3.2202 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=350 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}+\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (3 b e g-10 c d g+4 c e f)}{3 e^2 (d+e x)^3 (2 c d-b e)}+\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (3 b e g-10 c d g+4 c e f)}{6 e^2 (d+e x) (2 c d-b e)}+\frac{5 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-10 c d g+4 c e f)}{4 e^2}+\frac{5 \sqrt{c} (2 c d-b e) (3 b e g-10 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 e^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.2168, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^5 (2 c d-b e)}+\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (3 b e g-10 c d g+4 c e f)}{3 e^2 (d+e x)^3 (2 c d-b e)}+\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (3 b e g-10 c d g+4 c e f)}{6 e^2 (d+e x) (2 c d-b e)}+\frac{5 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-10 c d g+4 c e f)}{4 e^2}+\frac{5 \sqrt{c} (2 c d-b e) (3 b e g-10 c d g+4 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 131.842, size = 340, normalized size = 0.97 \[ - \frac{5 \sqrt{c} \left (b e - 2 c d\right ) \left (3 b e g - 10 c d g + 4 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{8 e^{2}} + \frac{5 c \left (3 b e g - 10 c d g + 4 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{4 e^{2}} - \frac{5 c \left (3 b e g - 10 c d g + 4 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{6 e^{2} \left (d + e x\right ) \left (b e - 2 c d\right )} - \frac{4 \left (\frac{3 b e g}{2} - 5 c d g + 2 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{3 e^{2} \left (d + e x\right )^{3} \left (b e - 2 c d\right )} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{3 e^{2} \left (d + e x\right )^{5} \left (b e - 2 c d\right )} \]

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)**(5/2)/(e*x+d)**5,x)

[Out]

-5*sqrt(c)*(b*e - 2*c*d)*(3*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*e**2) + 5*c*(3*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*e**2) -
5*c*(3*b*e*g - 10*c*d*g + 4*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**
(3/2)/(6*e**2*(d + e*x)*(b*e - 2*c*d)) - 4*(3*b*e*g/2 - 5*c*d*g + 2*c*e*f)*(-b*e
**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(3*e**2*(d + e*x)**3*(b*e - 2*c*d))
- 2*(d*g - e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(3*e**2*(d +
e*x)**5*(b*e - 2*c*d))

_______________________________________________________________________________________

Mathematica [C]  time = 1.20449, size = 260, normalized size = 0.74 \[ \frac{i ((d+e x) (c (d-e x)-b e))^{5/2} \left (2 i \sqrt{c (d-e x)-b e} \left (-3 c (d+e x)^2 (9 b e g+4 c (e f-5 d g))+8 (d+e x) (2 c d-b e) (-3 b e g+13 c d g-7 c e f)+8 (b e-2 c d)^2 (e f-d g)-6 c^2 e g x (d+e x)^2\right )+15 \sqrt{c} (d+e x)^{3/2} (2 c d-b e) (3 b e g-10 c d g+4 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )\right )}{24 e^2 (d+e x)^4 (c (d-e x)-b e)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((I/24)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((2*I)*Sqrt[-(b*e) + c*(d - e*x
)]*(8*(-2*c*d + b*e)^2*(e*f - d*g) + 8*(2*c*d - b*e)*(-7*c*e*f + 13*c*d*g - 3*b*
e*g)*(d + e*x) - 3*c*(9*b*e*g + 4*c*(e*f - 5*d*g))*(d + e*x)^2 - 6*c^2*e*g*x*(d
+ e*x)^2) + 15*Sqrt[c]*(2*c*d - b*e)*(4*c*e*f - 10*c*d*g + 3*b*e*g)*(d + e*x)^(3
/2)*Log[((-I)*e*(b + 2*c*x))/Sqrt[c] + 2*Sqrt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x)
]]))/(e^2*(d + e*x)^4*(-(b*e) + c*(d - e*x))^(5/2))

_______________________________________________________________________________________

Maple [B]  time = 0.029, size = 5227, normalized size = 14.9 \[ \text{output too large to display} \]

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)^(5/2)/(e*x+d)^5,x)

[Out]

result too large to display

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 1.84479, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^(5/2)*(g*x + f)/(e*x + d)^5,x, algorithm="fricas")

[Out]

[-1/48*(15*((4*(2*c^2*d*e^3 - b*c*e^4)*f - (20*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^
2*e^4)*g)*x^2 + 4*(2*c^2*d^3*e - b*c*d^2*e^2)*f - (20*c^2*d^4 - 16*b*c*d^3*e + 3
*b^2*d^2*e^2)*g + 2*(4*(2*c^2*d^2*e^2 - b*c*d*e^3)*f - (20*c^2*d^3*e - 16*b*c*d^
2*e^2 + 3*b^2*d*e^3)*g)*x)*sqrt(-c)*log(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 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x +
b*e)*sqrt(-c)) - 4*(6*c^2*e^3*g*x^3 + 3*(4*c^2*e^3*f - (16*c^2*d*e^2 - 9*b*c*e^3
)*g)*x^2 + 4*(23*c^2*d^2*e - 6*b*c*d*e^2 - 2*b^2*e^3)*f - (236*c^2*d^3 - 147*b*c
*d^2*e + 16*b^2*d*e^2)*g + 2*(4*(17*c^2*d*e^2 - 7*b*c*e^3)*f - (161*c^2*d^2*e -
103*b*c*d*e^2 + 12*b^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(e
^4*x^2 + 2*d*e^3*x + d^2*e^2), 1/24*(15*((4*(2*c^2*d*e^3 - b*c*e^4)*f - (20*c^2*
d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*g)*x^2 + 4*(2*c^2*d^3*e - b*c*d^2*e^2)*f - (
20*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2)*g + 2*(4*(2*c^2*d^2*e^2 - b*c*d*e^3)*
f - (20*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*g)*x)*sqrt(c)*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))) + 2*(6*c^2*e^3*
g*x^3 + 3*(4*c^2*e^3*f - (16*c^2*d*e^2 - 9*b*c*e^3)*g)*x^2 + 4*(23*c^2*d^2*e - 6
*b*c*d*e^2 - 2*b^2*e^3)*f - (236*c^2*d^3 - 147*b*c*d^2*e + 16*b^2*d*e^2)*g + 2*(
4*(17*c^2*d*e^2 - 7*b*c*e^3)*f - (161*c^2*d^2*e - 103*b*c*d*e^2 + 12*b^2*e^3)*g)
*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)]

_______________________________________________________________________________________

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)**(5/2)/(e*x+d)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)^(5/2)*(g*x + f)/(e*x + d)^5,x, algorithm="giac")

[Out]

Timed out