3.2242 \(\int \sqrt{d+e x} (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=267 \[ -\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{3465 c^4 e^2 (d+e x)^{5/2}}-\frac{8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{693 c^3 e^2 (d+e x)^{3/2}}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{99 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2} \]

[Out]

(-16*(2*c*d - b*e)^2*(11*c*e*f + c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(5/2))/(3465*c^4*e^2*(d + e*x)^(5/2)) - (8*(2*c*d - b*e)*(11*c*e*f + c*d
*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(693*c^3*e^2*(d + e*x
)^(3/2)) - (2*(11*c*e*f + c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(5/2))/(99*c^2*e^2*Sqrt[d + e*x]) - (2*g*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x
- c*e^2*x^2)^(5/2))/(11*c*e^2)

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Rubi [A]  time = 0.951657, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{3465 c^4 e^2 (d+e x)^{5/2}}-\frac{8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{693 c^3 e^2 (d+e x)^{3/2}}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-6 b e g+c d g+11 c e f)}{99 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{11 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

(-16*(2*c*d - b*e)^2*(11*c*e*f + c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(5/2))/(3465*c^4*e^2*(d + e*x)^(5/2)) - (8*(2*c*d - b*e)*(11*c*e*f + c*d
*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(693*c^3*e^2*(d + e*x
)^(3/2)) - (2*(11*c*e*f + c*d*g - 6*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(5/2))/(99*c^2*e^2*Sqrt[d + e*x]) - (2*g*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x
- c*e^2*x^2)^(5/2))/(11*c*e^2)

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Rubi in Sympy [A]  time = 89.7227, size = 257, normalized size = 0.96 \[ - \frac{2 g \sqrt{d + e x} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{11 c e^{2}} + \frac{2 \left (6 b e g - c d g - 11 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{99 c^{2} e^{2} \sqrt{d + e x}} - \frac{8 \left (b e - 2 c d\right ) \left (6 b e g - c d g - 11 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{693 c^{3} e^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{16 \left (b e - 2 c d\right )^{2} \left (6 b e g - c d g - 11 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{3465 c^{4} e^{2} \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*g*sqrt(d + e*x)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(11*c*e**2)
+ 2*(6*b*e*g - c*d*g - 11*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5
/2)/(99*c**2*e**2*sqrt(d + e*x)) - 8*(b*e - 2*c*d)*(6*b*e*g - c*d*g - 11*c*e*f)*
(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(693*c**3*e**2*(d + e*x)**(3/2
)) + 16*(b*e - 2*c*d)**2*(6*b*e*g - c*d*g - 11*c*e*f)*(-b*e**2*x - c*e**2*x**2 +
d*(-b*e + c*d))**(5/2)/(3465*c**4*e**2*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.348874, size = 183, normalized size = 0.69 \[ -\frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (-48 b^3 e^3 g+8 b^2 c e^2 (40 d g+11 e f+15 e g x)-2 b c^2 e \left (347 d^2 g+d e (286 f+340 g x)+5 e^2 x (22 f+21 g x)\right )+c^3 \left (422 d^3 g+d^2 e (1177 f+1055 g x)+10 d e^2 x (121 f+98 g x)+35 e^3 x^2 (11 f+9 g x)\right )\right )}{3465 c^4 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-48*b^3*e^3
*g + 8*b^2*c*e^2*(11*e*f + 40*d*g + 15*e*g*x) - 2*b*c^2*e*(347*d^2*g + 5*e^2*x*(
22*f + 21*g*x) + d*e*(286*f + 340*g*x)) + c^3*(422*d^3*g + 35*e^3*x^2*(11*f + 9*
g*x) + 10*d*e^2*x*(121*f + 98*g*x) + d^2*e*(1177*f + 1055*g*x))))/(3465*c^4*e^2*
Sqrt[d + e*x])

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Maple [A]  time = 0.013, size = 235, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -315\,{e}^{3}g{x}^{3}{c}^{3}+210\,b{c}^{2}{e}^{3}g{x}^{2}-980\,{c}^{3}d{e}^{2}g{x}^{2}-385\,{c}^{3}{e}^{3}f{x}^{2}-120\,{b}^{2}c{e}^{3}gx+680\,b{c}^{2}d{e}^{2}gx+220\,b{c}^{2}{e}^{3}fx-1055\,{c}^{3}{d}^{2}egx-1210\,{c}^{3}d{e}^{2}fx+48\,{b}^{3}{e}^{3}g-320\,{b}^{2}cd{e}^{2}g-88\,{b}^{2}c{e}^{3}f+694\,b{c}^{2}{d}^{2}eg+572\,b{c}^{2}d{e}^{2}f-422\,{c}^{3}{d}^{3}g-1177\,f{d}^{2}{c}^{3}e \right ) }{3465\,{c}^{4}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3465*(c*e*x+b*e-c*d)*(-315*c^3*e^3*g*x^3+210*b*c^2*e^3*g*x^2-980*c^3*d*e^2*g*
x^2-385*c^3*e^3*f*x^2-120*b^2*c*e^3*g*x+680*b*c^2*d*e^2*g*x+220*b*c^2*e^3*f*x-10
55*c^3*d^2*e*g*x-1210*c^3*d*e^2*f*x+48*b^3*e^3*g-320*b^2*c*d*e^2*g-88*b^2*c*e^3*
f+694*b*c^2*d^2*e*g+572*b*c^2*d*e^2*f-422*c^3*d^3*g-1177*c^3*d^2*e*f)*(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c^4/e^2/(e*x+d)^(3/2)

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Maxima [A]  time = 0.750495, size = 678, normalized size = 2.54 \[ -\frac{2 \,{\left (35 \, c^{4} e^{4} x^{4} + 107 \, c^{4} d^{4} - 266 \, b c^{3} d^{3} e + 219 \, b^{2} c^{2} d^{2} e^{2} - 68 \, b^{3} c d e^{3} + 8 \, b^{4} e^{4} + 10 \,{\left (4 \, c^{4} d e^{3} + 5 \, b c^{3} e^{4}\right )} x^{3} - 3 \,{\left (26 \, c^{4} d^{2} e^{2} - 46 \, b c^{3} d e^{3} - b^{2} c^{2} e^{4}\right )} x^{2} - 2 \,{\left (52 \, c^{4} d^{3} e - 39 \, b c^{3} d^{2} e^{2} - 15 \, b^{2} c^{2} d e^{3} + 2 \, b^{3} c e^{4}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} f}{315 \,{\left (c^{3} e^{2} x + c^{3} d e\right )}} - \frac{2 \,{\left (315 \, c^{5} e^{5} x^{5} + 422 \, c^{5} d^{5} - 1538 \, b c^{4} d^{4} e + 2130 \, b^{2} c^{3} d^{3} e^{2} - 1382 \, b^{3} c^{2} d^{2} e^{3} + 416 \, b^{4} c d e^{4} - 48 \, b^{5} e^{5} + 70 \,{\left (5 \, c^{5} d e^{4} + 6 \, b c^{4} e^{5}\right )} x^{4} - 5 \,{\left (118 \, c^{5} d^{2} e^{3} - 214 \, b c^{4} d e^{4} - 3 \, b^{2} c^{3} e^{5}\right )} x^{3} - 6 \,{\left (118 \, c^{5} d^{3} e^{2} - 101 \, b c^{4} d^{2} e^{3} - 20 \, b^{2} c^{3} d e^{4} + 3 \, b^{3} c^{2} e^{5}\right )} x^{2} +{\left (211 \, c^{5} d^{4} e - 558 \, b c^{4} d^{3} e^{2} + 507 \, b^{2} c^{3} d^{2} e^{3} - 184 \, b^{3} c^{2} d e^{4} + 24 \, b^{4} c e^{5}\right )} x\right )} \sqrt{-c e x + c d - b e}{\left (e x + d\right )} g}{3465 \,{\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(35*c^4*e^4*x^4 + 107*c^4*d^4 - 266*b*c^3*d^3*e + 219*b^2*c^2*d^2*e^2 - 6
8*b^3*c*d*e^3 + 8*b^4*e^4 + 10*(4*c^4*d*e^3 + 5*b*c^3*e^4)*x^3 - 3*(26*c^4*d^2*e
^2 - 46*b*c^3*d*e^3 - b^2*c^2*e^4)*x^2 - 2*(52*c^4*d^3*e - 39*b*c^3*d^2*e^2 - 15
*b^2*c^2*d*e^3 + 2*b^3*c*e^4)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*f/(c^3*e^2*x
+ c^3*d*e) - 2/3465*(315*c^5*e^5*x^5 + 422*c^5*d^5 - 1538*b*c^4*d^4*e + 2130*b^
2*c^3*d^3*e^2 - 1382*b^3*c^2*d^2*e^3 + 416*b^4*c*d*e^4 - 48*b^5*e^5 + 70*(5*c^5*
d*e^4 + 6*b*c^4*e^5)*x^4 - 5*(118*c^5*d^2*e^3 - 214*b*c^4*d*e^4 - 3*b^2*c^3*e^5)
*x^3 - 6*(118*c^5*d^3*e^2 - 101*b*c^4*d^2*e^3 - 20*b^2*c^3*d*e^4 + 3*b^3*c^2*e^5
)*x^2 + (211*c^5*d^4*e - 558*b*c^4*d^3*e^2 + 507*b^2*c^3*d^2*e^3 - 184*b^3*c^2*d
*e^4 + 24*b^4*c*e^5)*x)*sqrt(-c*e*x + c*d - b*e)*(e*x + d)*g/(c^4*e^3*x + c^4*d*
e^2)

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Fricas [A]  time = 0.291511, size = 1130, normalized size = 4.23 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*c^6*e^7*g*x^7 + 35*(11*c^6*e^7*f + (10*c^6*d*e^6 + 21*b*c^5*e^7)*g)*
x^6 + 5*(11*(8*c^6*d*e^6 + 17*b*c^5*e^7)*f - (181*c^6*d^2*e^5 - 347*b*c^5*d*e^6
- 87*b^2*c^4*e^7)*g)*x^5 - (11*(113*c^6*d^2*e^5 - 213*b*c^5*d*e^6 - 53*b^2*c^4*e
^7)*f + (1058*c^6*d^3*e^4 + 54*b*c^5*d^2*e^5 - 1610*b^2*c^4*d*e^6 + 3*b^3*c^3*e^
7)*g)*x^4 - (11*(144*c^6*d^3*e^4 + 10*b*c^5*d^2*e^5 - 218*b^2*c^4*d*e^6 + b^3*c^
3*e^7)*f - (801*c^6*d^4*e^3 - 2926*b*c^5*d^3*e^4 + 2168*b^2*c^4*d^2*e^5 - 49*b^3
*c^3*d*e^6 + 6*b^4*c^2*e^7)*g)*x^3 + (11*(185*c^6*d^4*e^3 - 586*b*c^5*d^3*e^4 +
432*b^2*c^4*d^2*e^5 - 35*b^3*c^3*d*e^6 + 4*b^4*c^2*e^7)*f + (1130*c^6*d^5*e^2 -
2641*b*c^5*d^4*e^3 + 2058*b^2*c^4*d^3*e^4 - 737*b^3*c^3*d^2*e^5 + 214*b^4*c^2*d*
e^6 - 24*b^5*c*e^7)*g)*x^2 - 11*(107*c^6*d^6*e - 373*b*c^5*d^5*e^2 + 485*b^2*c^4
*d^4*e^3 - 287*b^3*c^3*d^3*e^4 + 76*b^4*c^2*d^2*e^5 - 8*b^5*c*d*e^6)*f - 2*(211*
c^6*d^7 - 980*b*c^5*d^6*e + 1834*b^2*c^4*d^5*e^2 - 1756*b^3*c^3*d^4*e^3 + 899*b^
4*c^2*d^3*e^4 - 232*b^5*c*d^2*e^5 + 24*b^6*d*e^6)*g + (11*(104*c^6*d^5*e^2 - 75*
b*c^5*d^4*e^3 - 218*b^2*c^4*d^3*e^4 + 253*b^3*c^3*d^2*e^5 - 72*b^4*c^2*d*e^6 + 8
*b^5*c*e^7)*f - (211*c^6*d^6*e - 1191*b*c^5*d^5*e^2 + 2603*b^2*c^4*d^4*e^3 - 282
1*b^3*c^3*d^3*e^4 + 1590*b^4*c^2*d^2*e^5 - 440*b^5*c*d*e^6 + 48*b^6*e^7)*g)*x)/(
sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)*c^4*e^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out