3.2224 \(\int \frac{(d+e x)^4 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=291 \[ \frac{(-5 b e g+8 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{7/2} e^2}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+8 c d g+2 c e f)}{c^3 e^2 (2 c d-b e)}-\frac{2 (d+e x)^2 (-5 b e g+8 c d g+2 c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^4 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^4)/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2)^(3/2)) - (2*(2*c*e*f + 8*c*d*g - 5*b*e*g)*(d + e*x)^2)/(3*c
^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - ((2*c*e*f + 8*
c*d*g - 5*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(c^3*e^2*(2*c*d - b*
e)) + ((2*c*e*f + 8*c*d*g - 5*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c
*d - b*e) - b*e^2*x - c*e^2*x^2])])/(2*c^(7/2)*e^2)

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Rubi [A]  time = 0.926118, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ \frac{(-5 b e g+8 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{7/2} e^2}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-5 b e g+8 c d g+2 c e f)}{c^3 e^2 (2 c d-b e)}-\frac{2 (d+e x)^2 (-5 b e g+8 c d g+2 c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^4 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^4*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^4)/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2)^(3/2)) - (2*(2*c*e*f + 8*c*d*g - 5*b*e*g)*(d + e*x)^2)/(3*c
^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - ((2*c*e*f + 8*
c*d*g - 5*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(c^3*e^2*(2*c*d - b*
e)) + ((2*c*e*f + 8*c*d*g - 5*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c
*d - b*e) - b*e^2*x - c*e^2*x^2])])/(2*c^(7/2)*e^2)

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Rubi in Sympy [A]  time = 107.979, size = 280, normalized size = 0.96 \[ \frac{2 \left (d + e x\right )^{4} \left (b e g - c d g - c e f\right )}{3 c e^{2} \left (b e - 2 c d\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} - \frac{4 \left (d + e x\right )^{2} \left (\frac{5 b e g}{2} - 4 c d g - c e f\right )}{3 c^{2} e^{2} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{2 \left (\frac{5 b e g}{2} - 4 c d g - c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{c^{3} e^{2} \left (b e - 2 c d\right )} - \frac{\left (5 b e g - 8 c d g - 2 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{2 c^{\frac{7}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

2*(d + e*x)**4*(b*e*g - c*d*g - c*e*f)/(3*c*e**2*(b*e - 2*c*d)*(-b*e**2*x - c*e*
*2*x**2 + d*(-b*e + c*d))**(3/2)) - 4*(d + e*x)**2*(5*b*e*g/2 - 4*c*d*g - c*e*f)
/(3*c**2*e**2*(b*e - 2*c*d)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))) - 2*
(5*b*e*g/2 - 4*c*d*g - c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(c*
*3*e**2*(b*e - 2*c*d)) - (5*b*e*g - 8*c*d*g - 2*c*e*f)*atan(-e*(-b - 2*c*x)/(2*s
qrt(c)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))))/(2*c**(7/2)*e**2)

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Mathematica [C]  time = 0.854485, size = 219, normalized size = 0.75 \[ \frac{2 \sqrt{c} (d+e x)^3 (c (d-e x)-b e) \left (-15 b^2 e^2 g+2 b c e (17 d g+3 e f-10 e g x)+c^2 \left (-19 d^2 g+d e (26 g x-4 f)+e^2 x (8 f-3 g x)\right )\right )+3 i (d+e x)^{5/2} (c (d-e x)-b e)^{5/2} (-5 b e g+8 c d g+2 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{6 c^{7/2} e^2 ((d+e x) (c (d-e x)-b e))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^4*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[c]*(d + e*x)^3*(-(b*e) + c*(d - e*x))*(-15*b^2*e^2*g + 2*b*c*e*(3*e*f +
17*d*g - 10*e*g*x) + c^2*(-19*d^2*g + e^2*x*(8*f - 3*g*x) + d*e*(-4*f + 26*g*x))
) + (3*I)*(2*c*e*f + 8*c*d*g - 5*b*e*g)*(d + e*x)^(5/2)*(-(b*e) + c*(d - e*x))^(
5/2)*Log[((-I)*e*(b + 2*c*x))/Sqrt[c] + 2*Sqrt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x
)]])/(6*c^(7/2)*e^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))

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Maple [B]  time = 0.024, size = 5032, normalized size = 17.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

result too large to display

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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((e*x + d)^4*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.58305, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2),x, algorithm="fricas")

[Out]

[-1/12*(4*(3*c^2*e^2*g*x^2 + 2*(2*c^2*d*e - 3*b*c*e^2)*f + (19*c^2*d^2 - 34*b*c*
d*e + 15*b^2*e^2)*g - 2*(4*c^2*e^2*f + (13*c^2*d*e - 10*b*c*e^2)*g)*x)*sqrt(-c*e
^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-c) + 3*((2*c^3*e^3*f + (8*c^3*d*e^2 - 5*
b*c^2*e^3)*g)*x^2 + 2*(c^3*d^2*e - 2*b*c^2*d*e^2 + b^2*c*e^3)*f + (8*c^3*d^3 - 2
1*b*c^2*d^2*e + 18*b^2*c*d*e^2 - 5*b^3*e^3)*g - 2*(2*(c^3*d*e^2 - b*c^2*e^3)*f +
(8*c^3*d^2*e - 13*b*c^2*d*e^2 + 5*b^2*c*e^3)*g)*x)*log(-4*sqrt(-c*e^2*x^2 - b*e
^2*x + c*d^2 - b*d*e)*(2*c^2*e*x + b*c*e) + (8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2
*d^2 + 4*b*c*d*e + b^2*e^2)*sqrt(-c)))/((c^5*e^4*x^2 + c^5*d^2*e^2 - 2*b*c^4*d*e
^3 + b^2*c^3*e^4 - 2*(c^5*d*e^3 - b*c^4*e^4)*x)*sqrt(-c)), -1/6*(2*(3*c^2*e^2*g*
x^2 + 2*(2*c^2*d*e - 3*b*c*e^2)*f + (19*c^2*d^2 - 34*b*c*d*e + 15*b^2*e^2)*g - 2
*(4*c^2*e^2*f + (13*c^2*d*e - 10*b*c*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^
2 - b*d*e)*sqrt(c) - 3*((2*c^3*e^3*f + (8*c^3*d*e^2 - 5*b*c^2*e^3)*g)*x^2 + 2*(c
^3*d^2*e - 2*b*c^2*d*e^2 + b^2*c*e^3)*f + (8*c^3*d^3 - 21*b*c^2*d^2*e + 18*b^2*c
*d*e^2 - 5*b^3*e^3)*g - 2*(2*(c^3*d*e^2 - b*c^2*e^3)*f + (8*c^3*d^2*e - 13*b*c^2
*d*e^2 + 5*b^2*c*e^3)*g)*x)*arctan(1/2*(2*c*e*x + b*e)/(sqrt(-c*e^2*x^2 - b*e^2*
x + c*d^2 - b*d*e)*sqrt(c))))/((c^5*e^4*x^2 + c^5*d^2*e^2 - 2*b*c^4*d*e^3 + b^2*
c^3*e^4 - 2*(c^5*d*e^3 - b*c^4*e^4)*x)*sqrt(c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{4} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.333383, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2),x, algorithm="giac")

[Out]

Done