Optimal. Leaf size=254 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (3 A c e-2 b B e+B c d) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 c^{3/2} e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{-b} B d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 c^{3/2} e \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 B \sqrt{b x+c x^2} \sqrt{d+e x}}{3 c} \]
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Rubi [A] time = 0.800953, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (3 A c e-2 b B e+B c d) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 c^{3/2} e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{-b} B d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 c^{3/2} e \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 B \sqrt{b x+c x^2} \sqrt{d+e x}}{3 c} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[d + e*x])/Sqrt[b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 82.0571, size = 231, normalized size = 0.91 \[ \frac{2 B \sqrt{d + e x} \sqrt{b x + c x^{2}}}{3 c} + \frac{2 B d \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 c^{\frac{3}{2}} e \sqrt{d + e x} \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (3 A c e - 2 B b e + B c d\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 c^{\frac{3}{2}} e \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(1/2)/(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [C] time = 1.6635, size = 263, normalized size = 1.04 \[ \frac{2 x \left (\frac{(b+c x) (d+e x) (3 A c e-2 b B e+B c d)}{c e x}+\frac{i \sqrt{x} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (2 b B-3 A c) (b e-c d) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )}{b}+i \sqrt{x} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (3 A c e-2 b B e+B c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+B (b+c x) (d+e x)\right )}{3 c \sqrt{x (b+c x)} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[d + e*x])/Sqrt[b*x + c*x^2],x]
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Maple [B] time = 0.03, size = 629, normalized size = 2.5 \[ -{\frac{2}{3\,x \left ( ce{x}^{2}+bex+cdx+bd \right ){c}^{3}e}\sqrt{ex+d}\sqrt{x \left ( cx+b \right ) } \left ( 3\,A{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}c{e}^{2}\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}-3\,A{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{2}de\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}-B{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}cde\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}+B{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{2}{d}^{2}\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}-2\,B{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{3}{e}^{2}\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}+3\,B{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}cde\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}-B{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{2}{d}^{2}\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{be-cd}}}\sqrt{-{\frac{cx}{b}}}-B{x}^{3}{c}^{3}{e}^{2}-B{x}^{2}b{c}^{2}{e}^{2}-B{x}^{2}{c}^{3}de-Bxb{c}^{2}de \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \sqrt{e x + d}}{\sqrt{c x^{2} + b x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/sqrt(c*x^2 + b*x),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )} \sqrt{e x + d}}{\sqrt{c x^{2} + b x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/sqrt(c*x^2 + b*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \sqrt{d + e x}}{\sqrt{x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**(1/2)/(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \sqrt{e x + d}}{\sqrt{c x^{2} + b x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/sqrt(c*x^2 + b*x),x, algorithm="giac")
[Out]