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3.2198 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=346 \[ -\frac{(2 c d-b e)^5 (5 b e g-2 c (6 e f-d g)) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{7/2} e^2}-\frac{(b+2 c x) (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (5 b e g-2 c (6 e f-d g))}{512 c^3 e}-\frac{(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (5 b e g-2 c (6 e f-d g))}{192 c^2 e}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-5 b e g-2 c d g+12 c e f)}{60 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6 c e^2 (d+e x)} \]

[Out]

-((2*c*d - b*e)^3*(5*b*e*g - 2*c*(6*e*f - d*g))*(b + 2*c*x)*Sqrt[d*(c*d - b*e) -
 b*e^2*x - c*e^2*x^2])/(512*c^3*e) - ((2*c*d - b*e)*(5*b*e*g - 2*c*(6*e*f - d*g)
)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(192*c^2*e) + ((12*c*
e*f - 2*c*d*g - 5*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(60*c*e^2)
 - (g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(6*c*e^2*(d + e*x)) - ((2*c*d
 - b*e)^5*(5*b*e*g - 2*c*(6*e*f - d*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])])/(1024*c^(7/2)*e^2)

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Rubi [A]  time = 1.01777, antiderivative size = 346, 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 \[ -\frac{(2 c d-b e)^5 (5 b e g-2 c (6 e f-d g)) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{1024 c^{7/2} e^2}-\frac{(b+2 c x) (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (5 b e g-2 c (6 e f-d g))}{512 c^3 e}-\frac{(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (5 b e g-2 c (6 e f-d g))}{192 c^2 e}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-5 b e g-2 c d g+12 c e f)}{60 c e^2}-\frac{g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{6 c e^2 (d+e x)} \]

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),x]

[Out]

-((2*c*d - b*e)^3*(5*b*e*g - 2*c*(6*e*f - d*g))*(b + 2*c*x)*Sqrt[d*(c*d - b*e) -
 b*e^2*x - c*e^2*x^2])/(512*c^3*e) - ((2*c*d - b*e)*(5*b*e*g - 2*c*(6*e*f - d*g)
)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(192*c^2*e) + ((12*c*
e*f - 2*c*d*g - 5*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(60*c*e^2)
 - (g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(6*c*e^2*(d + e*x)) - ((2*c*d
 - b*e)^5*(5*b*e*g - 2*c*(6*e*f - d*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])])/(1024*c^(7/2)*e^2)

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Rubi in Sympy [A]  time = 54.9983, size = 323, normalized size = 0.93 \[ - \frac{g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{6 c e^{2} \left (d + e x\right )} - \frac{\left (\frac{5 b e g}{2} + c d g - 6 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{30 c e^{2}} + \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (5 b e g + 2 c d g - 12 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{192 c^{2} e} + \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right )^{3} \left (5 b e g + 2 c d g - 12 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{512 c^{3} e} + \frac{\left (b e - 2 c d\right )^{5} \left (5 b e g + 2 c d g - 12 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{1024 c^{\frac{7}{2}} e^{2}} \]

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),x)

[Out]

-g*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(6*c*e**2*(d + e*x)) - (5*b
*e*g/2 + c*d*g - 6*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(30*
c*e**2) + (b + 2*c*x)*(b*e - 2*c*d)*(5*b*e*g + 2*c*d*g - 12*c*e*f)*(-b*e**2*x -
c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(192*c**2*e) + (b + 2*c*x)*(b*e - 2*c*d)**3
*(5*b*e*g + 2*c*d*g - 12*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(
512*c**3*e) + (b*e - 2*c*d)**5*(5*b*e*g + 2*c*d*g - 12*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))))/(1024*c**(7/2)*e*
*2)

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Mathematica [C]  time = 6.0921, size = 476, normalized size = 1.38 \[ \frac{((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{\sqrt{c} \left (150 b^5 e^5 g-20 b^4 c e^4 (62 d g+18 e f+5 e g x)+80 b^3 c^2 e^3 \left (47 d^2 g+d e (39 f+9 g x)+e^2 x (3 f+g x)\right )-96 b^2 c^3 e^2 \left (67 d^3 g+d^2 e (43 f+9 g x)-d e^2 x (109 f+68 g x)-e^3 x^2 (62 f+45 g x)\right )+32 b c^4 e \left (207 d^4 g-6 d^3 e (7 f+3 g x)-6 d^2 e^2 x (107 f+67 g x)+4 d e^3 x^2 (3 f+2 g x)+4 e^4 x^3 (63 f+50 g x)\right )-64 c^5 \left (48 d^5 g-3 d^4 e (16 f+5 g x)-6 d^3 e^2 x (25 f+16 g x)+2 d^2 e^3 x^2 (48 f+35 g x)+12 d e^4 x^3 (5 f+4 g x)-8 e^5 x^4 (6 f+5 g x)\right )\right )}{(d+e x)^2 (b e-c d+c e x)^2}-\frac{15 i (2 c d-b e)^5 (5 b e g+2 c (d g-6 e f)) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{(d+e x)^{5/2} (c (d-e x)-b e)^{5/2}}\right )}{15360 c^{7/2} e^2} \]

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),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((Sqrt[c]*(150*b^5*e^5*g - 20*b^4*c*e^
4*(18*e*f + 62*d*g + 5*e*g*x) + 80*b^3*c^2*e^3*(47*d^2*g + e^2*x*(3*f + g*x) + d
*e*(39*f + 9*g*x)) - 64*c^5*(48*d^5*g + 12*d*e^4*x^3*(5*f + 4*g*x) - 8*e^5*x^4*(
6*f + 5*g*x) - 3*d^4*e*(16*f + 5*g*x) - 6*d^3*e^2*x*(25*f + 16*g*x) + 2*d^2*e^3*
x^2*(48*f + 35*g*x)) + 32*b*c^4*e*(207*d^4*g + 4*d*e^3*x^2*(3*f + 2*g*x) - 6*d^3
*e*(7*f + 3*g*x) + 4*e^4*x^3*(63*f + 50*g*x) - 6*d^2*e^2*x*(107*f + 67*g*x)) - 9
6*b^2*c^3*e^2*(67*d^3*g + d^2*e*(43*f + 9*g*x) - e^3*x^2*(62*f + 45*g*x) - d*e^2
*x*(109*f + 68*g*x))))/((d + e*x)^2*(-(c*d) + b*e + c*e*x)^2) - ((15*I)*(2*c*d -
 b*e)^5*(5*b*e*g + 2*c*(-6*e*f + d*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))))/(15360*c^(7/2)*e^2)

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Maple [B]  time = 0.024, size = 3533, normalized size = 10.2 \[ \text{output too large to display} \]

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),x)

[Out]

1/5/e*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(5/2)*f-1/16*e*b^2/c*(-c*(d/e+
x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*f-3/8/e*d^6*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))*g+9/64*e^2*b^3/c*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(
1/2)*d*f-3/256*e^5*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))*f+15/32*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+1/4*d*c*(-c*(d/e+x)^2*e^2+(-b
*e^2+2*c*d*e)*(d/e+x))^(3/2)*x*f+3/8*d^3*c^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*
(d/e+x))^(1/2)*x*f-1/4/e*d^2*c*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)
*x*g-3/8/e*d^4*c^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*g-9/32*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+9/32*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-9/64*e*b^3/c*(-c*(d/e+x)^2*e^
2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d^2*g+9/16*b*c*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d
*e)*(d/e+x))^(1/2)*x*d^3*g-15/32*e^3*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-3/64*e^3*b^3/c*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*f+3/1
28*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)*d*g-15/16*g*c^2
/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(1/2))*b*d^5+5/96*g*e/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*b^2+15/32*g*e*
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b^2*d^2+15/64*g*e/c*(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(1/2)*b^3*d^2+5/24*g/e*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*
d^2-5/64*g*e^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^4*d+5/16*g/e*c^2*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4+5/16*g/e*c^3/(c*e^2)^(1/2)*arctan((c*e
^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^6-5/8*g*c*(-c*e^
2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b*d^3+5/256*g*e^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(1/2)*x*b^4+5/32*g/e*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^4+5/
1024*g*e^5/c^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(1/2))*b^6-25/32*g*e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*
b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^3*d^3+5/512*g*e^3/c^3*(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^5-3/16/e*d^4*c*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*
(d/e+x))^(1/2)*b*g+15/16*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^5*g+75
/64*g*e*c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(1/2))*b^2*d^4-5/32*g*e^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b
^3*d+75/256*g*e^3/c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b
*e^2*x-b*d*e+c*d^2)^(1/2))*b^4*d^2-15/256*g*e^4/c^2/(c*e^2)^(1/2)*arctan((c*e^2)
^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^5*d+3/64*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*d*g-9/16*e*b*c*(-c*(d/e+x)^2
*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*d^2*f-15/128*e^3*b^4/c/(c*e^2)^(1/2)*arct
an((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-15/16*e*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*f+3/256*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))*d*g+15/128*
e^4*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*f-15/16*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^4*g+15/16*e^2*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*f+9/32*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d^
3*g+1/8*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*b*f-1/5/e^2*(-c*(d/e
+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(5/2)*d*g-5/16*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(1/2)*b^2*d^3+3/16*d^3*c*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)
*b*f-3/128*e^3*b^4/c^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*f-9/32*
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-1/8/e*d^2*(-c*(d/e
+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*b*g+1/8*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*
c*d*e)*(d/e+x))^(3/2)*x*d*g+3/8*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))
*f+1/12*g/e/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*b+5/192*g*e/c^2*(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(3/2)*b^3+5/48*g/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*
b*d^2+1/16*b^2/c*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*d*g-1/8*e*b*(
-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x*f-5/48*g/c*(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(3/2)*b^2*d+1/6*g/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x-5/24
*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*b*d

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.32483, 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),x, algorithm="fricas")

[Out]

[1/30720*(4*(1280*c^5*e^5*g*x^5 + 128*(12*c^5*e^5*f - (12*c^5*d*e^4 - 25*b*c^4*e
^5)*g)*x^4 - 16*(12*(10*c^5*d*e^4 - 21*b*c^4*e^5)*f + (140*c^5*d^2*e^3 - 8*b*c^4
*d*e^4 - 135*b^2*c^3*e^5)*g)*x^3 - 8*(12*(32*c^5*d^2*e^3 - 2*b*c^4*d*e^4 - 31*b^
2*c^3*e^5)*f - (384*c^5*d^3*e^2 - 804*b*c^4*d^2*e^3 + 408*b^2*c^3*d*e^4 + 5*b^3*
c^2*e^5)*g)*x^2 + 12*(128*c^5*d^4*e - 56*b*c^4*d^3*e^2 - 172*b^2*c^3*d^2*e^3 + 1
30*b^3*c^2*d*e^4 - 15*b^4*c*e^5)*f - (1536*c^5*d^5 - 3312*b*c^4*d^4*e + 3216*b^2
*c^3*d^3*e^2 - 1880*b^3*c^2*d^2*e^3 + 620*b^4*c*d*e^4 - 75*b^5*e^5)*g + 2*(12*(2
00*c^5*d^3*e^2 - 428*b*c^4*d^2*e^3 + 218*b^2*c^3*d*e^4 + 5*b^3*c^2*e^5)*f + (240
*c^5*d^4*e - 144*b*c^4*d^3*e^2 - 216*b^2*c^3*d^2*e^3 + 180*b^3*c^2*d*e^4 - 25*b^
4*c*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-c) + 15*(12*(32*
c^6*d^5*e - 80*b*c^5*d^4*e^2 + 80*b^2*c^4*d^3*e^3 - 40*b^3*c^3*d^2*e^4 + 10*b^4*
c^2*d*e^5 - b^5*c*e^6)*f - (64*c^6*d^6 - 240*b^2*c^4*d^4*e^2 + 320*b^3*c^3*d^3*e
^3 - 180*b^4*c^2*d^2*e^4 + 48*b^5*c*d*e^5 - 5*b^6*e^6)*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)))/(sqrt(-c)*c^3*e^2), 1/15360*(2*(1280
*c^5*e^5*g*x^5 + 128*(12*c^5*e^5*f - (12*c^5*d*e^4 - 25*b*c^4*e^5)*g)*x^4 - 16*(
12*(10*c^5*d*e^4 - 21*b*c^4*e^5)*f + (140*c^5*d^2*e^3 - 8*b*c^4*d*e^4 - 135*b^2*
c^3*e^5)*g)*x^3 - 8*(12*(32*c^5*d^2*e^3 - 2*b*c^4*d*e^4 - 31*b^2*c^3*e^5)*f - (3
84*c^5*d^3*e^2 - 804*b*c^4*d^2*e^3 + 408*b^2*c^3*d*e^4 + 5*b^3*c^2*e^5)*g)*x^2 +
 12*(128*c^5*d^4*e - 56*b*c^4*d^3*e^2 - 172*b^2*c^3*d^2*e^3 + 130*b^3*c^2*d*e^4
- 15*b^4*c*e^5)*f - (1536*c^5*d^5 - 3312*b*c^4*d^4*e + 3216*b^2*c^3*d^3*e^2 - 18
80*b^3*c^2*d^2*e^3 + 620*b^4*c*d*e^4 - 75*b^5*e^5)*g + 2*(12*(200*c^5*d^3*e^2 -
428*b*c^4*d^2*e^3 + 218*b^2*c^3*d*e^4 + 5*b^3*c^2*e^5)*f + (240*c^5*d^4*e - 144*
b*c^4*d^3*e^2 - 216*b^2*c^3*d^2*e^3 + 180*b^3*c^2*d*e^4 - 25*b^4*c*e^5)*g)*x)*sq
rt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c) + 15*(12*(32*c^6*d^5*e - 80*b*c
^5*d^4*e^2 + 80*b^2*c^4*d^3*e^3 - 40*b^3*c^3*d^2*e^4 + 10*b^4*c^2*d*e^5 - b^5*c*
e^6)*f - (64*c^6*d^6 - 240*b^2*c^4*d^4*e^2 + 320*b^3*c^3*d^3*e^3 - 180*b^4*c^2*d
^2*e^4 + 48*b^5*c*d*e^5 - 5*b^6*e^6)*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^(7/2)*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),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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),x, algorithm="giac")

[Out]

Exception raised: TypeError

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