3.2199 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=354 \[ \frac{(2 c d-b e)^4 (-3 b e g-4 c d g+10 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{256 c^{5/2} e^2}+\frac{(b+2 c x) (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g-4 c d g+10 c e f)}{128 c^2 e}+\frac{(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g-4 c d g+10 c e f)}{48 c e}+\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2 (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-3 b e g-4 c d g+10 c e f)}{15 e^2 (2 c d-b e)} \]

[Out]

((2*c*d - b*e)^2*(10*c*e*f - 4*c*d*g - 3*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2])/(128*c^2*e) + ((10*c*e*f - 4*c*d*g - 3*b*e*g)*(b + 2*c*x)
*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(48*c*e) + ((10*c*e*f - 4*c*d*g -
3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(15*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))/(3*e^2*(2*c*d - b*e)*
(d + e*x)^2) + ((2*c*d - b*e)^4*(10*c*e*f - 4*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])])/(256*c^(5/2)*e^2)

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Rubi [A]  time = 1.22134, antiderivative size = 354, 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 \[ \frac{(2 c d-b e)^4 (-3 b e g-4 c d g+10 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{256 c^{5/2} e^2}+\frac{(b+2 c x) (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-3 b e g-4 c d g+10 c e f)}{128 c^2 e}+\frac{(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-3 b e g-4 c d g+10 c e f)}{48 c e}+\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{3 e^2 (d+e x)^2 (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-3 b e g-4 c d g+10 c e f)}{15 e^2 (2 c d-b e)} \]

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)^2,x]

[Out]

((2*c*d - b*e)^2*(10*c*e*f - 4*c*d*g - 3*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2])/(128*c^2*e) + ((10*c*e*f - 4*c*d*g - 3*b*e*g)*(b + 2*c*x)
*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(48*c*e) + ((10*c*e*f - 4*c*d*g -
3*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(15*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))/(3*e^2*(2*c*d - b*e)*
(d + e*x)^2) + ((2*c*d - b*e)^4*(10*c*e*f - 4*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])])/(256*c^(5/2)*e^2)

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Rubi in Sympy [A]  time = 49.7236, size = 335, normalized size = 0.95 \[ \frac{\left (3 b e g + 4 c d g - 10 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{15 e^{2} \left (b e - 2 c d\right )} + \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{3 e^{2} \left (d + e x\right )^{2} \left (b e - 2 c d\right )} - \frac{\left (b + 2 c x\right ) \left (3 b e g + 4 c d g - 10 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{48 c e} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right )^{2} \left (3 b e g + 4 c d g - 10 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{128 c^{2} e} - \frac{\left (b e - 2 c d\right )^{4} \left (3 b e g + 4 c d g - 10 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{256 c^{\frac{5}{2}} e^{2}} \]

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)**2,x)

[Out]

(3*b*e*g + 4*c*d*g - 10*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)
/(15*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)/(3*e**2*(d + e*x)**2*(b*e - 2*c*d)) - (b + 2*c*x)*(3*b*e*g + 4*c*d*g
- 10*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(48*c*e) - (b + 2*
c*x)*(b*e - 2*c*d)**2*(3*b*e*g + 4*c*d*g - 10*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**
2 + d*(-b*e + c*d))/(128*c**2*e) - (b*e - 2*c*d)**4*(3*b*e*g + 4*c*d*g - 10*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)
)))/(256*c**(5/2)*e**2)

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Mathematica [C]  time = 2.78421, size = 381, normalized size = 1.08 \[ \frac{(b e-c d+c e x)^2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{\sqrt{c} \left (-90 b^4 e^4 g+60 b^3 c e^3 (8 d g+5 e f+e g x)+8 b^2 c^2 e^2 \left (-199 d^2 g+d e (70 f+32 g x)+e^2 x (295 f+186 g x)\right )+16 b c^3 e \left (174 d^3 g-d^2 e (195 f+71 g x)-2 d e^2 x (125 f+82 g x)+2 e^3 x^2 (85 f+63 g x)\right )-32 c^4 \left (56 d^4 g-10 d^3 e (8 f+3 g x)-d^2 e^2 x (45 f+32 g x)+20 d e^3 x^2 (4 f+3 g x)-6 e^4 x^3 (5 f+4 g x)\right )\right )}{(b e-c d+c e x)^2}+\frac{15 i (b e-2 c d)^4 (-3 b e g-4 c d g+10 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{\sqrt{d+e x} (c (d-e x)-b e)^{5/2}}\right )}{3840 c^{5/2} e^2} \]

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)^2,x]

[Out]

((-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((Sqrt[c]*(-90*
b^4*e^4*g + 60*b^3*c*e^3*(5*e*f + 8*d*g + e*g*x) - 32*c^4*(56*d^4*g + 20*d*e^3*x
^2*(4*f + 3*g*x) - 10*d^3*e*(8*f + 3*g*x) - 6*e^4*x^3*(5*f + 4*g*x) - d^2*e^2*x*
(45*f + 32*g*x)) + 16*b*c^3*e*(174*d^3*g + 2*e^3*x^2*(85*f + 63*g*x) - d^2*e*(19
5*f + 71*g*x) - 2*d*e^2*x*(125*f + 82*g*x)) + 8*b^2*c^2*e^2*(-199*d^2*g + d*e*(7
0*f + 32*g*x) + e^2*x*(295*f + 186*g*x))))/(-(c*d) + b*e + c*e*x)^2 + ((15*I)*(-
2*c*d + b*e)^4*(10*c*e*f - 4*c*d*g - 3*b*e*g)*Log[((-I)*e*(b + 2*c*x))/Sqrt[c] +
2*Sqrt[d + e*x]*Sqrt[-(b*e) + c*(d - e*x)]])/(Sqrt[d + e*x]*(-(b*e) + c*(d - e*
x))^(5/2))))/(3840*c^(5/2)*e^2)

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Maple [B]  time = 0.03, size = 4215, normalized size = 11.9 \[ \text{output too large to display} \]

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)^2,x)

[Out]

2/3/e^2/(-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)*f+25/8*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^
5*g-25/16*e^4*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*f-25/8*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
^4*g+5/128*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))*d
*g-25/8*e^2*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*
f+15/16*e^3*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*f+5/12*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*d*g+25/16*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^3*g-15/16*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^2*g-15/8*e^2*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*f+15/8*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^3*g-3/128*g*e^2*b^4/c^2*(-c*(d/e+x)^2*e^
2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)+1/8*g/e*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(
d/e+x))^(3/2)*b+5/6*e*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*f+5/12*e*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*f+5/8*e*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*f+15/128*g*e^3*b^4/c/(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+15/16*g*e*b^2*c/(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+5/4*e*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*f-25/64*
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^2*g+25/64*e^
5/(-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*f+5/4*e*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))*f-5/128*e^6/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))*f+5/32*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*d*g+5/64*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)*d*g+15/16*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^3*g-15/
16*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)*
d^2*f-5/12*e^2*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*f-2/3/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)*d*g-5/4*c^3/(-b*e^2+2*c*d*e)*d^4*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*
e)*(d/e+x))^(1/2)*x*g-5/8*c^2/(-b*e^2+2*c*d*e)*d^4*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c
*d*e)*(d/e+x))^(1/2)*b*g-2/3/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)*d*g-5/32*e^4/(-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*f-5/64*e^4/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)*f+5/24*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)*d*g-15/32*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^2*g-5/6*c^2/(-b*e^2+2*c*d*e)*d^2*(-c*(d/e+x
)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x*g-5/12*c/(-b*e^2+2*c*d*e)*d^2*(-c*(d/e
+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*b*g+15/32*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)*d*f-3/64*g*e^2*b^3/c*(-c*(d/e+x)^
2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x+9/32*g*e*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2
*c*d*e)*(d/e+x))^(1/2)*x*d+1/4*g/e*d*c*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x
))^(3/2)*x+3/16*g/e*d^3*c*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*b-15
/16*g*b*c^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+3/8*g/e*d^3*c^2*(-c*(d/
e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x-9/16*g*b*c*(-c*(d/e+x)^2*e^2+(-b*e^
2+2*c*d*e)*(d/e+x))^(1/2)*x*d^2+3/8*g/e*d^5*c^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))+9/64*g*e*b^3/c*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d-3/25
6*g*e^4*b^5/c^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))-15/32*g*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^2-5/4*c^4/(-b*e^2+2*c*d*e)*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-5/24*e^2/(-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)*f+25/8*e^3*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*f-9/32*g*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(
d/e+x))^(1/2)*d^2-1/8*g*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x-1/
16*g*b^2/c*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)+2/3*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)*f+1/5*g/e^2*(-c*(d/e+x)^2*
e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(5/2)

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.0301, 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)^2,x, algorithm="fricas")

[Out]

[1/7680*(4*(384*c^4*e^4*g*x^4 + 48*(10*c^4*e^4*f - (20*c^4*d*e^3 - 21*b*c^3*e^4)
*g)*x^3 - 8*(10*(16*c^4*d*e^3 - 17*b*c^3*e^4)*f - (64*c^4*d^2*e^2 - 164*b*c^3*d*
e^3 + 93*b^2*c^2*e^4)*g)*x^2 + 10*(128*c^4*d^3*e - 156*b*c^3*d^2*e^2 + 28*b^2*c^
2*d*e^3 + 15*b^3*c*e^4)*f - (896*c^4*d^4 - 1392*b*c^3*d^3*e + 796*b^2*c^2*d^2*e^
2 - 240*b^3*c*d*e^3 + 45*b^4*e^4)*g + 2*(10*(36*c^4*d^2*e^2 - 100*b*c^3*d*e^3 +
59*b^2*c^2*e^4)*f + (240*c^4*d^3*e - 284*b*c^3*d^2*e^2 + 64*b^2*c^2*d*e^3 + 15*b
^3*c*e^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-c) - 15*(10*(16
*c^5*d^4*e - 32*b*c^4*d^3*e^2 + 24*b^2*c^3*d^2*e^3 - 8*b^3*c^2*d*e^4 + b^4*c*e^5
)*f - (64*c^5*d^5 - 80*b*c^4*d^4*e + 40*b^3*c^2*d^2*e^3 - 20*b^4*c*d*e^4 + 3*b^5
*e^5)*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)))/(sqr
t(-c)*c^2*e^2), 1/3840*(2*(384*c^4*e^4*g*x^4 + 48*(10*c^4*e^4*f - (20*c^4*d*e^3
- 21*b*c^3*e^4)*g)*x^3 - 8*(10*(16*c^4*d*e^3 - 17*b*c^3*e^4)*f - (64*c^4*d^2*e^2
- 164*b*c^3*d*e^3 + 93*b^2*c^2*e^4)*g)*x^2 + 10*(128*c^4*d^3*e - 156*b*c^3*d^2*
e^2 + 28*b^2*c^2*d*e^3 + 15*b^3*c*e^4)*f - (896*c^4*d^4 - 1392*b*c^3*d^3*e + 796
*b^2*c^2*d^2*e^2 - 240*b^3*c*d*e^3 + 45*b^4*e^4)*g + 2*(10*(36*c^4*d^2*e^2 - 100
*b*c^3*d*e^3 + 59*b^2*c^2*e^4)*f + (240*c^4*d^3*e - 284*b*c^3*d^2*e^2 + 64*b^2*c
^2*d*e^3 + 15*b^3*c*e^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c
) + 15*(10*(16*c^5*d^4*e - 32*b*c^4*d^3*e^2 + 24*b^2*c^3*d^2*e^3 - 8*b^3*c^2*d*e
^4 + b^4*c*e^5)*f - (64*c^5*d^5 - 80*b*c^4*d^4*e + 40*b^3*c^2*d^2*e^3 - 20*b^4*c
*d*e^4 + 3*b^5*e^5)*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^(5/2)*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)**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)^2,x, algorithm="giac")

[Out]

Timed out