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

Optimal. Leaf size=360 \[ \frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (b^2 e^2-3 c e x (2 c d-b e)-11 b c d e+8 c^2 d^2\right )}{35 c e^3}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 c^{3/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 c^{3/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x}}{7 e} \]

[Out]

(2*Sqrt[d + e*x]*(8*c^2*d^2 - 11*b*c*d*e + b^2*e^2 - 3*c*e*(2*c*d - b*e)*x)*Sqrt
[b*x + c*x^2])/(35*c*e^3) + (2*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2))/(7*e) - (4*Sqr
t[-b]*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - 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)])/(35*c^
(3/2)*e^4*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*d*(c*d - b*e)*(16*c
^2*d^2 - 16*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Ellip
ticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*c^(3/2)*e^4*Sqrt[d +
e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.06279, antiderivative size = 360, 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 \[ \frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (b^2 e^2-3 c e x (2 c d-b e)-11 b c d e+8 c^2 d^2\right )}{35 c e^3}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (-b^2 e^2-16 b c d e+16 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 c^{3/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{35 c^{3/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x}}{7 e} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(8*c^2*d^2 - 11*b*c*d*e + b^2*e^2 - 3*c*e*(2*c*d - b*e)*x)*Sqrt
[b*x + c*x^2])/(35*c*e^3) + (2*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2))/(7*e) - (4*Sqr
t[-b]*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - 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)])/(35*c^
(3/2)*e^4*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*d*(c*d - b*e)*(16*c
^2*d^2 - 16*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Ellip
ticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(35*c^(3/2)*e^4*Sqrt[d +
e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 118.485, size = 338, normalized size = 0.94 \[ \frac{2 \sqrt{d + e x} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{7 e} + \frac{4 \sqrt{d + e x} \sqrt{b x + c x^{2}} \left (\frac{b^{2} e^{2}}{2} - \frac{11 b c d e}{2} + 4 c^{2} d^{2} + \frac{3 c e x \left (b e - 2 c d\right )}{2}\right )}{35 c e^{3}} - \frac{2 \sqrt{x} \left (- d\right )^{\frac{3}{2}} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \left (b^{2} e^{2} + 16 b c d e - 16 c^{2} d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{e} \sqrt{x}}{\sqrt{- d}} \right )}\middle | \frac{c d}{b e}\right )}{35 c e^{\frac{9}{2}} \sqrt{d + e x} \sqrt{b x + c x^{2}}} - \frac{4 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (b e - 2 c d\right ) \left (b^{2} e^{2} + 4 b c d e - 4 c^{2} d^{2}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{35 c^{\frac{3}{2}} e^{4} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/(e*x+d)**(1/2),x)

[Out]

2*sqrt(d + e*x)*(b*x + c*x**2)**(3/2)/(7*e) + 4*sqrt(d + e*x)*sqrt(b*x + c*x**2)
*(b**2*e**2/2 - 11*b*c*d*e/2 + 4*c**2*d**2 + 3*c*e*x*(b*e - 2*c*d)/2)/(35*c*e**3
) - 2*sqrt(x)*(-d)**(3/2)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*(b*e - c*d)*(b**2*e**2
+ 16*b*c*d*e - 16*c**2*d**2)*elliptic_f(asin(sqrt(e)*sqrt(x)/sqrt(-d)), c*d/(b*
e))/(35*c*e**(9/2)*sqrt(d + e*x)*sqrt(b*x + c*x**2)) - 4*sqrt(x)*sqrt(-b)*sqrt(1
+ c*x/b)*sqrt(d + e*x)*(b*e - 2*c*d)*(b**2*e**2 + 4*b*c*d*e - 4*c**2*d**2)*elli
ptic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(35*c**(3/2)*e**4*sqrt(1 + e*x
/d)*sqrt(b*x + c*x**2))

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Mathematica [C]  time = 3.48495, size = 380, normalized size = 1.06 \[ \frac{2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (b^2 e^2+b c e (8 e x-11 d)+c^2 \left (8 d^2-6 d e x+5 e^2 x^2\right )\right )+\sqrt{\frac{b}{c}} \left (i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 b^3 e^3+3 b^2 c d e^2-13 b c^2 d^2 e+8 c^3 d^3\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (b^3 e^3+2 b^2 c d e^2-12 b c^2 d^2 e+8 c^3 d^3\right )\right )\right )}{35 b c e^4 x^2 (b+c x)^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(d + e*x)*(b^2*e^2 + b*c*e*(-11*d + 8*e*
x) + c^2*(8*d^2 - 6*d*e*x + 5*e^2*x^2)) + Sqrt[b/c]*(-2*Sqrt[b/c]*(8*c^3*d^3 - 1
2*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*(b + c*x)*(d + e*x) - (2*I)*b*e*(8*c^3*
d^3 - 12*b*c^2*d^2*e + 2*b^2*c*d*e^2 + b^3*e^3)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*
x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*b*e*(8*c^3*
d^3 - 13*b*c^2*d^2*e + 3*b^2*c*d*e^2 + 2*b^3*e^3)*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)])))/(35*b*c*e^
4*x^2*(b + c*x)^2*Sqrt[d + e*x])

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Maple [B]  time = 0.026, size = 918, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/(e*x+d)^(1/2),x)

[Out]

2/35*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(5*x^5*c^5*e^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^4*c*d*e^3+15*((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^2*d^2*e^2-32*((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^3*d^3*e+16*((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^4
*d^4+2*((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^5*e^4+2*((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*c*d*e^3-28*((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^2*d^2*e^2+40*((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^3*d^3*e-16*((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^4*d
^4+13*x^4*b*c^4*e^4-x^4*c^5*d*e^3+9*x^3*b^2*c^3*e^4-4*x^3*b*c^4*d*e^3+2*x^3*c^5*
d^2*e^2+x^2*b^3*c^2*e^4-2*x^2*b^2*c^3*d*e^3-9*x^2*b*c^4*d^2*e^2+8*x^2*c^5*d^3*e+
x*b^3*c^2*d*e^3-11*x*b^2*c^3*d^2*e^2+8*x*b*c^4*d^3*e)/c^3/e^4/x/(c*e*x^2+b*e*x+c
*d*x+b*d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(3/2)/(e*x+d)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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