3.2257 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=360 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (2 b e g-9 c d g+5 c e f)}{5 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (2 b e g-9 c d g+5 c e f)}{3 e^2 (d+e x)^{3/2}}-\frac{(2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-9 c d g+5 c e f)}{e^2 \sqrt{d+e x}}+\frac{(2 c d-b e)^{3/2} (2 b e g-9 c d g+5 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2} \]

[Out]

-(((2*c*d - b*e)*(5*c*e*f - 9*c*d*g + 2*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2])/(e^2*Sqrt[d + e*x])) - ((5*c*e*f - 9*c*d*g + 2*b*e*g)*(d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(d + e*x)^(3/2)) - ((5*c*e*f - 9*c*d*g + 2*b
*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*e^2*(2*c*d - b*e)*(d + e*x
)^(5/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(e^2*(2*c*d
- b*e)*(d + e*x)^(9/2)) + ((2*c*d - b*e)^(3/2)*(5*c*e*f - 9*c*d*g + 2*b*e*g)*Ar
cTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sqrt[d + e*x]
)])/e^2

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Rubi [A]  time = 1.35042, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (2 b e g-9 c d g+5 c e f)}{5 e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (2 b e g-9 c d g+5 c e f)}{3 e^2 (d+e x)^{3/2}}-\frac{(2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-9 c d g+5 c e f)}{e^2 \sqrt{d+e x}}+\frac{(2 c d-b e)^{3/2} (2 b e g-9 c d g+5 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2} \]

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)^(9/2),x]

[Out]

-(((2*c*d - b*e)*(5*c*e*f - 9*c*d*g + 2*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2])/(e^2*Sqrt[d + e*x])) - ((5*c*e*f - 9*c*d*g + 2*b*e*g)*(d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(d + e*x)^(3/2)) - ((5*c*e*f - 9*c*d*g + 2*b
*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*e^2*(2*c*d - b*e)*(d + e*x
)^(5/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(e^2*(2*c*d
- b*e)*(d + e*x)^(9/2)) + ((2*c*d - b*e)^(3/2)*(5*c*e*f - 9*c*d*g + 2*b*e*g)*Ar
cTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sqrt[d + e*x]
)])/e^2

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Rubi in Sympy [A]  time = 162.974, size = 340, normalized size = 0.94 \[ - \frac{\left (b e - 2 c d\right )^{\frac{3}{2}} \left (2 b e g - 9 c d g + 5 c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{e^{2}} + \frac{\left (b e - 2 c d\right ) \left (2 b e g - 9 c d g + 5 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \sqrt{d + e x}} - \frac{2 \left (b e g - \frac{9 c d g}{2} + \frac{5 c e f}{2}\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{\frac{3}{2}}} + \frac{\left (2 b e g - 9 c d g + 5 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{5 e^{2} \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )} - \frac{\left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{e^{2} \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right )} \]

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)**(9/2),x)

[Out]

-(b*e - 2*c*d)**(3/2)*(2*b*e*g - 9*c*d*g + 5*c*e*f)*atan(sqrt(-b*e**2*x - c*e**2
*x**2 + d*(-b*e + c*d))/(sqrt(d + e*x)*sqrt(b*e - 2*c*d)))/e**2 + (b*e - 2*c*d)*
(2*b*e*g - 9*c*d*g + 5*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(e*
*2*sqrt(d + e*x)) - 2*(b*e*g - 9*c*d*g/2 + 5*c*e*f/2)*(-b*e**2*x - c*e**2*x**2 +
d*(-b*e + c*d))**(3/2)/(3*e**2*(d + e*x)**(3/2)) + (2*b*e*g - 9*c*d*g + 5*c*e*f
)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(5*e**2*(d + e*x)**(5/2)*(b*
e - 2*c*d)) - (d*g - e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(e**
2*(d + e*x)**(9/2)*(b*e - 2*c*d))

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Mathematica [A]  time = 1.53525, size = 236, normalized size = 0.66 \[ \frac{((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{46 b^2 g+\frac{15 (b e-2 c d)^2 (d g-e f)}{e^2 (d+e x)}+\frac{2 c x (11 b e g-21 c d g+5 c e f)}{e}+b c \left (70 f-\frac{232 d g}{e}\right )+\frac{2 c^2 d (138 d g-65 e f)}{e^2}+6 c^2 g x^2}{15 (b e-c d+c e x)^2}-\frac{(2 c d-b e)^{3/2} (-2 b e g+9 c d g-5 c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{e^2 (c (d-e x)-b e)^{5/2}}\right )}{(d+e x)^{5/2}} \]

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)^(9/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((46*b^2*g + (2*c^2*d*(-65*e*f + 138*d
*g))/e^2 + b*c*(70*f - (232*d*g)/e) + (2*c*(5*c*e*f - 21*c*d*g + 11*b*e*g)*x)/e
+ 6*c^2*g*x^2 + (15*(-2*c*d + b*e)^2*(-(e*f) + d*g))/(e^2*(d + e*x)))/(15*(-(c*d
) + b*e + c*e*x)^2) - ((2*c*d - b*e)^(3/2)*(-5*c*e*f + 9*c*d*g - 2*b*e*g)*ArcTan
h[Sqrt[c*d - b*e - c*e*x]/Sqrt[2*c*d - b*e]])/(e^2*(-(b*e) + c*(d - e*x))^(5/2))
))/(d + e*x)^(5/2)

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Maple [B]  time = 0.041, size = 1136, normalized size = 3.2 \[ \text{result too large to display} \]

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)^(9/2),x)

[Out]

-1/15*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(15*b^2*e^3*f*(b*e-2*c*d)^(1/2)*(-c
*e*x-b*e+c*d)^(1/2)-255*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b^2*c
*d*e^3*g-300*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c^2*d*e^3*f-33
6*c^2*d^3*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+300*arctan((-c*e*x-b*e+c*d)
^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*e*f-540*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c
*d)^(1/2))*c^3*d^4*g+30*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b^3*e
^4*g+30*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^3*d*e^3*g+75*arctan((
-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*d*e^3*f-300*arctan((-c*e*x-b*e+c*
d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^2*e^2*f-255*arctan((-c*e*x-b*e+c*d)^(1/2)/(b
*e-2*c*d)^(1/2))*b^2*c*d^2*e^2*g+75*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1
/2))*x*b^2*c*e^4*f-6*x^3*c^2*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-70*x
*b*c*e^3*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-234*x*c^2*d^2*e*g*(b*e-2*c*d
)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+120*x*c^2*d*e^2*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c
*d)^(1/2)+292*b*c*d^2*e*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-130*b*c*d*e^2
*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+660*arctan((-c*e*x-b*e+c*d)^(1/2)/(b
*e-2*c*d)^(1/2))*x*b*c^2*d^2*e^2*g-22*x^2*b*c*e^3*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*
e+c*d)^(1/2)+36*x^2*c^2*d*e^2*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-10*x^2*
c^2*e^3*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-46*x*b^2*e^3*g*(b*e-2*c*d)^(1
/2)*(-c*e*x-b*e+c*d)^(1/2)-61*b^2*d*e^2*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/
2)+190*c^2*d^2*e*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-540*arctan((-c*e*x-b
*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^3*d^3*e*g+300*arctan((-c*e*x-b*e+c*d)^(1/2)
/(b*e-2*c*d)^(1/2))*x*c^3*d^2*e^2*f+660*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d
)^(1/2))*b*c^2*d^3*e*g+210*x*b*c*d*e^2*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2
))/(e*x+d)^(3/2)/(-c*e*x-b*e+c*d)^(1/2)/e^2/(b*e-2*c*d)^(1/2)

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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((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.31459, 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)^(9/2),x, algorithm="fricas")

[Out]

[-1/30*(12*c^3*e^4*g*x^4 + 4*(5*c^3*e^4*f - 7*(3*c^3*d*e^3 - 2*b*c^2*e^4)*g)*x^3
+ 15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*(5*(2*c^2*d*e
- b*c*e^2)*f - (18*c^2*d^2 - 13*b*c*d*e + 2*b^2*e^2)*g)*sqrt(e*x + d)*log(-(c*e
^2*x^2 - 3*c*d^2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x + 2*sqrt(-c*e^2*x^2 - b*e^2*x +
c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x + d))/(e^2*x^2 + 2*d*e*x + d^2)) - 4*
(5*(13*c^3*d*e^3 - 8*b*c^2*e^4)*f - (135*c^3*d^2*e^2 - 134*b*c^2*d*e^3 + 34*b^2*
c*e^4)*g)*x^2 + 10*(38*c^3*d^3*e - 64*b*c^2*d^2*e^2 + 29*b^2*c*d*e^3 - 3*b^3*e^4
)*f - 2*(336*c^3*d^4 - 628*b*c^2*d^3*e + 353*b^2*c*d^2*e^2 - 61*b^3*d*e^3)*g - 2
*(5*(14*c^3*d^2*e^2 + 12*b*c^2*d*e^3 - 11*b^2*c*e^4)*f - (102*c^3*d^3*e + 152*b*
c^2*d^2*e^2 - 195*b^2*c*d*e^3 + 46*b^3*e^4)*g)*x)/(sqrt(-c*e^2*x^2 - b*e^2*x + c
*d^2 - b*d*e)*sqrt(e*x + d)*e^2), -1/15*(6*c^3*e^4*g*x^4 + 2*(5*c^3*e^4*f - 7*(3
*c^3*d*e^3 - 2*b*c^2*e^4)*g)*x^3 - 15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)
*sqrt(-2*c*d + b*e)*(5*(2*c^2*d*e - b*c*e^2)*f - (18*c^2*d^2 - 13*b*c*d*e + 2*b^
2*e^2)*g)*sqrt(e*x + d)*arctan(-sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*
d - b*e)*sqrt(e*x + d)/((c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)*sqrt(-2*c*d + b*e)
)) - 2*(5*(13*c^3*d*e^3 - 8*b*c^2*e^4)*f - (135*c^3*d^2*e^2 - 134*b*c^2*d*e^3 +
34*b^2*c*e^4)*g)*x^2 + 5*(38*c^3*d^3*e - 64*b*c^2*d^2*e^2 + 29*b^2*c*d*e^3 - 3*b
^3*e^4)*f - (336*c^3*d^4 - 628*b*c^2*d^3*e + 353*b^2*c*d^2*e^2 - 61*b^3*d*e^3)*g
- (5*(14*c^3*d^2*e^2 + 12*b*c^2*d*e^3 - 11*b^2*c*e^4)*f - (102*c^3*d^3*e + 152*
b*c^2*d^2*e^2 - 195*b^2*c*d*e^3 + 46*b^3*e^4)*g)*x)/(sqrt(-c*e^2*x^2 - b*e^2*x +
c*d^2 - b*d*e)*sqrt(e*x + d)*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((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

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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)^(9/2),x, algorithm="giac")

[Out]

Timed out