3.396 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=298 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (4 c d-3 b e) (4 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{16 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} (-3 b e+8 c d+2 c e x)}{3 e^3 \sqrt{d+e x}}-\frac{2 \left (b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]

[Out]

(2*(8*c*d - 3*b*e + 2*c*e*x)*Sqrt[b*x + c*x^2])/(3*e^3*Sqrt[d + e*x]) - (2*(b*x
+ c*x^2)^(3/2))/(3*e*(d + e*x)^(3/2)) - (16*Sqrt[-b]*Sqrt[c]*(2*c*d - b*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*e^4*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*(4*c*d
- 3*b*e)*(4*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[Arc
Sin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*Sqrt[c]*e^4*Sqrt[d + e*x]*Sqrt
[b*x + c*x^2])

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Rubi [A]  time = 0.931366, antiderivative size = 298, 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 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (4 c d-3 b e) (4 c d-b e) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 \sqrt{c} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{16 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} (-3 b e+8 c d+2 c e x)}{3 e^3 \sqrt{d+e x}}-\frac{2 \left (b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/(d + e*x)^(5/2),x]

[Out]

(2*(8*c*d - 3*b*e + 2*c*e*x)*Sqrt[b*x + c*x^2])/(3*e^3*Sqrt[d + e*x]) - (2*(b*x
+ c*x^2)^(3/2))/(3*e*(d + e*x)^(3/2)) - (16*Sqrt[-b]*Sqrt[c]*(2*c*d - b*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*e^4*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*(4*c*d
- 3*b*e)*(4*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[Arc
Sin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*Sqrt[c]*e^4*Sqrt[d + e*x]*Sqrt
[b*x + c*x^2])

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Rubi in Sympy [A]  time = 99.1421, size = 272, normalized size = 0.91 \[ \frac{16 \sqrt{c} \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e - 2 c d\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{3 e^{4} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} - \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 e \left (d + e x\right )^{\frac{3}{2}}} - \frac{4 \sqrt{b x + c x^{2}} \left (\frac{3 b e}{2} - 4 c d - c e x\right )}{3 e^{3} \sqrt{d + e x}} + \frac{2 \sqrt{x} \sqrt{- d} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - 4 c d\right ) \left (3 b e - 4 c d\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{3 e^{\frac{9}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/(e*x+d)**(5/2),x)

[Out]

16*sqrt(c)*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(b*e - 2*c*d)*elliptic
_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(3*e**4*sqrt(1 + e*x/d)*sqrt(b*x +
c*x**2)) - 2*(b*x + c*x**2)**(3/2)/(3*e*(d + e*x)**(3/2)) - 4*sqrt(b*x + c*x**2
)*(3*b*e/2 - 4*c*d - c*e*x)/(3*e**3*sqrt(d + e*x)) + 2*sqrt(x)*sqrt(-d)*sqrt(1 +
c*x/b)*sqrt(1 + e*x/d)*(b*e - 4*c*d)*(3*b*e - 4*c*d)*elliptic_f(asin(sqrt(e)*sq
rt(x)/sqrt(-d)), c*d/(b*e))/(3*e**(9/2)*sqrt(d + e*x)*sqrt(b*x + c*x**2))

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Mathematica [C]  time = 1.79612, size = 279, normalized size = 0.94 \[ \frac{2 (x (b+c x))^{3/2} \left (\frac{e x (b+c x) \left (c \left (8 d^2+10 d e x+e^2 x^2\right )-b e (3 d+4 e x)\right )}{d+e x}-i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (5 b e-8 c d) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+8 i c e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-2 c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+8 (b+c x) (d+e x) (b e-2 c d)\right )}{3 e^4 x^2 (b+c x)^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(3/2)/(d + e*x)^(5/2),x]

[Out]

(2*(x*(b + c*x))^(3/2)*(8*(-2*c*d + b*e)*(b + c*x)*(d + e*x) + (e*x*(b + c*x)*(-
(b*e*(3*d + 4*e*x)) + c*(8*d^2 + 10*d*e*x + e^2*x^2)))/(d + e*x) + (8*I)*Sqrt[b/
c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*Ar
cSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*Sqrt[b/c]*c*e*(-8*c*d + 5*b*e)*Sqrt[1
+ b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c
*d)/(b*e)]))/(3*e^4*x^2*(b + c*x)^2*Sqrt[d + e*x])

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Maple [B]  time = 0.038, size = 1051, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/(e*x+d)^(5/2),x)

[Out]

2/3*(x*(c*x+b))^(1/2)*(3*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^
3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-16*EllipticF
(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c*d*e^2*((c*x+b)/b)^(1/2)*(-(e*x
+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+16*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*x*b*c^2*d^2*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b
)^(1/2)-8*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*e^3*((c*x+b)/
b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+24*EllipticE(((c*x+b)/b)^(1
/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c*d*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d)
)^(1/2)*(-c*x/b)^(1/2)-16*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b
*c^2*d^2*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+3*Ellip
ticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*d*e^2*((c*x+b)/b)^(1/2)*(-(e*x
+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-16*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*b^2*c*d^2*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^
(1/2)+16*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^2*d^3*((c*x+b)/b
)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-8*EllipticE(((c*x+b)/b)^(1/2
),(b*e/(b*e-c*d))^(1/2))*b^3*d*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)+24*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c*d^2
*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-16*EllipticE(((
c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^2*d^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(
b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+c^3*x^4*e^3-3*x^3*b*c^2*e^3+10*x^3*c^3*d*e^2-4*x^
2*b^2*c*e^3+7*x^2*b*c^2*d*e^2+8*x^2*c^3*d^2*e-3*b^2*c*d*e^2*x+8*b*c^2*d^2*e*x)/(
c*x+b)/x/(e*x+d)^(3/2)/c/e^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x)^(3/2)/((e^2*x^2 + 2*d*e*x + d^2)*sqrt(e*x + d)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(3/2)/(e*x+d)**(5/2),x)

[Out]

Integral((x*(b + c*x))**(3/2)/(d + e*x)**(5/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(3/2)/(e*x + d)^(5/2), x)