3.2223 \(\int \frac{(d+e x)^5 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=364 \[ \frac{5 (2 c d-b e) (-7 b e g+10 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{9/2} e^2}-\frac{5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+10 c d g+4 c e f)}{4 c^4 e^2}-\frac{5 (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+10 c d g+4 c e f)}{6 c^3 e^2 (2 c d-b e)}-\frac{2 (d+e x)^3 (-7 b e g+10 c d g+4 c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^5 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^5)/(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*(4*c*e*f + 10*c*d*g - 7*b*e*g)*(d + e*x)^3)/(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]) - (5*(4*c*e*f +
10*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c^4*e^2) - (5
*(4*c*e*f + 10*c*d*g - 7*b*e*g)*(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2])/(6*c^3*e^2*(2*c*d - b*e)) + (5*(2*c*d - b*e)*(4*c*e*f + 10*c*d*g - 7*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])])/
(8*c^(9/2)*e^2)

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Rubi [A]  time = 1.23287, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{5 (2 c d-b e) (-7 b e g+10 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{9/2} e^2}-\frac{5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+10 c d g+4 c e f)}{4 c^4 e^2}-\frac{5 (d+e x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+10 c d g+4 c e f)}{6 c^3 e^2 (2 c d-b e)}-\frac{2 (d+e x)^3 (-7 b e g+10 c d g+4 c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^5 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^5*(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)^5)/(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*(4*c*e*f + 10*c*d*g - 7*b*e*g)*(d + e*x)^3)/(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]) - (5*(4*c*e*f +
10*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c^4*e^2) - (5
*(4*c*e*f + 10*c*d*g - 7*b*e*g)*(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x
^2])/(6*c^3*e^2*(2*c*d - b*e)) + (5*(2*c*d - b*e)*(4*c*e*f + 10*c*d*g - 7*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])])/
(8*c^(9/2)*e^2)

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Rubi in Sympy [A]  time = 139.625, size = 355, normalized size = 0.98 \[ \frac{2 \left (d + e x\right )^{5} \left (b e g - c d g - c e f\right )}{3 c e^{2} \left (b e - 2 c d\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} - \frac{2 \left (d + e x\right )^{3} \left (7 b e g - 10 c d g - 4 c e f\right )}{3 c^{2} e^{2} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{5 \left (d + e x\right ) \left (7 b e g - 10 c d g - 4 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{6 c^{3} e^{2} \left (b e - 2 c d\right )} + \frac{5 \left (7 b e g - 10 c d g - 4 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{4 c^{4} e^{2}} + \frac{5 \left (b e - 2 c d\right ) \left (7 b e g - 10 c d g - 4 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{8 c^{\frac{9}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**5*(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)**5*(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)) - 2*(d + e*x)**3*(7*b*e*g - 10*c*d*g - 4*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))) - 5
*(d + e*x)*(7*b*e*g - 10*c*d*g - 4*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e
+ c*d))/(6*c**3*e**2*(b*e - 2*c*d)) + 5*(7*b*e*g - 10*c*d*g - 4*c*e*f)*sqrt(-b*
e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(4*c**4*e**2) + 5*(b*e - 2*c*d)*(7*b*e*g
- 10*c*d*g - 4*c*e*f)*atan(-e*(-b - 2*c*x)/(2*sqrt(c)*sqrt(-b*e**2*x - c*e**2*x*
*2 + d*(-b*e + c*d))))/(8*c**(9/2)*e**2)

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Mathematica [C]  time = 2.01702, size = 291, normalized size = 0.8 \[ \frac{\frac{2 (d+e x)^3 (b e-c d+c e x) \left (-105 b^3 e^3 g+10 b^2 c e^2 (43 d g+6 e f-14 e g x)+b c^2 e \left (-561 d^2 g+d e (438 g x-160 f)+e^2 x (80 f-21 g x)\right )+2 c^3 \left (118 d^3 g+23 d^2 e (2 f-7 g x)+4 d e^2 x (6 g x-17 f)+3 e^3 x^2 (2 f+g x)\right )\right )}{3 c^4 e^2}-\frac{5 i (d+e x)^{5/2} (b e-2 c d) (c (d-e x)-b e)^{5/2} (-7 b e g+10 c d g+4 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{c^{9/2} e^2}}{8 ((d+e x) (c (d-e x)-b e))^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*(d + e*x)^3*(-(c*d) + b*e + c*e*x)*(-105*b^3*e^3*g + 10*b^2*c*e^2*(6*e*f + 4
3*d*g - 14*e*g*x) + 2*c^3*(118*d^3*g + 23*d^2*e*(2*f - 7*g*x) + 3*e^3*x^2*(2*f +
g*x) + 4*d*e^2*x*(-17*f + 6*g*x)) + b*c^2*e*(-561*d^2*g + e^2*x*(80*f - 21*g*x)
+ d*e*(-160*f + 438*g*x))))/(3*c^4*e^2) - ((5*I)*(-2*c*d + b*e)*(4*c*e*f + 10*c
*d*g - 7*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)]])/(c^(9/2)*e^2))/(8*(
(d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))

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Maple [B]  time = 0.073, size = 6704, normalized size = 18.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^5*(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)^5*(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 = 3.25439, 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)^5*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2),x, algorithm="fricas")

[Out]

[-1/48*(4*(6*c^3*e^3*g*x^3 + 3*(4*c^3*e^3*f + (16*c^3*d*e^2 - 7*b*c^2*e^3)*g)*x^
2 + 4*(23*c^3*d^2*e - 40*b*c^2*d*e^2 + 15*b^2*c*e^3)*f + (236*c^3*d^3 - 561*b*c^
2*d^2*e + 430*b^2*c*d*e^2 - 105*b^3*e^3)*g - 2*(4*(17*c^3*d*e^2 - 10*b*c^2*e^3)*
f + (161*c^3*d^2*e - 219*b*c^2*d*e^2 + 70*b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e
^2*x + c*d^2 - b*d*e)*sqrt(-c) - 15*((4*(2*c^4*d*e^3 - b*c^3*e^4)*f + (20*c^4*d^
2*e^2 - 24*b*c^3*d*e^3 + 7*b^2*c^2*e^4)*g)*x^2 + 4*(2*c^4*d^3*e - 5*b*c^3*d^2*e^
2 + 4*b^2*c^2*d*e^3 - b^3*c*e^4)*f + (20*c^4*d^4 - 64*b*c^3*d^3*e + 75*b^2*c^2*d
^2*e^2 - 38*b^3*c*d*e^3 + 7*b^4*e^4)*g - 2*(4*(2*c^4*d^2*e^2 - 3*b*c^3*d*e^3 + b
^2*c^2*e^4)*f + (20*c^4*d^3*e - 44*b*c^3*d^2*e^2 + 31*b^2*c^2*d*e^3 - 7*b^3*c*e^
4)*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^6
*e^4*x^2 + c^6*d^2*e^2 - 2*b*c^5*d*e^3 + b^2*c^4*e^4 - 2*(c^6*d*e^3 - b*c^5*e^4)
*x)*sqrt(-c)), -1/24*(2*(6*c^3*e^3*g*x^3 + 3*(4*c^3*e^3*f + (16*c^3*d*e^2 - 7*b*
c^2*e^3)*g)*x^2 + 4*(23*c^3*d^2*e - 40*b*c^2*d*e^2 + 15*b^2*c*e^3)*f + (236*c^3*
d^3 - 561*b*c^2*d^2*e + 430*b^2*c*d*e^2 - 105*b^3*e^3)*g - 2*(4*(17*c^3*d*e^2 -
10*b*c^2*e^3)*f + (161*c^3*d^2*e - 219*b*c^2*d*e^2 + 70*b^2*c*e^3)*g)*x)*sqrt(-c
*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c) - 15*((4*(2*c^4*d*e^3 - b*c^3*e^4)*f
+ (20*c^4*d^2*e^2 - 24*b*c^3*d*e^3 + 7*b^2*c^2*e^4)*g)*x^2 + 4*(2*c^4*d^3*e - 5
*b*c^3*d^2*e^2 + 4*b^2*c^2*d*e^3 - b^3*c*e^4)*f + (20*c^4*d^4 - 64*b*c^3*d^3*e +
75*b^2*c^2*d^2*e^2 - 38*b^3*c*d*e^3 + 7*b^4*e^4)*g - 2*(4*(2*c^4*d^2*e^2 - 3*b*
c^3*d*e^3 + b^2*c^2*e^4)*f + (20*c^4*d^3*e - 44*b*c^3*d^2*e^2 + 31*b^2*c^2*d*e^3
- 7*b^3*c*e^4)*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^6*e^4*x^2 + c^6*d^2*e^2 - 2*b*c^5*d*e^3 + b^2*c^4*e^
4 - 2*(c^6*d*e^3 - b*c^5*e^4)*x)*sqrt(c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

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GIAC/XCAS [A]  time = 0.327872, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^5*(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