∖int(f+gx)∖left(cd2−bde−be2x−ce2x2∖right)5∕2 ____________________________________________________________________(d+ex)3∖,dx

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3.2200 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=354 \[ \frac{5 (2 c d-b e)^3 (-b e g-6 c d g+8 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{128 c^{3/2} e^2}+\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^3 (2 c d-b e)}+\frac{5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-6 c d g+8 c e f)}{24 e^2}+\frac{(-b e+c d-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-6 c d g+8 c e f)}{4 e^2 (2 c d-b e)}+\frac{5 (b+2 c x) (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-6 c d g+8 c e f)}{64 c e} \]

[Out]

(5*(2*c*d - b*e)*(8*c*e*f - 6*c*d*g - b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*

e^2*x - c*e^2*x^2])/(64*c*e) + (5*(8*c*e*f - 6*c*d*g - b*e*g)*(d*(c*d - b*e) - b

*e^2*x - c*e^2*x^2)^(3/2))/(24*e^2) + ((8*c*e*f - 6*c*d*g - b*e*g)*(c*d - b*e -

c*e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(4*e^2*(2*c*d - b*e)) + (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)^3) + (5*(2*c*d - b*e)^3*(8*c*e*f - 6*c*d*g - 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])])/(128*c^(3/2)*e^2)

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Rubi [A] time = 1.36636, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.159 \[ \frac{5 (2 c d-b e)^3 (-b e g-6 c d g+8 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{128 c^{3/2} e^2}+\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^3 (2 c d-b e)}+\frac{5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-6 c d g+8 c e f)}{24 e^2}+\frac{(-b e+c d-c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-b e g-6 c d g+8 c e f)}{4 e^2 (2 c d-b e)}+\frac{5 (b+2 c x) (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-b e g-6 c d g+8 c e f)}{64 c e} \]

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)^3,x]

[Out]

(5*(2*c*d - b*e)*(8*c*e*f - 6*c*d*g - b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*

e^2*x - c*e^2*x^2])/(64*c*e) + (5*(8*c*e*f - 6*c*d*g - b*e*g)*(d*(c*d - b*e) - b

*e^2*x - c*e^2*x^2)^(3/2))/(24*e^2) + ((8*c*e*f - 6*c*d*g - b*e*g)*(c*d - b*e -

c*e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(4*e^2*(2*c*d - b*e)) + (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)^3) + (5*(2*c*d - b*e)^3*(8*c*e*f - 6*c*d*g - 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])])/(128*c^(3/2)*e^2)

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Rubi in Sympy [A] time = 161.135, size = 333, normalized size = 0.94 \[ - \frac{5 \left (b e g + 6 c d g - 8 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{24 e^{2}} - \frac{\left (b e - c d + c e x\right ) \left (b e g + 6 c d g - 8 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{4 e^{2} \left (b e - 2 c d\right )} + \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{e^{2} \left (d + e x\right )^{3} \left (b e - 2 c d\right )} + \frac{5 \left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (b e g + 6 c d g - 8 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{64 c e} + \frac{5 \left (b e - 2 c d\right )^{3} \left (b e g + 6 c d g - 8 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{128 c^{\frac{3}{2}} e^{2}} \]

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)**3,x)

[Out]

-5*(b*e*g + 6*c*d*g - 8*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)

/(24*e**2) - (b*e - c*d + c*e*x)*(b*e*g + 6*c*d*g - 8*c*e*f)*(-b*e**2*x - c*e**2

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

+ 2*c*x)*(b*e - 2*c*d)*(b*e*g + 6*c*d*g - 8*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2

+ d*(-b*e + c*d))/(64*c*e) + 5*(b*e - 2*c*d)**3*(b*e*g + 6*c*d*g - 8*c*e*f)*ata

n(-e*(-b - 2*c*x)/(2*sqrt(c)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))))/(1

28*c**(3/2)*e**2)

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Mathematica [C] time = 2.40496, size = 295, normalized size = 0.83 \[ \frac{((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{\sqrt{c} \left (30 b^3 e^3 g+4 b^2 c e^2 (-118 d g+132 e f+59 e g x)+8 b c^2 e \left (173 d^2 g-2 d e (106 f+51 g x)+2 e^2 x (26 f+17 g x)\right )-16 c^3 \left (72 d^3 g-d^2 e (88 f+45 g x)+12 d e^2 x (3 f+2 g x)-2 e^3 x^2 (4 f+3 g x)\right )\right )}{(d+e x)^2 (b e-c d+c e x)^2}-\frac{15 i (2 c d-b e)^3 (b e g+6 c d g-8 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{(d+e x)^{5/2} (c (d-e x)-b e)^{5/2}}\right )}{384 c^{3/2} e^2} \]

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)^3,x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((Sqrt[c]*(30*b^3*e^3*g + 4*b^2*c*e^2*

(132*e*f - 118*d*g + 59*e*g*x) - 16*c^3*(72*d^3*g + 12*d*e^2*x*(3*f + 2*g*x) - 2

*e^3*x^2*(4*f + 3*g*x) - d^2*e*(88*f + 45*g*x)) + 8*b*c^2*e*(173*d^2*g + 2*e^2*x

*(26*f + 17*g*x) - 2*d*e*(106*f + 51*g*x))))/((d + e*x)^2*(-(c*d) + b*e + c*e*x)

^2) - ((15*I)*(2*c*d - b*e)^3*(-8*c*e*f + 6*c*d*g + b*e*g)*Log[((-I)*e*(b + 2*c*

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

*e) + c*(d - e*x))^(5/2))))/(384*c^(3/2)*e^2)

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Maple [B] time = 0.029, size = 4726, normalized size = 13.4 \[ \text{output too large to display} \]

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)^3,x)

[Out]

-15*e^3*c^3/(-b*e^2+2*c*d*e)^2*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/

2)*x*d^2*f+25/8*e^6*c/(-b*e^2+2*c*d*e)^2*b^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*

(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(

1/2))*d*f+25*e^2*c^4/(-b*e^2+2*c*d*e)^2*b/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+

d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2

))*d^5*g-25*e^3*c^4/(-b*e^2+2*c*d*e)^2*b/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d

/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)

)*d^4*f-15/8*g*e*c^2/(-b*e^2+2*c*d*e)*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+

x))^(1/2)*x*d^2+15*e^2*c^3/(-b*e^2+2*c*d*e)^2*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*

e)*(d/e+x))^(1/2)*x*d^3*g-25*e^3*c^3/(-b*e^2+2*c*d*e)^2*b^2/(c*e^2)^(1/2)*arctan

((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*

d*e)*(d/e+x))^(1/2))*d^4*g+25*e^4*c^3/(-b*e^2+2*c*d*e)^2*b^2/(c*e^2)^(1/2)*arcta

n((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c

*d*e)*(d/e+x))^(1/2))*d^3*f+15/2*e^4*c^2/(-b*e^2+2*c*d*e)^2*b^2*(-c*(d/e+x)^2*e^

2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*d*f+25/2*e^4*c^2/(-b*e^2+2*c*d*e)^2*b^3/(c*e

^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*

e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d^3*g-25/8*e^5*c/(-b*e^2+2*c*d*e)^2*b^4/(c*

e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2

*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d^2*g-5/24*g*e/(-b*e^2+2*c*d*e)*b^2*(-c*(d

/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)+5/8*g*c^2/(-b*e^2+2*c*d*e)*d^3*(-c*(

d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*b+5/4*g*c^4/(-b*e^2+2*c*d*e)*d^5/(c

*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^

2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))-2/e^4/(-b*e^2+2*c*d*e)/(d/e+x)^3*(-c*(d/e

+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(7/2)*d*g+16/3/e*c/(-b*e^2+2*c*d*e)^2/(d/e+x

)^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(7/2)*f-5/16*e^7/(-b*e^2+2*c*d*e

)^2*b^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-

c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*f-5/3*e^3*c/(-b*e^2+2*c*d*e)^2*

b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*f+5/8*e^4/(-b*e^2+2*c*d*e)

^2*b^4*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d*g-5/8*e^5/(-b*e^2+2*c

*d*e)^2*b^4*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*f-16/3*c^2/(-b*e^2

+2*c*d*e)^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(5/2)*d*g+2/3*g/e^3/(-b*

e^2+2*c*d*e)/(d/e+x)^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(7/2)+2/3*g/e

*c/(-b*e^2+2*c*d*e)*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(5/2)-20/3*e*c^3

/(-b*e^2+2*c*d*e)^2*d^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x*g+16

/3*e*c^2/(-b*e^2+2*c*d*e)^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(5/2)*f-

5/128*g*e^5/c/(-b*e^2+2*c*d*e)*b^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2

*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))+2/e^

3/(-b*e^2+2*c*d*e)/(d/e+x)^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(7/2)*f

+20/3*e^2*c^3/(-b*e^2+2*c*d*e)^2*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(

3/2)*x*f-10/3*e*c^2/(-b*e^2+2*c*d*e)^2*d^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d

/e+x))^(3/2)*b*g+10/3*e^2*c^2/(-b*e^2+2*c*d*e)^2*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c

*d*e)*(d/e+x))^(3/2)*b*f-10*e*c^4/(-b*e^2+2*c*d*e)^2*d^4*(-c*(d/e+x)^2*e^2+(-b*e

^2+2*c*d*e)*(d/e+x))^(1/2)*x*g-15/4*e^3*c/(-b*e^2+2*c*d*e)^2*b^3*(-c*(d/e+x)^2*e

^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d^2*g+15/4*e^4*c/(-b*e^2+2*c*d*e)^2*b^3*(-c*(

d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d*f-5/12*g*e*c/(-b*e^2+2*c*d*e)*b*(

-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x-5/4*e^5*c/(-b*e^2+2*c*d*e)^2*

b^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*f-15/2*e^3*c^2/(-b*e^2+2

*c*d*e)^2*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*d^2*g-25/8*g*e

*c^3/(-b*e^2+2*c*d*e)*b/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*

c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d^4+10/3*e^2*c^

2/(-b*e^2+2*c*d*e)^2*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x*d*g+1

5/16*g*e^2*c/(-b*e^2+2*c*d*e)*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1

/2)*x*d-25/16*g*e^3*c/(-b*e^2+2*c*d*e)*b^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x

+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/

2))*d^2+5/4*e^4*c/(-b*e^2+2*c*d*e)^2*b^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e

+x))^(1/2)*x*d*g-25/2*e^5*c^2/(-b*e^2+2*c*d*e)^2*b^3/(c*e^2)^(1/2)*arctan((c*e^2

)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d

/e+x))^(1/2))*d^2*f+25/8*g*e^2*c^2/(-b*e^2+2*c*d*e)*b^2/(c*e^2)^(1/2)*arctan((c*

e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)

*(d/e+x))^(1/2))*d^3+25/64*g*e^4/(-b*e^2+2*c*d*e)*b^4/(c*e^2)^(1/2)*arctan((c*e^

2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(

d/e+x))^(1/2))*d+10*e^2*c^4/(-b*e^2+2*c*d*e)^2*d^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c

*d*e)*(d/e+x))^(1/2)*x*f-5*e*c^3/(-b*e^2+2*c*d*e)^2*d^4*(-c*(d/e+x)^2*e^2+(-b*e^

2+2*c*d*e)*(d/e+x))^(1/2)*b*g+5*e^2*c^3/(-b*e^2+2*c*d*e)^2*d^3*(-c*(d/e+x)^2*e^2

+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*b*f-10*e*c^5/(-b*e^2+2*c*d*e)^2*d^6/(c*e^2)^(1/

2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b

*e^2+2*c*d*e)*(d/e+x))^(1/2))*g+10*e^2*c^5/(-b*e^2+2*c*d*e)^2*d^5/(c*e^2)^(1/2)*

arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e^2+(-b*e^

2+2*c*d*e)*(d/e+x))^(1/2))*f-16/3/e^2*c/(-b*e^2+2*c*d*e)^2/(d/e+x)^2*(-c*(d/e+x)

^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(7/2)*d*g+5/16*e^6/(-b*e^2+2*c*d*e)^2*b^5/(c*e^

2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*(d/e+x)^2*e

^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2))*d*g-10/3*e^3*c^2/(-b*e^2+2*c*d*e)^2*b*(-c*(d

/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x*f+5/3*e^2*c/(-b*e^2+2*c*d*e)^2*b^2

*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*d*g+15/2*e^2*c^2/(-b*e^2+2*c*

d*e)^2*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d^3*g-15/2*e^3*c^2/

(-b*e^2+2*c*d*e)^2*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d^2*f-1

5/16*g*e*c/(-b*e^2+2*c*d*e)*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2

)*d^2-5/64*g*e^3/c/(-b*e^2+2*c*d*e)*b^4*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+

x))^(1/2)-5/32*g*e^3/(-b*e^2+2*c*d*e)*b^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/

e+x))^(1/2)*x+15/32*g*e^2/(-b*e^2+2*c*d*e)*b^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e

)*(d/e+x))^(1/2)*d+5/6*g*c^2/(-b*e^2+2*c*d*e)*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*

e)*(d/e+x))^(3/2)*x+5/12*g*c/(-b*e^2+2*c*d*e)*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*

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

*d*e)*(d/e+x))^(1/2)*x

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A] time = 1.04245, 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)^3,x, algorithm="fricas")

[Out]

[1/768*(4*(48*c^3*e^3*g*x^3 + 8*(8*c^3*e^3*f - (24*c^3*d*e^2 - 17*b*c^2*e^3)*g)*

x^2 + 8*(88*c^3*d^2*e - 106*b*c^2*d*e^2 + 33*b^2*c*e^3)*f - (576*c^3*d^3 - 692*b

*c^2*d^2*e + 236*b^2*c*d*e^2 - 15*b^3*e^3)*g - 2*(8*(18*c^3*d*e^2 - 13*b*c^2*e^3

)*f - (180*c^3*d^2*e - 204*b*c^2*d*e^2 + 59*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*(8*(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2

*c^2*d*e^3 - b^3*c*e^4)*f - (48*c^4*d^4 - 64*b*c^3*d^3*e + 24*b^2*c^2*d^2*e^2 -

b^4*e^4)*g)*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)))/(s

qrt(-c)*c*e^2), 1/384*(2*(48*c^3*e^3*g*x^3 + 8*(8*c^3*e^3*f - (24*c^3*d*e^2 - 17

*b*c^2*e^3)*g)*x^2 + 8*(88*c^3*d^2*e - 106*b*c^2*d*e^2 + 33*b^2*c*e^3)*f - (576*

c^3*d^3 - 692*b*c^2*d^2*e + 236*b^2*c*d*e^2 - 15*b^3*e^3)*g - 2*(8*(18*c^3*d*e^2

- 13*b*c^2*e^3)*f - (180*c^3*d^2*e - 204*b*c^2*d*e^2 + 59*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*(8*(8*c^4*d^3*e - 12*b*c^3*d

^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*f - (48*c^4*d^4 - 64*b*c^3*d^3*e + 24*b^2*

c^2*d^2*e^2 - b^4*e^4)*g)*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^(3/2)*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A] time = 0.833775, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

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)^3,x, algorithm="giac")

[Out]

sage0*x

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