3.2254 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=193 \[ -\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{693 c^3 e^2 (d+e x)^{7/2}}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{99 c^2 e^2 (d+e x)^{5/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}} \]

[Out]

(-4*(2*c*d - b*e)*(11*c*e*f - 3*c*d*g - 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^
2*x^2)^(7/2))/(693*c^3*e^2*(d + e*x)^(7/2)) - (2*(11*c*e*f - 3*c*d*g - 4*b*e*g)*
(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(99*c^2*e^2*(d + e*x)^(5/2)) - (2*g
*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(11*c*e^2*(d + e*x)^(3/2))

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Rubi [A]  time = 0.762113, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{4 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{693 c^3 e^2 (d+e x)^{7/2}}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-4 b e g-3 c d g+11 c e f)}{99 c^2 e^2 (d+e x)^{5/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 c e^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(3/2),x]

[Out]

(-4*(2*c*d - b*e)*(11*c*e*f - 3*c*d*g - 4*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^
2*x^2)^(7/2))/(693*c^3*e^2*(d + e*x)^(7/2)) - (2*(11*c*e*f - 3*c*d*g - 4*b*e*g)*
(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(99*c^2*e^2*(d + e*x)^(5/2)) - (2*g
*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(11*c*e^2*(d + e*x)^(3/2))

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Rubi in Sympy [A]  time = 65.2461, size = 185, normalized size = 0.96 \[ - \frac{2 g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{11 c e^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{2 \left (4 b e g + 3 c d g - 11 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{99 c^{2} e^{2} \left (d + e x\right )^{\frac{5}{2}}} - \frac{4 \left (b e - 2 c d\right ) \left (4 b e g + 3 c d g - 11 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{693 c^{3} e^{2} \left (d + e x\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(3/2),x)

[Out]

-2*g*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(11*c*e**2*(d + e*x)**(3/
2)) + 2*(4*b*e*g + 3*c*d*g - 11*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d)
)**(7/2)/(99*c**2*e**2*(d + e*x)**(5/2)) - 4*(b*e - 2*c*d)*(4*b*e*g + 3*c*d*g -
11*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(693*c**3*e**2*(d +
e*x)**(7/2))

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Mathematica [A]  time = 0.233547, size = 121, normalized size = 0.63 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (8 b^2 e^2 g-2 b c e (19 d g+11 e f+14 e g x)+c^2 \left (30 d^2 g+d e (121 f+105 g x)+7 e^2 x (11 f+9 g x)\right )\right )}{693 c^3 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^(3/2),x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(8*b^2*e^2*g
- 2*b*c*e*(11*e*f + 19*d*g + 14*e*g*x) + c^2*(30*d^2*g + 7*e^2*x*(11*f + 9*g*x)
+ d*e*(121*f + 105*g*x))))/(693*c^3*e^2*Sqrt[d + e*x])

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Maple [A]  time = 0.008, size = 139, normalized size = 0.7 \[{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 63\,g{x}^{2}{c}^{2}{e}^{2}-28\,bc{e}^{2}gx+105\,{c}^{2}degx+77\,{c}^{2}{e}^{2}fx+8\,{b}^{2}{e}^{2}g-38\,bcdeg-22\,bc{e}^{2}f+30\,{c}^{2}{d}^{2}g+121\,{c}^{2}def \right ) }{693\,{c}^{3}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^(3/2),x)

[Out]

2/693*(c*e*x+b*e-c*d)*(63*c^2*e^2*g*x^2-28*b*c*e^2*g*x+105*c^2*d*e*g*x+77*c^2*e^
2*f*x+8*b^2*e^2*g-38*b*c*d*e*g-22*b*c*e^2*f+30*c^2*d^2*g+121*c^2*d*e*f)*(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/c^3/e^2/(e*x+d)^(5/2)

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Maxima [A]  time = 0.738787, size = 628, normalized size = 3.25 \[ \frac{2 \,{\left (7 \, c^{4} e^{4} x^{4} - 11 \, c^{4} d^{4} + 35 \, b c^{3} d^{3} e - 39 \, b^{2} c^{2} d^{2} e^{2} + 17 \, b^{3} c d e^{3} - 2 \, b^{4} e^{4} -{\left (10 \, c^{4} d e^{3} - 19 \, b c^{3} e^{4}\right )} x^{3} - 3 \,{\left (4 \, c^{4} d^{2} e^{2} + b c^{3} d e^{3} - 5 \, b^{2} c^{2} e^{4}\right )} x^{2} +{\left (26 \, c^{4} d^{3} e - 51 \, b c^{3} d^{2} e^{2} + 24 \, b^{2} c^{2} d e^{3} + b^{3} c e^{4}\right )} x\right )} \sqrt{-c e x + c d - b e} f}{63 \, c^{2} e} + \frac{2 \,{\left (63 \, c^{5} e^{5} x^{5} - 30 \, c^{5} d^{5} + 128 \, b c^{4} d^{4} e - 212 \, b^{2} c^{3} d^{3} e^{2} + 168 \, b^{3} c^{2} d^{2} e^{3} - 62 \, b^{4} c d e^{4} + 8 \, b^{5} e^{5} - 7 \,{\left (12 \, c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} x^{4} -{\left (96 \, c^{5} d^{2} e^{3} + 17 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \,{\left (54 \, c^{5} d^{3} e^{2} - 107 \, b c^{4} d^{2} e^{3} + 52 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} x^{2} -{\left (15 \, c^{5} d^{4} e - 49 \, b c^{4} d^{3} e^{2} + 57 \, b^{2} c^{3} d^{2} e^{3} - 27 \, b^{3} c^{2} d e^{4} + 4 \, b^{4} c e^{5}\right )} x\right )} \sqrt{-c e x + c d - b e} g}{693 \, c^{3} e^{2}} \]

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)^(5/2)*(g*x + f)/(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

2/63*(7*c^4*e^4*x^4 - 11*c^4*d^4 + 35*b*c^3*d^3*e - 39*b^2*c^2*d^2*e^2 + 17*b^3*
c*d*e^3 - 2*b^4*e^4 - (10*c^4*d*e^3 - 19*b*c^3*e^4)*x^3 - 3*(4*c^4*d^2*e^2 + b*c
^3*d*e^3 - 5*b^2*c^2*e^4)*x^2 + (26*c^4*d^3*e - 51*b*c^3*d^2*e^2 + 24*b^2*c^2*d*
e^3 + b^3*c*e^4)*x)*sqrt(-c*e*x + c*d - b*e)*f/(c^2*e) + 2/693*(63*c^5*e^5*x^5 -
30*c^5*d^5 + 128*b*c^4*d^4*e - 212*b^2*c^3*d^3*e^2 + 168*b^3*c^2*d^2*e^3 - 62*b
^4*c*d*e^4 + 8*b^5*e^5 - 7*(12*c^5*d*e^4 - 23*b*c^4*e^5)*x^4 - (96*c^5*d^2*e^3 +
17*b*c^4*d*e^4 - 113*b^2*c^3*e^5)*x^3 + 3*(54*c^5*d^3*e^2 - 107*b*c^4*d^2*e^3 +
52*b^2*c^3*d*e^4 + b^3*c^2*e^5)*x^2 - (15*c^5*d^4*e - 49*b*c^4*d^3*e^2 + 57*b^2
*c^3*d^2*e^3 - 27*b^3*c^2*d*e^4 + 4*b^4*c*e^5)*x)*sqrt(-c*e*x + c*d - b*e)*g/(c^
3*e^2)

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Fricas [A]  time = 0.292498, size = 1130, normalized size = 5.85 \[ \text{result too large to display} \]

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)^(5/2)*(g*x + f)/(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

-2/693*(63*c^6*e^7*g*x^7 + 7*(11*c^6*e^7*f - 4*(3*c^6*d*e^6 - 8*b*c^5*e^7)*g)*x^
6 - (22*(5*c^6*d*e^6 - 13*b*c^5*e^7)*f + (159*c^6*d^2*e^5 + 38*b*c^5*d*e^6 - 274
*b^2*c^4*e^7)*g)*x^5 - (11*(19*c^6*d^2*e^5 + 6*b*c^5*d*e^6 - 34*b^2*c^4*e^7)*f -
2*(123*c^6*d^3*e^4 - 331*b*c^5*d^2*e^5 + 150*b^2*c^4*d*e^6 + 58*b^3*c^3*e^7)*g)
*x^4 + (44*(9*c^6*d^3*e^4 - 23*b*c^5*d^2*e^5 + 10*b^2*c^4*d*e^6 + 4*b^3*c^3*e^7)
*f + (81*c^6*d^4*e^3 + 132*b*c^5*d^3*e^4 - 508*b^2*c^4*d^2*e^5 + 296*b^3*c^3*d*e
^6 - b^4*c^2*e^7)*g)*x^3 + (11*(c^6*d^4*e^3 + 52*b*c^5*d^3*e^4 - 108*b^2*c^4*d^2
*e^5 + 56*b^3*c^3*d*e^6 - b^4*c^2*e^7)*f - 4*(48*c^6*d^5*e^2 - 149*b*c^5*d^4*e^3
+ 160*b^2*c^4*d^3*e^4 - 66*b^3*c^3*d^2*e^5 + 8*b^4*c^2*d*e^6 - b^5*c*e^7)*g)*x^
2 + 11*(11*c^6*d^6*e - 46*b*c^5*d^5*e^2 + 74*b^2*c^4*d^4*e^3 - 56*b^3*c^3*d^3*e^
4 + 19*b^4*c^2*d^2*e^5 - 2*b^5*c*d*e^6)*f + 2*(15*c^6*d^7 - 79*b*c^5*d^6*e + 170
*b^2*c^4*d^5*e^2 - 190*b^3*c^3*d^4*e^3 + 115*b^4*c^2*d^3*e^4 - 35*b^5*c*d^2*e^5
+ 4*b^6*d*e^6)*g - (22*(13*c^6*d^5*e^2 - 33*b*c^5*d^4*e^3 + 20*b^2*c^4*d^3*e^4 +
8*b^3*c^3*d^2*e^5 - 9*b^4*c^2*d*e^6 + b^5*c*e^7)*f - (15*c^6*d^6*e - 94*b*c^5*d
^5*e^2 + 234*b^2*c^4*d^4*e^3 - 296*b^3*c^3*d^3*e^4 + 199*b^4*c^2*d^2*e^5 - 66*b^
5*c*d*e^6 + 8*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^3*e^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(3/2),x)

[Out]

Timed out

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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)^(5/2)*(g*x + f)/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

Timed out