3.2244 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=118 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g-3 c d g+7 c e f)}{35 c^2 e^2 (d+e x)^{5/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}} \]

[Out]

(-2*(7*c*e*f - 3*c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(
35*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)^(5/2))/
(7*c*e^2*(d + e*x)^(3/2))

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Rubi [A]  time = 0.486596, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g-3 c d g+7 c e f)}{35 c^2 e^2 (d+e x)^{5/2}}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(7*c*e*f - 3*c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(
35*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)^(5/2))/
(7*c*e^2*(d + e*x)^(3/2))

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Rubi in Sympy [A]  time = 42.4216, size = 110, normalized size = 0.93 \[ - \frac{2 g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{7 c e^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{2 \left (2 b e g + 3 c d g - 7 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{35 c^{2} e^{2} \left (d + e x\right )^{\frac{5}{2}}} \]

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

[Out]

-2*g*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(7*c*e**2*(d + e*x)**(3/2
)) + 2*(2*b*e*g + 3*c*d*g - 7*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))*
*(5/2)/(35*c**2*e**2*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.13386, size = 78, normalized size = 0.66 \[ -\frac{2 (b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} (c (2 d g+7 e f+5 e g x)-2 b e g)}{35 c^2 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 79, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -5\,cegx+2\,beg-2\,cdg-7\,fce \right ) }{35\,{c}^{2}{e}^{2}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/35*(c*e*x+b*e-c*d)*(-5*c*e*g*x+2*b*e*g-2*c*d*g-7*c*e*f)*(-c*e^2*x^2-b*e^2*x-b
*d*e+c*d^2)^(3/2)/c^2/e^2/(e*x+d)^(3/2)

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Maxima [A]  time = 0.730979, size = 266, normalized size = 2.25 \[ -\frac{2 \,{\left (c^{2} e^{2} x^{2} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \,{\left (c^{2} d e - b c e^{2}\right )} x\right )} \sqrt{-c e x + c d - b e} f}{5 \, c e} - \frac{2 \,{\left (5 \, c^{3} e^{3} x^{3} + 2 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3} - 8 \,{\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} +{\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x\right )} \sqrt{-c e x + c d - b e} g}{35 \, c^{2} e^{2}} \]

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

[Out]

-2/5*(c^2*e^2*x^2 + c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2*(c^2*d*e - b*c*e^2)*x)*sqr
t(-c*e*x + c*d - b*e)*f/(c*e) - 2/35*(5*c^3*e^3*x^3 + 2*c^3*d^3 - 6*b*c^2*d^2*e
+ 6*b^2*c*d*e^2 - 2*b^3*e^3 - 8*(c^3*d*e^2 - b*c^2*e^3)*x^2 + (c^3*d^2*e - 2*b*c
^2*d*e^2 + b^2*c*e^3)*x)*sqrt(-c*e*x + c*d - b*e)*g/(c^2*e^2)

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Fricas [A]  time = 0.286397, size = 582, normalized size = 4.93 \[ \frac{2 \,{\left (5 \, c^{4} e^{5} g x^{5} +{\left (7 \, c^{4} e^{5} f -{\left (8 \, c^{4} d e^{4} - 13 \, b c^{3} e^{5}\right )} g\right )} x^{4} -{\left (7 \,{\left (2 \, c^{4} d e^{4} - 3 \, b c^{3} e^{5}\right )} f +{\left (4 \, c^{4} d^{2} e^{3} + 5 \, b c^{3} d e^{4} - 9 \, b^{2} c^{2} e^{5}\right )} g\right )} x^{3} -{\left (21 \,{\left (b c^{3} d e^{4} - b^{2} c^{2} e^{5}\right )} f -{\left (10 \, c^{4} d^{3} e^{2} - 21 \, b c^{3} d^{2} e^{3} + 12 \, b^{2} c^{2} d e^{4} - b^{3} c e^{5}\right )} g\right )} x^{2} - 7 \,{\left (c^{4} d^{4} e - 3 \, b c^{3} d^{3} e^{2} + 3 \, b^{2} c^{2} d^{2} e^{3} - b^{3} c d e^{4}\right )} f - 2 \,{\left (c^{4} d^{5} - 4 \, b c^{3} d^{4} e + 6 \, b^{2} c^{2} d^{3} e^{2} - 4 \, b^{3} c d^{2} e^{3} + b^{4} d e^{4}\right )} g +{\left (7 \,{\left (2 \, c^{4} d^{3} e^{2} - 3 \, b c^{3} d^{2} e^{3} + b^{3} c e^{5}\right )} f -{\left (c^{4} d^{4} e - 5 \, b c^{3} d^{3} e^{2} + 9 \, b^{2} c^{2} d^{2} e^{3} - 7 \, b^{3} c d e^{4} + 2 \, b^{4} e^{5}\right )} g\right )} x\right )}}{35 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{2} e^{2}} \]

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

[Out]

2/35*(5*c^4*e^5*g*x^5 + (7*c^4*e^5*f - (8*c^4*d*e^4 - 13*b*c^3*e^5)*g)*x^4 - (7*
(2*c^4*d*e^4 - 3*b*c^3*e^5)*f + (4*c^4*d^2*e^3 + 5*b*c^3*d*e^4 - 9*b^2*c^2*e^5)*
g)*x^3 - (21*(b*c^3*d*e^4 - b^2*c^2*e^5)*f - (10*c^4*d^3*e^2 - 21*b*c^3*d^2*e^3
+ 12*b^2*c^2*d*e^4 - b^3*c*e^5)*g)*x^2 - 7*(c^4*d^4*e - 3*b*c^3*d^3*e^2 + 3*b^2*
c^2*d^2*e^3 - b^3*c*d*e^4)*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 + (7*(2*c^4*d^3*e^2 - 3*b*c^3*d^2*e^3 + b^3*c*e^5
)*f - (c^4*d^4*e - 5*b*c^3*d^3*e^2 + 9*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4 + 2*b^4*e
^5)*g)*x)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)*c^2*e^2)

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

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)**(3/2),x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: AttributeError} \]

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)^(3/2),x, algorithm="giac")

[Out]

Exception raised: AttributeError