3.1266 \(\int \frac{(A+B x) \sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=254 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (3 A c e-2 b B e+B c d) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 c^{3/2} e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{-b} B d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 c^{3/2} e \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 B \sqrt{b x+c x^2} \sqrt{d+e x}}{3 c} \]

[Out]

(2*B*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*c) + (2*Sqrt[-b]*(B*c*d - 2*b*B*e + 3*A
*c*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])
/Sqrt[-b]], (b*e)/(c*d)])/(3*c^(3/2)*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2
*Sqrt[-b]*B*d*(c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[
ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*c^(3/2)*e*Sqrt[d + e*x]*Sqr
t[b*x + c*x^2])

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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]

(2*B*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*c) + (2*Sqrt[-b]*(B*c*d - 2*b*B*e + 3*A
*c*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])
/Sqrt[-b]], (b*e)/(c*d)])/(3*c^(3/2)*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2
*Sqrt[-b]*B*d*(c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[
ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*c^(3/2)*e*Sqrt[d + e*x]*Sqr
t[b*x + c*x^2])

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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]

2*B*sqrt(d + e*x)*sqrt(b*x + c*x**2)/(3*c) + 2*B*d*sqrt(x)*sqrt(-b)*sqrt(1 + c*x
/b)*sqrt(1 + e*x/d)*(b*e - c*d)*elliptic_f(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(
c*d))/(3*c**(3/2)*e*sqrt(d + e*x)*sqrt(b*x + c*x**2)) + 2*sqrt(x)*sqrt(-b)*sqrt(
1 + c*x/b)*sqrt(d + e*x)*(3*A*c*e - 2*B*b*e + B*c*d)*elliptic_e(asin(sqrt(c)*sqr
t(x)/sqrt(-b)), b*e/(c*d))/(3*c**(3/2)*e*sqrt(1 + e*x/d)*sqrt(b*x + c*x**2))

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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]

[Out]

(2*x*(B*(b + c*x)*(d + e*x) + ((B*c*d - 2*b*B*e + 3*A*c*e)*(b + c*x)*(d + e*x))/
(c*e*x) + I*Sqrt[b/c]*(B*c*d - 2*b*B*e + 3*A*c*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(
e*x)]*Sqrt[x]*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + (I*Sqrt[b/c
]*(2*b*B - 3*A*c)*(-(c*d) + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*Sqrt[x]*Ell
ipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/b))/(3*c*Sqrt[x*(b + c*x)]*Sq
rt[d + e*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]

-2/3*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(3*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-
c*d))^(1/2))*b^2*c*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(
1/2)-3*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^2*d*e*((c*x+b)/b
)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-B*EllipticF(((c*x+b)/b)^(1/2
),(b*e/(b*e-c*d))^(1/2))*b^2*c*d*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)+B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^2*d^2*
((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-2*B*EllipticE(((c*
x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-
c*d))^(1/2)*(-c*x/b)^(1/2)+3*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2)
)*b^2*c*d*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-B*Elli
pticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^2*d^2*((c*x+b)/b)^(1/2)*(-(e*
x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-B*x^3*c^3*e^2-B*x^2*b*c^2*e^2-B*x^2*c^3*d
*e-B*x*b*c^2*d*e)/x/(c*e*x^2+b*e*x+c*d*x+b*d)/c^3/e

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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]

integrate((B*x + A)*sqrt(e*x + d)/sqrt(c*x^2 + b*x), x)

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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]

integral((B*x + A)*sqrt(e*x + d)/sqrt(c*x^2 + b*x), x)

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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]

Integral((A + B*x)*sqrt(d + e*x)/sqrt(x*(b + c*x)), x)

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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]

integrate((B*x + A)*sqrt(e*x + d)/sqrt(c*x^2 + b*x), x)