^2+b\right ) \sqrt{c+d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.174857, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {932, 168, 538, 537} \[ -\frac{2 \sqrt{\frac{f^2 (c+d x)}{c f^2+d}} \Pi \left (\frac{2 b}{a f^2+b};\sin ^{-1}\left (\frac{\sqrt{1-f^2 x}}{\sqrt{2}}\right )|\frac{2 d}{c f^2+d}\right )}{\left (a f^2+b\right ) \sqrt{c+d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 932
Rule 168
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{1-f^4 x^2}} \, dx &=\int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{1-f^2 x} \sqrt{1+f^2 x}} \, dx\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2} \left (b+a f^2-b x^2\right ) \sqrt{c+\frac{d}{f^2}-\frac{d x^2}{f^2}}} \, dx,x,\sqrt{1-f^2 x}\right )\right )\\ &=-\frac{\left (2 \sqrt{\frac{f^2 (c+d x)}{d+c f^2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2} \left (b+a f^2-b x^2\right ) \sqrt{1-\frac{d x^2}{\left (c+\frac{d}{f^2}\right ) f^2}}} \, dx,x,\sqrt{1-f^2 x}\right )}{\sqrt{c+d x}}\\ &=-\frac{2 \sqrt{\frac{f^2 (c+d x)}{d+c f^2}} \Pi \left (\frac{2 b}{b+a f^2};\sin ^{-1}\left (\frac{\sqrt{1-f^2 x}}{\sqrt{2}}\right )|\frac{2 d}{d+c f^2}\right )}{\left (b+a f^2\right ) \sqrt{c+d x}}\\ \end{align*}
Mathematica [C] time = 0.107044, size = 218, normalized size = 2.53 \[ \frac{2 i (c+d x) \sqrt{\frac{d \left (f^2 x-1\right )}{f^2 (c+d x)}} \sqrt{\frac{d \left (f^2 x+1\right )}{f^2 (c+d x)}} \left (\text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-c-\frac{d}{f^2}}}{\sqrt{c+d x}}\right ),\frac{c f^2-d}{c f^2+d}\right )-\Pi \left (\frac{(b c-a d) f^2}{b \left (c f^2+d\right )};i \sinh ^{-1}\left (\frac{\sqrt{-c-\frac{d}{f^2}}}{\sqrt{c+d x}}\right )|\frac{c f^2-d}{c f^2+d}\right )\right )}{\sqrt{1-f^4 x^2} \sqrt{-c-\frac{d}{f^2}} (a d-b c)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.024, size = 205, normalized size = 2.4 \begin{align*} -2\,{\frac{ \left ( c{f}^{2}-d \right ) \sqrt{-{f}^{4}{x}^{2}+1}\sqrt{dx+c}}{ \left ( ad-bc \right ){f}^{2} \left ( d{f}^{4}{x}^{3}+c{f}^{4}{x}^{2}-dx-c \right ) }{\it EllipticPi} \left ( \sqrt{{\frac{{f}^{2} \left ( dx+c \right ) }{c{f}^{2}-d}}},-{\frac{ \left ( c{f}^{2}-d \right ) b}{ \left ( ad-bc \right ){f}^{2}}},\sqrt{{\frac{c{f}^{2}-d}{c{f}^{2}+d}}} \right ) \sqrt{-{\frac{ \left ({f}^{2}x+1 \right ) d}{c{f}^{2}-d}}}\sqrt{-{\frac{ \left ({f}^{2}x-1 \right ) d}{c{f}^{2}+d}}}\sqrt{{\frac{{f}^{2} \left ( dx+c \right ) }{c{f}^{2}-d}}}} \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{1}{\sqrt{-f^{4} x^{2} + 1}{\left (b x + a\right )} \sqrt{d x + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{\sqrt{- \left (f^{2} x - 1\right ) \left (f^{2} x + 1\right )} \left (a + b x\right ) \sqrt{c + d x}}\, 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{1}{\sqrt{-f^{4} x^{2} + 1}{\left (b x + a\right )} \sqrt{d x + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]