3.407 \(\int \frac{(d+e x)^{5/2}}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=303 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{8 e \sqrt{b x+c x^2} \sqrt{d+e x} (2 c d-b e)}{15 c^2}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2}}{5 c} \]

[Out]

(8*e*(2*c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(15*c^2) + (2*e*(d + e*x)^(3
/2)*Sqrt[b*x + c*x^2])/(5*c) + (2*Sqrt[-b]*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)
*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt
[-b]], (b*e)/(c*d)])/(15*c^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (8*Sqrt[
-b]*d*(c*d - b*e)*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Elli
pticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*c^(5/2)*Sqrt[d + e*x
]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.01518, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 c^{5/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{8 e \sqrt{b x+c x^2} \sqrt{d+e x} (2 c d-b e)}{15 c^2}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2}}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)/Sqrt[b*x + c*x^2],x]

[Out]

(8*e*(2*c*d - b*e)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(15*c^2) + (2*e*(d + e*x)^(3
/2)*Sqrt[b*x + c*x^2])/(5*c) + (2*Sqrt[-b]*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)
*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt
[-b]], (b*e)/(c*d)])/(15*c^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (8*Sqrt[
-b]*d*(c*d - b*e)*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Elli
pticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*c^(5/2)*Sqrt[d + e*x
]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 109.792, size = 279, normalized size = 0.92 \[ \frac{2 e \left (d + e x\right )^{\frac{3}{2}} \sqrt{b x + c x^{2}}}{5 c} - \frac{8 e \sqrt{d + e x} \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{15 c^{2}} - \frac{8 d \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - 2 c d\right ) \left (b e - c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{15 c^{\frac{5}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (8 b^{2} e^{2} - 23 b c d e + 23 c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{15 c^{\frac{5}{2}} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)/(c*x**2+b*x)**(1/2),x)

[Out]

2*e*(d + e*x)**(3/2)*sqrt(b*x + c*x**2)/(5*c) - 8*e*sqrt(d + e*x)*(b*e - 2*c*d)*
sqrt(b*x + c*x**2)/(15*c**2) - 8*d*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(1 + e*x
/d)*(b*e - 2*c*d)*(b*e - c*d)*elliptic_f(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*
d))/(15*c**(5/2)*sqrt(d + e*x)*sqrt(b*x + c*x**2)) + 2*sqrt(x)*sqrt(-b)*sqrt(1 +
c*x/b)*sqrt(d + e*x)*(8*b**2*e**2 - 23*b*c*d*e + 23*c**2*d**2)*elliptic_e(asin(
sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(15*c**(5/2)*sqrt(1 + e*x/d)*sqrt(b*x + c*
x**2))

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Mathematica [C]  time = 1.94261, size = 314, normalized size = 1.04 \[ \frac{2 \sqrt{x} \left (\frac{(b+c x) (d+e x) \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )}{c \sqrt{x}}+i e x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-\frac{i x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (8 b^3 e^3-27 b^2 c d e^2+34 b c^2 d^2 e-15 c^3 d^3\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )}{b}+e \sqrt{x} (b+c x) (d+e x) (-4 b e+11 c d+3 c e x)\right )}{15 c^2 \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)/Sqrt[b*x + c*x^2],x]

[Out]

(2*Sqrt[x]*(((23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*(b + c*x)*(d + e*x))/(c*Sqrt[
x]) + e*Sqrt[x]*(b + c*x)*(d + e*x)*(11*c*d - 4*b*e + 3*c*e*x) + I*Sqrt[b/c]*e*(
23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*Ellip
ticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (I*Sqrt[b/c]*(-15*c^3*d^3 + 34
*b*c^2*d^2*e - 27*b^2*c*d*e^2 + 8*b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x
*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/b))/(15*c^2*Sqrt[x*(b + c
*x)]*Sqrt[d + e*x])

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Maple [B]  time = 0.029, size = 682, normalized size = 2.3 \[ -{\frac{2}{15\,x \left ( ce{x}^{2}+bex+cdx+bd \right ){c}^{4}}\sqrt{ex+d}\sqrt{x \left ( cx+b \right ) } \left ( 4\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{3}cd{e}^{2}-12\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}{c}^{2}{d}^{2}e+8\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{3}{d}^{3}+8\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{4}{e}^{3}-31\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{3}cd{e}^{2}+46\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}{c}^{2}{d}^{2}e-23\,\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) b{c}^{3}{d}^{3}-3\,{x}^{4}{c}^{4}{e}^{3}+{x}^{3}b{c}^{3}{e}^{3}-14\,{x}^{3}{c}^{4}d{e}^{2}+4\,{x}^{2}{b}^{2}{c}^{2}{e}^{3}-10\,{x}^{2}b{c}^{3}d{e}^{2}-11\,{x}^{2}{c}^{4}{d}^{2}e+4\,x{b}^{2}{c}^{2}d{e}^{2}-11\,xb{c}^{3}{d}^{2}e \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(c*x^2+b*x)^(1/2),x)

[Out]

-2/15*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d)
)^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*
d*e^2-12*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF
(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^2*e+8*((c*x+b)/b)^(1/2)*(-(e
*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*
d))^(1/2))*b*c^3*d^3+8*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(
1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*e^3-31*((c*x+b)/b)^(
1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*
e/(b*e-c*d))^(1/2))*b^3*c*d*e^2+46*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^2*
e-23*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^3-3*x^4*c^4*e^3+x^3*b*c^3*e^3-14*x
^3*c^4*d*e^2+4*x^2*b^2*c^2*e^3-10*x^2*b*c^3*d*e^2-11*x^2*c^4*d^2*e+4*x*b^2*c^2*d
*e^2-11*x*b*c^3*d^2*e)/x/(c*e*x^2+b*e*x+c*d*x+b*d)/c^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + b x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/sqrt(c*x^2 + b*x),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(5/2)/sqrt(c*x^2 + b*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{e x + d}}{\sqrt{c x^{2} + b x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/sqrt(c*x^2 + b*x),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{\frac{5}{2}}}{\sqrt{x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(5/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Integral((d + e*x)**(5/2)/sqrt(x*(b + c*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + b x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/sqrt(c*x^2 + b*x),x, algorithm="giac")

[Out]

integrate((e*x + d)^(5/2)/sqrt(c*x^2 + b*x), x)