3.2275 \(\int \frac{f+g x}{(d+e x)^{5/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx\)
Optimal. Leaf size=387 \[ \frac{5 c^2 \sqrt{d+e x} (-6 b e g+5 c d g+7 c e f)}{8 e^2 (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 c^2 (-6 b e g+5 c d g+7 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 (2 c d-b e)^{9/2}}-\frac{5 c (-6 b e g+5 c d g+7 c e f)}{24 e^2 \sqrt{d+e x} (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{-6 b e g+5 c d g+7 c e f}{12 e^2 (d+e x)^{3/2} (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{e f-d g}{3 e^2 (d+e x)^{5/2} (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]
[Out]
-(e*f - d*g)/(3*e^2*(2*c*d - b*e)*(d + e*x)^(5/2)*Sqrt[d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2]) - (7*c*e*f + 5*c*d*g - 6*b*e*g)/(12*e^2*(2*c*d - b*e)^2*(d + e*x)^(
3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (5*c*(7*c*e*f + 5*c*d*g - 6*b*
e*g))/(24*e^2*(2*c*d - b*e)^3*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2]) + (5*c^2*(7*c*e*f + 5*c*d*g - 6*b*e*g)*Sqrt[d + e*x])/(8*e^2*(2*c*d - b*e
)^4*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (5*c^2*(7*c*e*f + 5*c*d*g - 6*b
*e*g)*ArcTanh[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])])/(8*e^2*(2*c*d - b*e)^(9/2))
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Rubi [A] time = 1.49334, antiderivative size = 387, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.109
\[ \frac{5 c^2 \sqrt{d+e x} (-6 b e g+5 c d g+7 c e f)}{8 e^2 (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 c^2 (-6 b e g+5 c d g+7 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 (2 c d-b e)^{9/2}}-\frac{5 c (-6 b e g+5 c d g+7 c e f)}{24 e^2 \sqrt{d+e x} (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{-6 b e g+5 c d g+7 c e f}{12 e^2 (d+e x)^{3/2} (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{e f-d g}{3 e^2 (d+e x)^{5/2} (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(f + g*x)/((d + e*x)^(5/2)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2)),x]
[Out]
-(e*f - d*g)/(3*e^2*(2*c*d - b*e)*(d + e*x)^(5/2)*Sqrt[d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2]) - (7*c*e*f + 5*c*d*g - 6*b*e*g)/(12*e^2*(2*c*d - b*e)^2*(d + e*x)^(
3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (5*c*(7*c*e*f + 5*c*d*g - 6*b*
e*g))/(24*e^2*(2*c*d - b*e)^3*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2]) + (5*c^2*(7*c*e*f + 5*c*d*g - 6*b*e*g)*Sqrt[d + e*x])/(8*e^2*(2*c*d - b*e
)^4*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (5*c^2*(7*c*e*f + 5*c*d*g - 6*b
*e*g)*ArcTanh[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])])/(8*e^2*(2*c*d - b*e)^(9/2))
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Rubi in Sympy [A] time = 162.9, size = 369, normalized size = 0.95 \[ - \frac{5 c^{2} \sqrt{d + e x} \left (6 b e g - 5 c d g - 7 c e f\right )}{8 e^{2} \left (b e - 2 c d\right )^{4} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{5 c^{2} \left (6 b e g - 5 c d g - 7 c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{8 e^{2} \left (b e - 2 c d\right )^{\frac{9}{2}}} - \frac{5 c \left (6 b e g - 5 c d g - 7 c e f\right )}{24 e^{2} \sqrt{d + e x} \left (b e - 2 c d\right )^{3} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{6 b e g - 5 c d g - 7 c e f}{12 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right )^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{d g - e f}{3 e^{2} \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)/(e*x+d)**(5/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)
[Out]
-5*c**2*sqrt(d + e*x)*(6*b*e*g - 5*c*d*g - 7*c*e*f)/(8*e**2*(b*e - 2*c*d)**4*sqr
t(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))) - 5*c**2*(6*b*e*g - 5*c*d*g - 7*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)))/(8*e**2*(b*e - 2*c*d)**(9/2)) - 5*c*(6*b*e*g - 5*c*d*g - 7*c*e*f)/(24
*e**2*sqrt(d + e*x)*(b*e - 2*c*d)**3*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*
d))) + (6*b*e*g - 5*c*d*g - 7*c*e*f)/(12*e**2*(d + e*x)**(3/2)*(b*e - 2*c*d)**2*
sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))) - (d*g - e*f)/(3*e**2*(d + e*x)*
*(5/2)*(b*e - 2*c*d)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d)))
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Mathematica [A] time = 4.01052, size = 296, normalized size = 0.76 \[ \frac{(d+e x)^{3/2} \left (\frac{(c (d-e x)-b e) \left (48 c^2 (d+e x)^3 (-b e g+c d g+c e f)+3 c (d+e x)^2 (b e-c d+c e x) (-14 b e g+9 c d g+19 c e f)-2 (d+e x) (2 c d-b e) (c (d-e x)-b e) (-6 b e g+c d g+11 c e f)+8 (b e-2 c d)^2 (d g-e f) (c (d-e x)-b e)\right )}{(d+e x)^3 (b e-2 c d)^4}-\frac{15 c^2 (c (d-e x)-b e)^{3/2} (-6 b e g+5 c d g+7 c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(2 c d-b e)^{9/2}}\right )}{24 e^2 ((d+e x) (c (d-e x)-b e))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(f + g*x)/((d + e*x)^(5/2)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2)),x]
[Out]
((d + e*x)^(3/2)*(((-(b*e) + c*(d - e*x))*(48*c^2*(c*e*f + c*d*g - b*e*g)*(d + e
*x)^3 + 3*c*(19*c*e*f + 9*c*d*g - 14*b*e*g)*(d + e*x)^2*(-(c*d) + b*e + c*e*x) +
8*(-2*c*d + b*e)^2*(-(e*f) + d*g)*(-(b*e) + c*(d - e*x)) - 2*(2*c*d - b*e)*(11*
c*e*f + c*d*g - 6*b*e*g)*(d + e*x)*(-(b*e) + c*(d - e*x))))/((-2*c*d + b*e)^4*(d
+ e*x)^3) - (15*c^2*(7*c*e*f + 5*c*d*g - 6*b*e*g)*(-(b*e) + c*(d - e*x))^(3/2)*
ArcTanh[Sqrt[c*d - b*e - c*e*x]/Sqrt[2*c*d - b*e]])/(2*c*d - b*e)^(9/2)))/(24*e^
2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2))
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Maple [B] time = 0.053, size = 1224, 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)/(e*x+d)^(5/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)
[Out]
1/24/(e*x+d)^(7/2)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(90*arctan((-c*e*x-b*e
+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x^3*b*c^2*e^4*g+270*arctan
((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x*b*c^2*d^2*e^
2*g-4*(b*e-2*c*d)^(1/2)*b^3*d*e^3*g+85*(b*e-2*c*d)^(1/2)*c^3*d^3*e*f-75*arctan((
-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x^3*c^3*d*e^3*g-
225*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x^2*
c^3*d^2*e^2*g-315*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c
*d)^(1/2)*x^2*c^3*d*e^3*f+185*(b*e-2*c*d)^(1/2)*x^2*b*c^2*d*e^3*g-225*arctan((-c
*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x*c^3*d^3*e*g-315*
arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x*c^3*d^
2*e^2*f+118*(b*e-2*c*d)^(1/2)*x*b^2*c*d*e^3*g+12*(b*e-2*c*d)^(1/2)*x*b*c^2*d^2*e
^2*g-126*(b*e-2*c*d)^(1/2)*x*b*c^2*d*e^3*f+90*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e
-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*b*c^2*d^3*e*g-105*(b*e-2*c*d)^(1/2)*x^3*c^
3*e^4*f+40*(b*e-2*c*d)^(1/2)*b^2*c*d^2*e^2*g+62*(b*e-2*c*d)^(1/2)*b^2*c*d*e^3*f-
8*(b*e-2*c*d)^(1/2)*b^3*e^4*f-49*(b*e-2*c*d)^(1/2)*c^3*d^4*g+13*(b*e-2*c*d)^(1/2
)*b*c^2*d^3*e*g-187*(b*e-2*c*d)^(1/2)*b*c^2*d^2*e^2*f-12*(b*e-2*c*d)^(1/2)*x*b^3
*e^4*g-75*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2
)*c^3*d^4*g-75*(b*e-2*c*d)^(1/2)*x^3*c^3*d*e^3*g+30*(b*e-2*c*d)^(1/2)*x^2*b^2*c*
e^4*g-35*(b*e-2*c*d)^(1/2)*x^2*b*c^2*e^4*f-175*(b*e-2*c*d)^(1/2)*x^2*c^3*d^2*e^2
*g-245*(b*e-2*c*d)^(1/2)*x^2*c^3*d*e^3*f+14*(b*e-2*c*d)^(1/2)*x*b^2*c*e^4*f-85*(
b*e-2*c*d)^(1/2)*x*c^3*d^3*e*g-119*(b*e-2*c*d)^(1/2)*x*c^3*d^2*e^2*f-105*arctan(
(-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c^3*d^3*e*f-105
*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x^3*c^3
*e^4*f+90*(b*e-2*c*d)^(1/2)*x^3*b*c^2*e^4*g+270*arctan((-c*e*x-b*e+c*d)^(1/2)/(b
*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x^2*b*c^2*d*e^3*g)/(c*e*x+b*e-c*d)/e^2/(
b*e-2*c*d)^(9/2)
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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((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(e*x + d)^(5/2)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.367607, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(e*x + d)^(5/2)),x, algorithm="fricas")
[Out]
[1/48*(2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(15*(7*c^3*e^4*f + (5*c^3*d*
e^3 - 6*b*c^2*e^4)*g)*x^3 + 5*(7*(7*c^3*d*e^3 + b*c^2*e^4)*f + (35*c^3*d^2*e^2 -
37*b*c^2*d*e^3 - 6*b^2*c*e^4)*g)*x^2 - (85*c^3*d^3*e - 187*b*c^2*d^2*e^2 + 62*b
^2*c*d*e^3 - 8*b^3*e^4)*f + (49*c^3*d^4 - 13*b*c^2*d^3*e - 40*b^2*c*d^2*e^2 + 4*
b^3*d*e^3)*g + (7*(17*c^3*d^2*e^2 + 18*b*c^2*d*e^3 - 2*b^2*c*e^4)*f + (85*c^3*d^
3*e - 12*b*c^2*d^2*e^2 - 118*b^2*c*d*e^3 + 12*b^3*e^4)*g)*x)*sqrt(2*c*d - b*e)*s
qrt(e*x + d) + 15*((7*c^4*e^6*f + (5*c^4*d*e^5 - 6*b*c^3*e^6)*g)*x^5 + (7*(3*c^4
*d*e^5 + b*c^3*e^6)*f + (15*c^4*d^2*e^4 - 13*b*c^3*d*e^5 - 6*b^2*c^2*e^6)*g)*x^4
+ 2*(7*(c^4*d^2*e^4 + 2*b*c^3*d*e^5)*f + (5*c^4*d^3*e^3 + 4*b*c^3*d^2*e^4 - 12*
b^2*c^2*d*e^5)*g)*x^3 - 2*(7*(c^4*d^3*e^3 - 3*b*c^3*d^2*e^4)*f + (5*c^4*d^4*e^2
- 21*b*c^3*d^3*e^3 + 18*b^2*c^2*d^2*e^4)*g)*x^2 - 7*(c^4*d^5*e - b*c^3*d^4*e^2)*
f - (5*c^4*d^6 - 11*b*c^3*d^5*e + 6*b^2*c^2*d^4*e^2)*g - (7*(3*c^4*d^4*e^2 - 4*b
*c^3*d^3*e^3)*f + (15*c^4*d^5*e - 38*b*c^3*d^4*e^2 + 24*b^2*c^2*d^3*e^3)*g)*x)*l
og((2*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 - 3*c*d^2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x)*sqrt(2*c*d - b*e))/(e^2*x^2
+ 2*d*e*x + d^2)))/((16*c^5*d^9*e^2 - 48*b*c^4*d^8*e^3 + 56*b^2*c^3*d^7*e^4 - 3
2*b^3*c^2*d^6*e^5 + 9*b^4*c*d^5*e^6 - b^5*d^4*e^7 - (16*c^5*d^4*e^7 - 32*b*c^4*d
^3*e^8 + 24*b^2*c^3*d^2*e^9 - 8*b^3*c^2*d*e^10 + b^4*c*e^11)*x^5 - (48*c^5*d^5*e
^6 - 80*b*c^4*d^4*e^7 + 40*b^2*c^3*d^3*e^8 - 5*b^4*c*d*e^10 + b^5*e^11)*x^4 - 2*
(16*c^5*d^6*e^5 - 40*b^2*c^3*d^4*e^7 + 40*b^3*c^2*d^3*e^8 - 15*b^4*c*d^2*e^9 + 2
*b^5*d*e^10)*x^3 + 2*(16*c^5*d^7*e^4 - 80*b*c^4*d^6*e^5 + 120*b^2*c^3*d^5*e^6 -
80*b^3*c^2*d^4*e^7 + 25*b^4*c*d^3*e^8 - 3*b^5*d^2*e^9)*x^2 + (48*c^5*d^8*e^3 - 1
60*b*c^4*d^7*e^4 + 200*b^2*c^3*d^6*e^5 - 120*b^3*c^2*d^5*e^6 + 35*b^4*c*d^4*e^7
- 4*b^5*d^3*e^8)*x)*sqrt(2*c*d - b*e)), 1/24*(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2
- b*d*e)*(15*(7*c^3*e^4*f + (5*c^3*d*e^3 - 6*b*c^2*e^4)*g)*x^3 + 5*(7*(7*c^3*d*e
^3 + b*c^2*e^4)*f + (35*c^3*d^2*e^2 - 37*b*c^2*d*e^3 - 6*b^2*c*e^4)*g)*x^2 - (85
*c^3*d^3*e - 187*b*c^2*d^2*e^2 + 62*b^2*c*d*e^3 - 8*b^3*e^4)*f + (49*c^3*d^4 - 1
3*b*c^2*d^3*e - 40*b^2*c*d^2*e^2 + 4*b^3*d*e^3)*g + (7*(17*c^3*d^2*e^2 + 18*b*c^
2*d*e^3 - 2*b^2*c*e^4)*f + (85*c^3*d^3*e - 12*b*c^2*d^2*e^2 - 118*b^2*c*d*e^3 +
12*b^3*e^4)*g)*x)*sqrt(-2*c*d + b*e)*sqrt(e*x + d) - 15*((7*c^4*e^6*f + (5*c^4*d
*e^5 - 6*b*c^3*e^6)*g)*x^5 + (7*(3*c^4*d*e^5 + b*c^3*e^6)*f + (15*c^4*d^2*e^4 -
13*b*c^3*d*e^5 - 6*b^2*c^2*e^6)*g)*x^4 + 2*(7*(c^4*d^2*e^4 + 2*b*c^3*d*e^5)*f +
(5*c^4*d^3*e^3 + 4*b*c^3*d^2*e^4 - 12*b^2*c^2*d*e^5)*g)*x^3 - 2*(7*(c^4*d^3*e^3
- 3*b*c^3*d^2*e^4)*f + (5*c^4*d^4*e^2 - 21*b*c^3*d^3*e^3 + 18*b^2*c^2*d^2*e^4)*g
)*x^2 - 7*(c^4*d^5*e - b*c^3*d^4*e^2)*f - (5*c^4*d^6 - 11*b*c^3*d^5*e + 6*b^2*c^
2*d^4*e^2)*g - (7*(3*c^4*d^4*e^2 - 4*b*c^3*d^3*e^3)*f + (15*c^4*d^5*e - 38*b*c^3
*d^4*e^2 + 24*b^2*c^2*d^3*e^3)*g)*x)*arctan(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)/(c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)))/
((16*c^5*d^9*e^2 - 48*b*c^4*d^8*e^3 + 56*b^2*c^3*d^7*e^4 - 32*b^3*c^2*d^6*e^5 +
9*b^4*c*d^5*e^6 - b^5*d^4*e^7 - (16*c^5*d^4*e^7 - 32*b*c^4*d^3*e^8 + 24*b^2*c^3*
d^2*e^9 - 8*b^3*c^2*d*e^10 + b^4*c*e^11)*x^5 - (48*c^5*d^5*e^6 - 80*b*c^4*d^4*e^
7 + 40*b^2*c^3*d^3*e^8 - 5*b^4*c*d*e^10 + b^5*e^11)*x^4 - 2*(16*c^5*d^6*e^5 - 40
*b^2*c^3*d^4*e^7 + 40*b^3*c^2*d^3*e^8 - 15*b^4*c*d^2*e^9 + 2*b^5*d*e^10)*x^3 + 2
*(16*c^5*d^7*e^4 - 80*b*c^4*d^6*e^5 + 120*b^2*c^3*d^5*e^6 - 80*b^3*c^2*d^4*e^7 +
25*b^4*c*d^3*e^8 - 3*b^5*d^2*e^9)*x^2 + (48*c^5*d^8*e^3 - 160*b*c^4*d^7*e^4 + 2
00*b^2*c^3*d^6*e^5 - 120*b^3*c^2*d^5*e^6 + 35*b^4*c*d^4*e^7 - 4*b^5*d^3*e^8)*x)*
sqrt(-2*c*d + b*e))]
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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)/(e*x+d)**(5/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)
[Out]
Timed out
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(e*x + d)^(5/2)),x, algorithm="giac")
[Out]
[undef, undef, undef, undef, undef, undef, undef, 1]