3.2191 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=138 \[ \frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-2 c (6 d g+e f))}{35 e^2 (d+e x)^5 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 e^2 (d+e x)^6 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(7*e^2*(2*c*d - b*e
)*(d + e*x)^6) + (2*(7*b*e*g - 2*c*(e*f + 6*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(5/2))/(35*e^2*(2*c*d - b*e)^2*(d + e*x)^5)

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Rubi [A]  time = 0.505281, 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 \[ \frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (7 b e g-2 c (6 d g+e f))}{35 e^2 (d+e x)^5 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 e^2 (d+e x)^6 (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^6,x]

[Out]

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(7*e^2*(2*c*d - b*e
)*(d + e*x)^6) + (2*(7*b*e*g - 2*c*(e*f + 6*d*g))*(d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2)^(5/2))/(35*e^2*(2*c*d - b*e)^2*(d + e*x)^5)

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Rubi in Sympy [A]  time = 50.3393, size = 126, normalized size = 0.91 \[ \frac{2 \left (7 b e g - 12 c d g - 2 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{35 e^{2} \left (d + e x\right )^{5} \left (b e - 2 c d\right )^{2}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{7 e^{2} \left (d + e x\right )^{6} \left (b e - 2 c d\right )} \]

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)**(3/2)/(e*x+d)**6,x)

[Out]

2*(7*b*e*g - 12*c*d*g - 2*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/
2)/(35*e**2*(d + e*x)**5*(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))**(5/2)/(7*e**2*(d + e*x)**6*(b*e - 2*c*d))

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Mathematica [A]  time = 0.330206, size = 104, normalized size = 0.75 \[ -\frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (2 c \left (d^2 g+6 d e (f+g x)+e^2 f x\right )-b e (2 d g+5 e f+7 e g x)\right )}{35 e^2 (d+e x)^4 (b e-2 c d)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^6,x]

[Out]

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-(b*e*(5*e*
f + 2*d*g + 7*e*g*x)) + 2*c*(d^2*g + e^2*f*x + 6*d*e*(f + g*x))))/(35*e^2*(-2*c*
d + b*e)^2*(d + e*x)^4)

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Maple [A]  time = 0.014, size = 128, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 7\,b{e}^{2}gx-12\,cdegx-2\,c{e}^{2}fx+2\,bdeg+5\,b{e}^{2}f-2\,c{d}^{2}g-12\,cdef \right ) }{35\, \left ( ex+d \right ) ^{5}{e}^{2} \left ({b}^{2}{e}^{2}-4\,bcde+4\,{c}^{2}{d}^{2} \right ) } \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^(3/2)/(e*x+d)^6,x)

[Out]

-2/35*(c*e*x+b*e-c*d)*(7*b*e^2*g*x-12*c*d*e*g*x-2*c*e^2*f*x+2*b*d*e*g+5*b*e^2*f-
2*c*d^2*g-12*c*d*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^5/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)^(3/2)*(g*x + f)/(e*x + d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.62775, size = 626, normalized size = 4.54 \[ -\frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left ({\left (2 \, c^{3} e^{4} f +{\left (12 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}\right )} g\right )} x^{3} +{\left ({\left (8 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} f - 2 \,{\left (11 \, c^{3} d^{2} e^{2} - 18 \, b c^{2} d e^{3} + 7 \, b^{2} c e^{4}\right )} g\right )} x^{2} +{\left (12 \, c^{3} d^{3} e - 29 \, b c^{2} d^{2} e^{2} + 22 \, b^{2} c d e^{3} - 5 \, b^{3} e^{4}\right )} f + 2 \,{\left (c^{3} d^{4} - 3 \, b c^{2} d^{3} e + 3 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} g -{\left (2 \,{\left (11 \, c^{3} d^{2} e^{2} - 15 \, b c^{2} d e^{3} + 4 \, b^{2} c e^{4}\right )} f -{\left (8 \, c^{3} d^{3} e - 23 \, b c^{2} d^{2} e^{2} + 22 \, b^{2} c d e^{3} - 7 \, b^{3} e^{4}\right )} g\right )} x\right )}}{35 \,{\left (4 \, c^{2} d^{6} e^{2} - 4 \, b c d^{5} e^{3} + b^{2} d^{4} e^{4} +{\left (4 \, c^{2} d^{2} e^{6} - 4 \, b c d e^{7} + b^{2} e^{8}\right )} x^{4} + 4 \,{\left (4 \, c^{2} d^{3} e^{5} - 4 \, b c d^{2} e^{6} + b^{2} d e^{7}\right )} x^{3} + 6 \,{\left (4 \, c^{2} d^{4} e^{4} - 4 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6}\right )} x^{2} + 4 \,{\left (4 \, c^{2} d^{5} e^{3} - 4 \, b c d^{4} e^{4} + b^{2} d^{3} e^{5}\right )} x\right )}} \]

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)^(3/2)*(g*x + f)/(e*x + d)^6,x, algorithm="fricas")

[Out]

-2/35*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((2*c^3*e^4*f + (12*c^3*d*e^3 -
7*b*c^2*e^4)*g)*x^3 + ((8*c^3*d*e^3 - b*c^2*e^4)*f - 2*(11*c^3*d^2*e^2 - 18*b*c
^2*d*e^3 + 7*b^2*c*e^4)*g)*x^2 + (12*c^3*d^3*e - 29*b*c^2*d^2*e^2 + 22*b^2*c*d*e
^3 - 5*b^3*e^4)*f + 2*(c^3*d^4 - 3*b*c^2*d^3*e + 3*b^2*c*d^2*e^2 - b^3*d*e^3)*g
- (2*(11*c^3*d^2*e^2 - 15*b*c^2*d*e^3 + 4*b^2*c*e^4)*f - (8*c^3*d^3*e - 23*b*c^2
*d^2*e^2 + 22*b^2*c*d*e^3 - 7*b^3*e^4)*g)*x)/(4*c^2*d^6*e^2 - 4*b*c*d^5*e^3 + b^
2*d^4*e^4 + (4*c^2*d^2*e^6 - 4*b*c*d*e^7 + b^2*e^8)*x^4 + 4*(4*c^2*d^3*e^5 - 4*b
*c*d^2*e^6 + b^2*d*e^7)*x^3 + 6*(4*c^2*d^4*e^4 - 4*b*c*d^3*e^5 + b^2*d^2*e^6)*x^
2 + 4*(4*c^2*d^5*e^3 - 4*b*c*d^4*e^4 + b^2*d^3*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)**(3/2)/(e*x+d)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 10.5968, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

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)*(g*x + f)/(e*x + d)^6,x, algorithm="giac")

[Out]

sage0*x