Optimal. Leaf size=96 \[ \frac{2 \sqrt{a} \sqrt{c+d x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right )|-\frac{a d}{(b c-a d) (1-e)}\right )}{b \sqrt{1-e} \sqrt{\frac{b (c+d x)}{b c-a d}}} \]
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Rubi [A] time = 0.382393, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 \sqrt{a} \sqrt{c+d x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right )|-\frac{a d}{(b c-a d) (1-e)}\right )}{b \sqrt{1-e} \sqrt{\frac{b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x]/(Sqrt[a + b*x]*Sqrt[e + (b*(-1 + e)*x)/a]),x]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(1/2)/(b*x+a)**(1/2)/(e+b*(-1+e)*x/a)**(1/2),x)
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Mathematica [B] time = 1.99785, size = 200, normalized size = 2.08 \[ \frac{2 \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} \left (b \sqrt{a+b x} (c+d x) \sqrt{a-\frac{b c}{d}} \sqrt{\frac{a e+b (e-1) x}{(e-1) (a+b x)}}-(a+b x) (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{a-\frac{b c}{d}}}{\sqrt{a+b x}}\right )|\frac{a d}{(b c-a d) (e-1)}\right )\right )}{b^2 \sqrt{c+d x} \sqrt{a-\frac{b c}{d}} \sqrt{\frac{b (e-1) x}{a}+e}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c + d*x]/(Sqrt[a + b*x]*Sqrt[e + (b*(-1 + e)*x)/a]),x]
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Maple [B] time = 0.047, size = 822, normalized size = 8.6 \[ -2\,{\frac{\sqrt{bx+a}\sqrt{dx+c}}{ \left ( d{x}^{2}b+adx+bcx+ac \right ) \left ( -1+e \right ) ^{2}{b}^{2}d}\sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( -1+e \right ) }{a}}}\sqrt{-{\frac{ \left ( dx+c \right ) b \left ( -1+e \right ) }{ade-bce+bc}}} \left ({\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){a}^{2}{d}^{2}{e}^{2}-2\,{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ) abcd{e}^{2}+{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){b}^{2}{c}^{2}{e}^{2}-{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){a}^{2}{d}^{2}e+3\,{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ) abcde-2\,{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){b}^{2}{c}^{2}e+{\it EllipticE} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){a}^{2}{d}^{2}e-{\it EllipticE} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ) abcde-{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ) abcd+{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){b}^{2}{c}^{2}+{\it EllipticE} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ) abcd \right ){\frac{1}{\sqrt{{\frac{bxe+ae-bx}{a}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(1/2)/(b*x+a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x + c}}{\sqrt{b x + a} \sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(sqrt(b*x + a)*sqrt(b*(e - 1)*x/a + e)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{d x + c}}{\sqrt{b x + a} \sqrt{\frac{a e +{\left (b e - b\right )} x}{a}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(sqrt(b*x + a)*sqrt(b*(e - 1)*x/a + e)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x}}{\sqrt{a + b x} \sqrt{e + \frac{b e x}{a} - \frac{b x}{a}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(1/2)/(b*x+a)**(1/2)/(e+b*(-1+e)*x/a)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x + c}}{\sqrt{b x + a} \sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(sqrt(b*x + a)*sqrt(b*(e - 1)*x/a + e)),x, algorithm="giac")
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