3.2205 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=138 \[ \frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-2 c (8 d g+e f))}{63 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (d+e x)^8 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9*e^2*(2*c*d - b*e
)*(d + e*x)^8) + (2*(9*b*e*g - 2*c*(e*f + 8*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(7/2))/(63*e^2*(2*c*d - b*e)^2*(d + e*x)^7)

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Rubi [A]  time = 0.507758, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-2 c (8 d g+e f))}{63 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (d+e x)^8 (2 c d-b e)} \]

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)^8,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9*e^2*(2*c*d - b*e
)*(d + e*x)^8) + (2*(9*b*e*g - 2*c*(e*f + 8*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(7/2))/(63*e^2*(2*c*d - b*e)^2*(d + e*x)^7)

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Rubi in Sympy [A]  time = 50.1047, size = 126, normalized size = 0.91 \[ \frac{2 \left (9 b e g - 16 c d g - 2 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{63 e^{2} \left (d + e x\right )^{7} \left (b e - 2 c d\right )^{2}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{9 e^{2} \left (d + e x\right )^{8} \left (b e - 2 c d\right )} \]

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)**8,x)

[Out]

2*(9*b*e*g - 16*c*d*g - 2*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/
2)/(63*e**2*(d + e*x)**7*(b*e - 2*c*d)**2) - 2*(d*g - e*f)*(-b*e**2*x - c*e**2*x
**2 + d*(-b*e + c*d))**(7/2)/(9*e**2*(d + e*x)**8*(b*e - 2*c*d))

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Mathematica [A]  time = 0.53411, size = 104, normalized size = 0.75 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (2 c \left (d^2 g+8 d e (f+g x)+e^2 f x\right )-b e (2 d g+7 e f+9 e g x)\right )}{63 e^2 (d+e x)^5 (b e-2 c d)^2} \]

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)^8,x]

[Out]

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-(b*e*(7*e*f
+ 2*d*g + 9*e*g*x)) + 2*c*(d^2*g + e^2*f*x + 8*d*e*(f + g*x))))/(63*e^2*(-2*c*d
+ b*e)^2*(d + e*x)^5)

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Maple [A]  time = 0.014, size = 128, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 9\,b{e}^{2}gx-16\,cdegx-2\,c{e}^{2}fx+2\,bdeg+7\,b{e}^{2}f-2\,c{d}^{2}g-16\,cdef \right ) }{63\, \left ( ex+d \right ) ^{7}{e}^{2} \left ({b}^{2}{e}^{2}-4\,bcde+4\,{c}^{2}{d}^{2} \right ) } \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^8,x)

[Out]

-2/63*(c*e*x+b*e-c*d)*(9*b*e^2*g*x-16*c*d*e*g*x-2*c*e^2*f*x+2*b*d*e*g+7*b*e^2*f-
2*c*d^2*g-16*c*d*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^7/e^2/(b^2*
e^2-4*b*c*d*e+4*c^2*d^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 14.145, size = 872, normalized size = 6.32 \[ \frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left ({\left (2 \, c^{4} e^{5} f +{\left (16 \, c^{4} d e^{4} - 9 \, b c^{3} e^{5}\right )} g\right )} x^{4} +{\left ({\left (10 \, c^{4} d e^{4} - b c^{3} e^{5}\right )} f -{\left (46 \, c^{4} d^{2} e^{3} - 73 \, b c^{3} d e^{4} + 27 \, b^{2} c^{2} e^{5}\right )} g\right )} x^{3} - 3 \,{\left ({\left (14 \, c^{4} d^{2} e^{3} - 19 \, b c^{3} d e^{4} + 5 \, b^{2} c^{2} e^{5}\right )} f -{\left (14 \, c^{4} d^{3} e^{2} - 37 \, b c^{3} d^{2} e^{3} + 32 \, b^{2} c^{2} d e^{4} - 9 \, b^{3} c e^{5}\right )} g\right )} x^{2} -{\left (16 \, c^{4} d^{4} e - 55 \, b c^{3} d^{3} e^{2} + 69 \, b^{2} c^{2} d^{2} e^{3} - 37 \, b^{3} c d e^{4} + 7 \, b^{4} e^{5}\right )} f - 2 \,{\left (c^{4} d^{5} - 4 \, b c^{3} d^{4} e + 6 \, b^{2} c^{2} d^{3} e^{2} - 4 \, b^{3} c d^{2} e^{3} + b^{4} d e^{4}\right )} g +{\left ({\left (46 \, c^{4} d^{3} e^{2} - 111 \, b c^{3} d^{2} e^{3} + 84 \, b^{2} c^{2} d e^{4} - 19 \, b^{3} c e^{5}\right )} f -{\left (10 \, c^{4} d^{4} e - 39 \, b c^{3} d^{3} e^{2} + 57 \, b^{2} c^{2} d^{2} e^{3} - 37 \, b^{3} c d e^{4} + 9 \, b^{4} e^{5}\right )} g\right )} x\right )}}{63 \,{\left (4 \, c^{2} d^{7} e^{2} - 4 \, b c d^{6} e^{3} + b^{2} d^{5} e^{4} +{\left (4 \, c^{2} d^{2} e^{7} - 4 \, b c d e^{8} + b^{2} e^{9}\right )} x^{5} + 5 \,{\left (4 \, c^{2} d^{3} e^{6} - 4 \, b c d^{2} e^{7} + b^{2} d e^{8}\right )} x^{4} + 10 \,{\left (4 \, c^{2} d^{4} e^{5} - 4 \, b c d^{3} e^{6} + b^{2} d^{2} e^{7}\right )} x^{3} + 10 \,{\left (4 \, c^{2} d^{5} e^{4} - 4 \, b c d^{4} e^{5} + b^{2} d^{3} e^{6}\right )} x^{2} + 5 \,{\left (4 \, c^{2} d^{6} e^{3} - 4 \, b c d^{5} e^{4} + b^{2} d^{4} e^{5}\right )} x\right )}} \]

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)^(5/2)*(g*x + f)/(e*x + d)^8,x, algorithm="fricas")

[Out]

2/63*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((2*c^4*e^5*f + (16*c^4*d*e^4 -
9*b*c^3*e^5)*g)*x^4 + ((10*c^4*d*e^4 - b*c^3*e^5)*f - (46*c^4*d^2*e^3 - 73*b*c^3
*d*e^4 + 27*b^2*c^2*e^5)*g)*x^3 - 3*((14*c^4*d^2*e^3 - 19*b*c^3*d*e^4 + 5*b^2*c^
2*e^5)*f - (14*c^4*d^3*e^2 - 37*b*c^3*d^2*e^3 + 32*b^2*c^2*d*e^4 - 9*b^3*c*e^5)*
g)*x^2 - (16*c^4*d^4*e - 55*b*c^3*d^3*e^2 + 69*b^2*c^2*d^2*e^3 - 37*b^3*c*d*e^4
+ 7*b^4*e^5)*f - 2*(c^4*d^5 - 4*b*c^3*d^4*e + 6*b^2*c^2*d^3*e^2 - 4*b^3*c*d^2*e^
3 + b^4*d*e^4)*g + ((46*c^4*d^3*e^2 - 111*b*c^3*d^2*e^3 + 84*b^2*c^2*d*e^4 - 19*
b^3*c*e^5)*f - (10*c^4*d^4*e - 39*b*c^3*d^3*e^2 + 57*b^2*c^2*d^2*e^3 - 37*b^3*c*
d*e^4 + 9*b^4*e^5)*g)*x)/(4*c^2*d^7*e^2 - 4*b*c*d^6*e^3 + b^2*d^5*e^4 + (4*c^2*d
^2*e^7 - 4*b*c*d*e^8 + b^2*e^9)*x^5 + 5*(4*c^2*d^3*e^6 - 4*b*c*d^2*e^7 + b^2*d*e
^8)*x^4 + 10*(4*c^2*d^4*e^5 - 4*b*c*d^3*e^6 + b^2*d^2*e^7)*x^3 + 10*(4*c^2*d^5*e
^4 - 4*b*c*d^4*e^5 + b^2*d^3*e^6)*x^2 + 5*(4*c^2*d^6*e^3 - 4*b*c*d^5*e^4 + b^2*d
^4*e^5)*x)

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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)**8,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)^8,x, algorithm="giac")

[Out]

Timed out