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

Optimal. Leaf size=457 \[ \frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-4 b^3 e^3-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-15 b c^2 d^2 e+8 c^3 d^3\right )}{315 c^2 e^3}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (2 c d-b e)}{21 c e} \]

[Out]

(2*Sqrt[d + e*x]*(8*c^3*d^3 - 15*b*c^2*d^2*e + 3*b^2*c*d*e^2 - 4*b^3*e^3 - 6*c*e
*(c^2*d^2 - b*c*d*e + 2*b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(315*c^2*e^3) - (2*(2*c*d
- b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2))/(21*c*e) + (2*(d + e*x)^(3/2)*(b*x +
c*x^2)^(3/2))/(9*e) - (2*Sqrt[-b]*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e
^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*Elliptic
E[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(315*c^(5/2)*e^4*Sqrt[1 + (e
*x)/d]*Sqrt[b*x + c*x^2]) + (8*Sqrt[-b]*d*(c*d - b*e)*(2*c*d - b*e)*(2*c^2*d^2 -
2*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcS
in[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(315*c^(5/2)*e^4*Sqrt[d + e*x]*Sqr
t[b*x + c*x^2])

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Rubi [A]  time = 1.55323, antiderivative size = 457, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ \frac{8 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (-b^2 e^2-2 b c d e+2 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (-4 b^3 e^3-6 c e x \left (2 b^2 e^2-b c d e+c^2 d^2\right )+3 b^2 c d e^2-15 b c^2 d^2 e+8 c^3 d^3\right )}{315 c^2 e^3}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{315 c^{5/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{3/2} (d+e x)^{3/2}}{9 e}-\frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} (2 c d-b e)}{21 c e} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(8*c^3*d^3 - 15*b*c^2*d^2*e + 3*b^2*c*d*e^2 - 4*b^3*e^3 - 6*c*e
*(c^2*d^2 - b*c*d*e + 2*b^2*e^2)*x)*Sqrt[b*x + c*x^2])/(315*c^2*e^3) - (2*(2*c*d
- b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^(3/2))/(21*c*e) + (2*(d + e*x)^(3/2)*(b*x +
c*x^2)^(3/2))/(9*e) - (2*Sqrt[-b]*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e
^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*Elliptic
E[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(315*c^(5/2)*e^4*Sqrt[1 + (e
*x)/d]*Sqrt[b*x + c*x^2]) + (8*Sqrt[-b]*d*(c*d - b*e)*(2*c*d - b*e)*(2*c^2*d^2 -
2*b*c*d*e - b^2*e^2)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcS
in[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(315*c^(5/2)*e^4*Sqrt[d + e*x]*Sqr
t[b*x + c*x^2])

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Rubi in Sympy [A]  time = 165.823, size = 440, normalized size = 0.96 \[ \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (b x + c x^{2}\right )^{\frac{3}{2}}}{9 e} + \frac{2 \sqrt{d + e x} \left (b e - 2 c d\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{21 c e} - \frac{8 \sqrt{d + e x} \sqrt{b x + c x^{2}} \left (b^{3} e^{3} - \frac{3 b^{2} c d e^{2}}{4} + \frac{15 b c^{2} d^{2} e}{4} - 2 c^{3} d^{3} + \frac{3 c e x \left (2 b^{2} e^{2} - b c d e + c^{2} d^{2}\right )}{2}\right )}{315 c^{2} e^{3}} - \frac{8 d \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (b e - 2 c d\right ) \left (b e - c d\right ) \left (b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{315 c^{\frac{5}{2}} e^{4} \sqrt{d + e x} \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (8 b^{4} e^{4} - 7 b^{3} c d e^{3} - 9 b^{2} c^{2} d^{2} e^{2} + 32 b c^{3} d^{3} e - 16 c^{4} d^{4}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{315 c^{\frac{5}{2}} e^{4} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

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*(d + e*x)**(3/2)*(b*x + c*x**2)**(3/2)/(9*e) + 2*sqrt(d + e*x)*(b*e - 2*c*d)*(
b*x + c*x**2)**(3/2)/(21*c*e) - 8*sqrt(d + e*x)*sqrt(b*x + c*x**2)*(b**3*e**3 -
3*b**2*c*d*e**2/4 + 15*b*c**2*d**2*e/4 - 2*c**3*d**3 + 3*c*e*x*(2*b**2*e**2 - b*
c*d*e + c**2*d**2)/2)/(315*c**2*e**3) - 8*d*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqr
t(1 + e*x/d)*(b*e - 2*c*d)*(b*e - c*d)*(b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)*ell
iptic_f(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(315*c**(5/2)*e**4*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*
*4*e**4 - 7*b**3*c*d*e**3 - 9*b**2*c**2*d**2*e**2 + 32*b*c**3*d**3*e - 16*c**4*d
**4)*elliptic_e(asin(sqrt(c)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(315*c**(5/2)*e**4*sq
rt(1 + e*x/d)*sqrt(b*x + c*x**2))

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Mathematica [C]  time = 4.13246, size = 463, normalized size = 1.01 \[ \frac{2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (-4 b^3 e^3+3 b^2 c e^2 (d+e x)+b c^2 e \left (-15 d^2+11 d e x+50 e^2 x^2\right )+c^3 \left (8 d^3-6 d^2 e x+5 d e^2 x^2+35 e^3 x^3\right )\right )-\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^4 e^4+11 b^3 c d e^3+6 b^2 c^2 d^2 e^2-17 b c^3 d^3 e+8 c^4 d^4\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (-8 b^4 e^4+7 b^3 c d e^3+9 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right )\right )\right )}{315 b c^2 e^4 x^2 (b+c x)^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(3/2)*(b*e*x*(b + c*x)*(d + e*x)*(-4*b^3*e^3 + 3*b^2*c*e^2*(d +
e*x) + b*c^2*e*(-15*d^2 + 11*d*e*x + 50*e^2*x^2) + c^3*(8*d^3 - 6*d^2*e*x + 5*d
*e^2*x^2 + 35*e^3*x^3)) - Sqrt[b/c]*(Sqrt[b/c]*(16*c^4*d^4 - 32*b*c^3*d^3*e + 9*
b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*(b + c*x)*(d + e*x) + I*b*e*(16*c^4
*d^4 - 32*b*c^3*d^3*e + 9*b^2*c^2*d^2*e^2 + 7*b^3*c*d*e^3 - 8*b^4*e^4)*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^4*d^4 - 17*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3
- 8*b^4*e^4)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sq
rt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(315*b*c^2*e^4*x^2*(b + c*x)^2*Sqrt[d + e*x])

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Maple [B]  time = 0.025, size = 1170, 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]

-2/315*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(-15*((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
*c*d*e^4+4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipti
cF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*c*d*e^4-4*((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^2*d^2*e^3+41*((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^3*d^3*e^2-4
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^2*c^4*d^4*e-8*x^2*c^6*d^4*e-85*x^5*b*c^5*e^
5-40*x^5*c^6*d*e^4-53*x^4*b^2*c^4*e^5+x^4*c^6*d^2*e^3+x^3*b^3*c^3*e^5-2*x^3*c^6*
d^3*e^2+4*x^2*b^4*c^2*e^5-101*x^4*b*c^5*d*e^4+40*((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^4*d^4*e-67*x^3*b^2*c^4*d*e^4+5*x^3*b*c^5*d^2*e^3-2*x^2*b^3*c^3*d*e^4+x^2
*b^2*c^4*d^2*e^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^6*e^5-2*((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^2*d^2*e^3-24*((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^3*d^3*e
^2+13*x^2*b*c^5*d^3*e^2+4*x*b^4*c^2*d*e^4-3*x*b^3*c^3*d^2*e^3+15*x*b^2*c^4*d^3*e
^2-16*((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^5*d^5+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^5*d^5-8*x*b*c^5*d^4*e-35*x^6*c^6*e^5)/c^4/x/(c*e*x^2+b*e*x+c*d*x+b*d)/e
^4

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

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. \[{\rm integral}\left ({\left (c x^{2} + b x\right )}^{\frac{3}{2}} \sqrt{e x + d}, x\right ) \]

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

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. \[ \int{\left (c x^{2} + b x\right )}^{\frac{3}{2}} \sqrt{e x + d}\,{d x} \]

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)