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

Optimal. Leaf size=521 \[ \frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{231 c^2 e}-\frac{16 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{1155 c^3 e^3}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 e \left (b x+c x^2\right )^{5/2} \sqrt{d+e x}}{11 c} \]

[Out]

(2*Sqrt[d + e*x]*(8*c^4*d^4 - 19*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 19*b^3*c*d*e^
3 + 8*b^4*e^4 - 3*c*e*(2*c*d - b*e)*(c^2*d^2 - b*c*d*e + 8*b^2*e^2)*x)*Sqrt[b*x
+ c*x^2])/(1155*c^3*e^3) + (2*Sqrt[d + e*x]*(c^2*d^2 + 13*b*c*d*e - 6*b^2*e^2 +
14*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(231*c^2*e) + (2*e*Sqrt[d + e*x]*(b
*x + c*x^2)^(5/2))/(11*c) - (16*Sqrt[-b]*(c*d - 2*b*e)*(2*c*d - b*e)*(c*d + b*e)
*(c^2*d^2 - 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)])/(1155*c^(7/2)*e^4*Sqrt[1 + (e
*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*d*(c*d - b*e)*(16*c^4*d^4 - 32*b*c^3*d^3
*e + 3*b^2*c^2*d^2*e^2 + 13*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*S
qrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(11
55*c^(7/2)*e^4*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.87641, antiderivative size = 521, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ \frac{2 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} \left (-6 b^2 e^2+14 c e x (2 c d-b e)+13 b c d e+c^2 d^2\right )}{231 c^2 e}-\frac{16 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} (c d-2 b e) (2 c d-b e) (b e+c d) \left (b^2 e^2-b c d e+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (8 b^4 e^4-19 b^3 c d e^3-3 c e x (2 c d-b e) \left (8 b^2 e^2-b c d e+c^2 d^2\right )+6 b^2 c^2 d^2 e^2-19 b c^3 d^3 e+8 c^4 d^4\right )}{1155 c^3 e^3}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (-8 b^4 e^4+13 b^3 c d e^3+3 b^2 c^2 d^2 e^2-32 b c^3 d^3 e+16 c^4 d^4\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{1155 c^{7/2} e^4 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 e \left (b x+c x^2\right )^{5/2} \sqrt{d+e x}}{11 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(8*c^4*d^4 - 19*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 19*b^3*c*d*e^
3 + 8*b^4*e^4 - 3*c*e*(2*c*d - b*e)*(c^2*d^2 - b*c*d*e + 8*b^2*e^2)*x)*Sqrt[b*x
+ c*x^2])/(1155*c^3*e^3) + (2*Sqrt[d + e*x]*(c^2*d^2 + 13*b*c*d*e - 6*b^2*e^2 +
14*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(231*c^2*e) + (2*e*Sqrt[d + e*x]*(b
*x + c*x^2)^(5/2))/(11*c) - (16*Sqrt[-b]*(c*d - 2*b*e)*(2*c*d - b*e)*(c*d + b*e)
*(c^2*d^2 - 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)])/(1155*c^(7/2)*e^4*Sqrt[1 + (e
*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*d*(c*d - b*e)*(16*c^4*d^4 - 32*b*c^3*d^3
*e + 3*b^2*c^2*d^2*e^2 + 13*b^3*c*d*e^3 - 8*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x)/b]*S
qrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(11
55*c^(7/2)*e^4*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 5.16485, size = 559, normalized size = 1.07 \[ \frac{2 (x (b+c x))^{3/2} \left (b e x (b+c x) (d+e x) \left (8 b^4 e^4-b^3 c e^3 (19 d+6 e x)+b^2 c^2 e^2 \left (6 d^2+14 d e x+5 e^2 x^2\right )+b c^3 e \left (-19 d^3+14 d^2 e x+205 d e^2 x^2+140 e^3 x^3\right )+c^4 \left (8 d^4-6 d^3 e x+5 d^2 e^2 x^2+140 d e^3 x^3+105 e^4 x^4\right )\right )+\sqrt{\frac{b}{c}} \left (i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (16 b^5 e^5-48 b^4 c d e^4+35 b^3 c^2 d^2 e^3+10 b^2 c^3 d^3 e^2-21 b c^4 d^4 e+8 c^5 d^5\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-8 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (2 b^5 e^5-5 b^4 c d e^4+2 b^3 c^2 d^2 e^3+2 b^2 c^3 d^3 e^2-5 b c^4 d^4 e+2 c^5 d^5\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-8 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (2 b^5 e^5-5 b^4 c d e^4+2 b^3 c^2 d^2 e^3+2 b^2 c^3 d^3 e^2-5 b c^4 d^4 e+2 c^5 d^5\right )\right )\right )}{1155 b c^3 e^4 x^2 (b+c x)^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(3/2)*(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)*(8*b^4*e^4 - b^3*c*e^3*(19*d +
6*e*x) + b^2*c^2*e^2*(6*d^2 + 14*d*e*x + 5*e^2*x^2) + b*c^3*e*(-19*d^3 + 14*d^2
*e*x + 205*d*e^2*x^2 + 140*e^3*x^3) + c^4*(8*d^4 - 6*d^3*e*x + 5*d^2*e^2*x^2 + 1
40*d*e^3*x^3 + 105*e^4*x^4)) + Sqrt[b/c]*(-8*Sqrt[b/c]*(2*c^5*d^5 - 5*b*c^4*d^4*
e + 2*b^2*c^3*d^3*e^2 + 2*b^3*c^2*d^2*e^3 - 5*b^4*c*d*e^4 + 2*b^5*e^5)*(b + c*x)
*(d + e*x) - (8*I)*b*e*(2*c^5*d^5 - 5*b*c^4*d^4*e + 2*b^2*c^3*d^3*e^2 + 2*b^3*c^
2*d^2*e^3 - 5*b^4*c*d*e^4 + 2*b^5*e^5)*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^5*d^5 - 21*
b*c^4*d^4*e + 10*b^2*c^3*d^3*e^2 + 35*b^3*c^2*d^2*e^3 - 48*b^4*c*d*e^4 + 16*b^5*
e^5)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/S
qrt[x]], (c*d)/(b*e)])))/(1155*b*c^3*e^4*x^2*(b + c*x)^2*Sqrt[d + e*x])

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Maple [B]  time = 0.058, size = 1359, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1155*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(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^7*
e^6+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^6*d^6-16*((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^6*d^6-21*((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^5*c^2*d^2*e^4+590*x^5*b*c^
6*d*e^5-19*x*b^4*c^3*d^2*e^4+6*x*b^3*c^4*d^3*e^3-19*x*b^2*c^5*d^4*e^2+8*x*b*c^6*
d^5*e+56*((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^2*d^2*e^4-48*((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^5*d^5*e+35*((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^4*d^4*e^2+
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^6*c*d*e^5-56*((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^4*d^4*e^2+10*((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^3*d^3*e^3+56*((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^5*d^5*e-56*((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
*c*d*e^5+105*x^7*c^7*e^6+364*x^4*b^2*c^5*d*e^5+364*x^4*b*c^6*d^2*e^4-6*x^3*b^3*c
^4*d*e^5+239*x^3*b^2*c^5*d^2*e^4-6*x^3*b*c^6*d^3*e^3-17*x^2*b^4*c^3*d*e^5+x^2*b^
3*c^4*d^2*e^4+x^2*b^2*c^5*d^3*e^3-17*x^2*b*c^6*d^4*e^2+8*x*b^5*c^2*d*e^5+245*x^6
*b*c^6*e^6+245*x^6*c^7*d*e^5+145*x^5*b^2*c^5*e^6+145*x^5*c^7*d^2*e^4-x^4*b^3*c^4
*e^6-x^4*c^7*d^3*e^3+2*x^3*b^4*c^3*e^6+2*x^3*c^7*d^4*e^2+8*x^2*b^5*c^2*e^6+8*x^2
*c^7*d^5*e)/c^5/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. \[ \int{\left (c x^{2} + b x\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.864291, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done