Optimal. Leaf size=246 \[ \frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x}}{3 e} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.680304, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304 \[ \frac{4 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^2 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^2 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x}}{3 e} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[b*x + c*x^2]/Sqrt[d + e*x],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 76.5903, size = 221, normalized size = 0.9 \[ \frac{2 \sqrt{d + e x} \sqrt{b x + c x^{2}}}{3 e} - \frac{4 d \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 \sqrt{c} e^{2} \sqrt{d + e x} \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e - 2 c d\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 \sqrt{c} e^{2} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(1/2)/(e*x+d)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 2.60991, size = 226, normalized size = 0.92 \[ \frac{2 \left (-i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-c d) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+(b+c x) (d+e x) (b e-2 c d+c e x)\right )}{3 c e^2 \sqrt{x (b+c x)} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[b*x + c*x^2]/Sqrt[d + e*x],x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.038, size = 467, normalized size = 1.9 \[ -{\frac{2}{3\,x \left ( ce{x}^{2}+bex+cdx+bd \right ){e}^{2}{c}^{2}}\sqrt{x \left ( cx+b \right ) }\sqrt{ex+d} \left ( 2\,{b}^{2}d\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) ec-2\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{2}{d}^{2}+\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{3}{e}^{2}-3\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}cde+2\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{2}{d}^{2}-{x}^{3}{c}^{3}{e}^{2}-{x}^{2}b{c}^{2}{e}^{2}-{x}^{2}{c}^{3}de-xb{c}^{2}de \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^(1/2)/(e*x+d)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x}}{\sqrt{e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)/sqrt(e*x + d),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2} + b x}}{\sqrt{e x + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)/sqrt(e*x + d),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{\sqrt{d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**(1/2)/(e*x+d)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2} + b x}}{\sqrt{e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)/sqrt(e*x + d),x, algorithm="giac")
[Out]