3.2270 \(\int \frac{(d+e x)^{5/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=217 \[ \frac{4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{3 c^3 e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{3 c^2 e^2 (2 c d-b e)}+\frac{2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(5/2))/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b
*e) - b*e^2*x - c*e^2*x^2]) + (4*(3*c*e*f + 5*c*d*g - 4*b*e*g)*Sqrt[d*(c*d - b*e
) - b*e^2*x - c*e^2*x^2])/(3*c^3*e^2*Sqrt[d + e*x]) + (2*(3*c*e*f + 5*c*d*g - 4*
b*e*g)*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(3*c^2*e^2*(2*c*
d - b*e))

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Rubi [A]  time = 0.705782, antiderivative size = 217, 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 \[ \frac{4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{3 c^3 e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{3 c^2 e^2 (2 c d-b e)}+\frac{2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(5/2))/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b
*e) - b*e^2*x - c*e^2*x^2]) + (4*(3*c*e*f + 5*c*d*g - 4*b*e*g)*Sqrt[d*(c*d - b*e
) - b*e^2*x - c*e^2*x^2])/(3*c^3*e^2*Sqrt[d + e*x]) + (2*(3*c*e*f + 5*c*d*g - 4*
b*e*g)*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(3*c^2*e^2*(2*c*
d - b*e))

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Rubi in Sympy [A]  time = 80.8729, size = 206, normalized size = 0.95 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (b e g - c d g - c e f\right )}{c e^{2} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{2 \sqrt{d + e x} \left (4 b e g - 5 c d g - 3 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{3 c^{2} e^{2} \left (b e - 2 c d\right )} - \frac{4 \left (4 b e g - 5 c d g - 3 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{3 c^{3} e^{2} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(d + e*x)**(5/2)*(b*e*g - c*d*g - c*e*f)/(c*e**2*(b*e - 2*c*d)*sqrt(-b*e**2*x
- c*e**2*x**2 + d*(-b*e + c*d))) + 2*sqrt(d + e*x)*(4*b*e*g - 5*c*d*g - 3*c*e*f)
*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(3*c**2*e**2*(b*e - 2*c*d)) - 4*
(4*b*e*g - 5*c*d*g - 3*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(3*
c**3*e**2*sqrt(d + e*x))

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Mathematica [A]  time = 0.154233, size = 105, normalized size = 0.48 \[ -\frac{2 \sqrt{d+e x} \left (-8 b^2 e^2 g+2 b c e (11 d g+3 e f-2 e g x)+c^2 \left (-14 d^2 g+d e (7 g x-9 f)+e^2 x (3 f+g x)\right )\right )}{3 c^3 e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 139, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -g{x}^{2}{c}^{2}{e}^{2}+4\,bc{e}^{2}gx-7\,{c}^{2}degx-3\,{c}^{2}{e}^{2}fx+8\,{b}^{2}{e}^{2}g-22\,bcdeg-6\,bc{e}^{2}f+14\,{c}^{2}{d}^{2}g+9\,{c}^{2}def \right ) }{3\,{c}^{3}{e}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(c*e*x+b*e-c*d)*(-c^2*e^2*g*x^2+4*b*c*e^2*g*x-7*c^2*d*e*g*x-3*c^2*e^2*f*x+8
*b^2*e^2*g-22*b*c*d*e*g-6*b*c*e^2*f+14*c^2*d^2*g+9*c^2*d*e*f)*(e*x+d)^(3/2)/c^3/
e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)

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Maxima [A]  time = 0.736334, size = 151, normalized size = 0.7 \[ -\frac{2 \,{\left (c e x - 3 \, c d + 2 \, b e\right )} f}{\sqrt{-c e x + c d - b e} c^{2} e} - \frac{2 \,{\left (c^{2} e^{2} x^{2} - 14 \, c^{2} d^{2} + 22 \, b c d e - 8 \, b^{2} e^{2} +{\left (7 \, c^{2} d e - 4 \, b c e^{2}\right )} x\right )} g}{3 \, \sqrt{-c e x + c d - b e} c^{3} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(c*e*x - 3*c*d + 2*b*e)*f/(sqrt(-c*e*x + c*d - b*e)*c^2*e) - 2/3*(c^2*e^2*x^2
- 14*c^2*d^2 + 22*b*c*d*e - 8*b^2*e^2 + (7*c^2*d*e - 4*b*c*e^2)*x)*g/(sqrt(-c*e
*x + c*d - b*e)*c^3*e^2)

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Fricas [A]  time = 0.272152, size = 258, normalized size = 1.19 \[ -\frac{2 \,{\left (c^{2} e^{3} g x^{3} +{\left (3 \, c^{2} e^{3} f + 4 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} - 3 \,{\left (3 \, c^{2} d^{2} e - 2 \, b c d e^{2}\right )} f - 2 \,{\left (7 \, c^{2} d^{3} - 11 \, b c d^{2} e + 4 \, b^{2} d e^{2}\right )} g -{\left (6 \,{\left (c^{2} d e^{2} - b c e^{3}\right )} f +{\left (7 \, c^{2} d^{2} e - 18 \, b c d e^{2} + 8 \, b^{2} e^{3}\right )} g\right )} x\right )}}{3 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{3} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(c^2*e^3*g*x^3 + (3*c^2*e^3*f + 4*(2*c^2*d*e^2 - b*c*e^3)*g)*x^2 - 3*(3*c^2
*d^2*e - 2*b*c*d*e^2)*f - 2*(7*c^2*d^3 - 11*b*c*d^2*e + 4*b^2*d*e^2)*g - (6*(c^2
*d*e^2 - b*c*e^3)*f + (7*c^2*d^2*e - 18*b*c*d*e^2 + 8*b^2*e^3)*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((e*x+d)**(5/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.643812, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x