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

Optimal. Leaf size=666 \[ \frac{10 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} \left (-18 b^3 e^3-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right )}{9009 c^2 e^3}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (24 b^4 e^4+49 b^3 c d e^3+79 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (24 b^5 e^5-17 b^4 c d e^4-22 b^3 c^2 d^2 e^3+303 b^2 c^3 d^3 e^2-3 c e x \left (-24 b^4 e^4+11 b^3 c d e^3+21 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )-368 b c^4 d^4 e+128 c^5 d^5\right )}{9009 c^3 e^5}-\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac{10 \left (b x+c x^2\right )^{5/2} \sqrt{d+e x} (2 c d-b e)}{143 c e} \]

[Out]

(2*Sqrt[d + e*x]*(128*c^5*d^5 - 368*b*c^4*d^4*e + 303*b^2*c^3*d^3*e^2 - 22*b^3*c
^2*d^2*e^3 - 17*b^4*c*d*e^4 + 24*b^5*e^5 - 3*c*e*(32*c^4*d^4 - 64*b*c^3*d^3*e +
21*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3 - 24*b^4*e^4)*x)*Sqrt[b*x + c*x^2])/(9009*c^
3*e^5) + (10*Sqrt[d + e*x]*(16*c^3*d^3 - 31*b*c^2*d^2*e + 9*b^2*c*d*e^2 - 18*b^3
*e^3 - 14*c*e*(c^2*d^2 - b*c*d*e + 3*b^2*e^2)*x)*(b*x + c*x^2)^(3/2))/(9009*c^2*
e^3) - (10*(2*c*d - b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^(5/2))/(143*c*e) + (2*(d +
e*x)^(3/2)*(b*x + c*x^2)^(5/2))/(13*e) - (4*Sqrt[-b]*(128*c^6*d^6 - 384*b*c^5*d^
5*e + 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^3 - 21*b^4*c^2*d^2*e^4 - 20*b^5*c*d
*e^5 + 24*b^6*e^6)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqr
t[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(9009*c^(7/2)*e^6*Sqrt[1 + (e*x)/d]*Sqrt[
b*x + c*x^2]) + (2*Sqrt[-b]*d*(c*d - b*e)*(2*c*d - b*e)*(128*c^4*d^4 - 256*b*c^3
*d^3*e + 79*b^2*c^2*d^2*e^2 + 49*b^3*c*d*e^3 + 24*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x
)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)
])/(9009*c^(7/2)*e^6*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 2.17609, antiderivative size = 666, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ \frac{10 \left (b x+c x^2\right )^{3/2} \sqrt{d+e x} \left (-18 b^3 e^3-14 c e x \left (3 b^2 e^2-b c d e+c^2 d^2\right )+9 b^2 c d e^2-31 b c^2 d^2 e+16 c^3 d^3\right )}{9009 c^2 e^3}+\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) (2 c d-b e) \left (24 b^4 e^4+49 b^3 c d e^3+79 b^2 c^2 d^2 e^2-256 b c^3 d^3 e+128 c^4 d^4\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (24 b^5 e^5-17 b^4 c d e^4-22 b^3 c^2 d^2 e^3+303 b^2 c^3 d^3 e^2-3 c e x \left (-24 b^4 e^4+11 b^3 c d e^3+21 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )-368 b c^4 d^4 e+128 c^5 d^5\right )}{9009 c^3 e^5}-\frac{4 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{9009 c^{7/2} e^6 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \left (b x+c x^2\right )^{5/2} (d+e x)^{3/2}}{13 e}-\frac{10 \left (b x+c x^2\right )^{5/2} \sqrt{d+e x} (2 c d-b e)}{143 c e} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(128*c^5*d^5 - 368*b*c^4*d^4*e + 303*b^2*c^3*d^3*e^2 - 22*b^3*c
^2*d^2*e^3 - 17*b^4*c*d*e^4 + 24*b^5*e^5 - 3*c*e*(32*c^4*d^4 - 64*b*c^3*d^3*e +
21*b^2*c^2*d^2*e^2 + 11*b^3*c*d*e^3 - 24*b^4*e^4)*x)*Sqrt[b*x + c*x^2])/(9009*c^
3*e^5) + (10*Sqrt[d + e*x]*(16*c^3*d^3 - 31*b*c^2*d^2*e + 9*b^2*c*d*e^2 - 18*b^3
*e^3 - 14*c*e*(c^2*d^2 - b*c*d*e + 3*b^2*e^2)*x)*(b*x + c*x^2)^(3/2))/(9009*c^2*
e^3) - (10*(2*c*d - b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^(5/2))/(143*c*e) + (2*(d +
e*x)^(3/2)*(b*x + c*x^2)^(5/2))/(13*e) - (4*Sqrt[-b]*(128*c^6*d^6 - 384*b*c^5*d^
5*e + 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^3 - 21*b^4*c^2*d^2*e^4 - 20*b^5*c*d
*e^5 + 24*b^6*e^6)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqr
t[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(9009*c^(7/2)*e^6*Sqrt[1 + (e*x)/d]*Sqrt[
b*x + c*x^2]) + (2*Sqrt[-b]*d*(c*d - b*e)*(2*c*d - b*e)*(128*c^4*d^4 - 256*b*c^3
*d^3*e + 79*b^2*c^2*d^2*e^2 + 49*b^3*c*d*e^3 + 24*b^4*e^4)*Sqrt[x]*Sqrt[1 + (c*x
)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)
])/(9009*c^(7/2)*e^6*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((c*x**2+b*x)**(5/2)*(e*x+d)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 5.8778, size = 663, normalized size = 1. \[ \frac{2 (x (b+c x))^{5/2} \left (b e x (b+c x) (d+e x) \left (24 b^5 e^5-b^4 c e^4 (17 d+18 e x)+b^3 c^2 e^3 \left (-22 d^2+12 d e x+15 e^2 x^2\right )+b^2 c^3 e^2 \left (303 d^3-218 d^2 e x+178 d e^2 x^2+1113 e^3 x^3\right )+b c^4 e \left (-368 d^4+272 d^3 e x-225 d^2 e^2 x^2+196 d e^3 x^3+1701 e^4 x^4\right )+c^5 \left (128 d^5-96 d^4 e x+80 d^3 e^2 x^2-70 d^2 e^3 x^3+63 d e^4 x^4+693 e^5 x^5\right )\right )+\sqrt{\frac{b}{c}} \left (i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (48 b^6 e^6-64 b^5 c d e^5-25 b^4 c^2 d^2 e^4-70 b^3 c^3 d^3 e^3+383 b^2 c^4 d^4 e^2-400 b c^5 d^5 e+128 c^6 d^6\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-2 i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-2 \sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (24 b^6 e^6-20 b^5 c d e^5-21 b^4 c^2 d^2 e^4-46 b^3 c^3 d^3 e^3+343 b^2 c^4 d^4 e^2-384 b c^5 d^5 e+128 c^6 d^6\right )\right )\right )}{9009 b c^3 e^6 x^3 (b+c x)^3 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(5/2)*(b*e*x*(b + c*x)*(d + e*x)*(24*b^5*e^5 - b^4*c*e^4*(17*d
+ 18*e*x) + b^3*c^2*e^3*(-22*d^2 + 12*d*e*x + 15*e^2*x^2) + b^2*c^3*e^2*(303*d^3
- 218*d^2*e*x + 178*d*e^2*x^2 + 1113*e^3*x^3) + b*c^4*e*(-368*d^4 + 272*d^3*e*x
- 225*d^2*e^2*x^2 + 196*d*e^3*x^3 + 1701*e^4*x^4) + c^5*(128*d^5 - 96*d^4*e*x +
80*d^3*e^2*x^2 - 70*d^2*e^3*x^3 + 63*d*e^4*x^4 + 693*e^5*x^5)) + Sqrt[b/c]*(-2*
Sqrt[b/c]*(128*c^6*d^6 - 384*b*c^5*d^5*e + 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*
e^3 - 21*b^4*c^2*d^2*e^4 - 20*b^5*c*d*e^5 + 24*b^6*e^6)*(b + c*x)*(d + e*x) - (2
*I)*b*e*(128*c^6*d^6 - 384*b*c^5*d^5*e + 343*b^2*c^4*d^4*e^2 - 46*b^3*c^3*d^3*e^
3 - 21*b^4*c^2*d^2*e^4 - 20*b^5*c*d*e^5 + 24*b^6*e^6)*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*(
128*c^6*d^6 - 400*b*c^5*d^5*e + 383*b^2*c^4*d^4*e^2 - 70*b^3*c^3*d^3*e^3 - 25*b^
4*c^2*d^2*e^4 - 64*b^5*c*d*e^5 + 48*b^6*e^6)*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)])))/(9009*b*c^3*e^6
*x^3*(b + c*x)^3*Sqrt[d + e*x])

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Maple [B]  time = 0.046, size = 1728, 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)^(5/2)*(e*x+d)^(1/2),x)

[Out]

2/9009*(x*(c*x+b))^(1/2)*(e*x+d)^(1/2)*(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^7*
c*d*e^6+3188*x^5*b^2*c^6*d*e^6-36*x^5*b*c^7*d^2*e^5+1318*x^4*b^3*c^5*d*e^6-69*x^
4*b^2*c^6*d^2*e^5+63*x^2*b^3*c^5*d^3*e^4+207*x^2*b^2*c^6*d^4*e^3-336*x^2*b*c^7*d
^5*e^2+24*x*b^6*c^2*d*e^6-17*x*b^5*c^3*d^2*e^5-22*x*b^4*c^4*d^3*e^4+303*x*b^3*c^
5*d^4*e^3-368*x*b^2*c^6*d^5*e^2+128*x*b*c^7*d^6*e+57*x^4*b*c^7*d^3*e^4-8*x^3*b^4
*c^4*d*e^6-50*x^3*b^3*c^5*d^2*e^5+132*x^3*b^2*c^6*d^3*e^4-112*x^3*b*c^7*d^4*e^3-
11*x^2*b^5*c^3*d*e^6-27*x^2*b^4*c^4*d^2*e^5+256*((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^7*d^7-256*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ell
ipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^7*d^7+2653*x^6*b*c^7*d*e^6-2
0*((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^3*d^3*e^4-395*((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^4*d^4*e^3+1054*((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^5*d^5*e^2-89
6*((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^6*d^6*e-88*((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*c*d*e^6-2*((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^6*c^2*d^2*e^5-50*((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^3*d^3*e^4+778*((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^4*d^4*e^3-1454*((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^5*d^5*e^2+1024*((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^6*d^6*e+2814*x^6*b^2*c^6*e^7-7*x^6*c^8*d^2*e^5+11
28*x^5*b^3*c^5*e^7+10*x^5*c^8*d^3*e^4-3*x^4*b^4*c^4*e^7-16*x^4*c^8*d^4*e^3+6*x^3
*b^5*c^3*e^7+32*x^3*c^8*d^5*e^2+24*x^2*b^6*c^2*e^7+128*x^2*c^8*d^6*e+2394*x^7*b*
c^7*e^7+756*x^7*c^8*d*e^6+693*x^8*c^8*e^7+48*((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^
8*e^7-23*((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^2*d^2*e^5)/c^5/x/(c*e*x^2+b*e*x+
c*d*x+b*d)/e^6

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{e x + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + b^2*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. \[ \int \left (x \left (b + c x\right )\right )^{\frac{5}{2}} \sqrt{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.868137, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done