3.2283 \(\int \frac{f+g x}{\sqrt{d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=378 \[ -\frac{e f-d g}{2 e^2 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{5 c \sqrt{d+e x} (-4 b e g+c d g+7 c e f)}{4 e^2 (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 (-4 b e g+c d g+7 c e f)}{12 e^2 \sqrt{d+e x} (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{\sqrt{d+e x} (-4 b e g+c d g+7 c e f)}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{5 c (-4 b e g+c d g+7 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{9/2}} \]
[Out]
-(e*f - d*g)/(2*e^2*(2*c*d - b*e)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2)^(3/2)) + ((7*c*e*f + c*d*g - 4*b*e*g)*Sqrt[d + e*x])/(6*e^2*(2*c*d - b*e)^
2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (5*(7*c*e*f + c*d*g - 4*b*e*g))
/(12*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*(7*c*e*f + c*d*g - 4*b*e*g)*Sqrt[d + e*x])/(4*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*(7*c*e*f + c*d*g - 4*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])])/(
4*e^2*(2*c*d - b*e)^(9/2))
_______________________________________________________________________________________
Rubi [A] time = 1.42608, antiderivative size = 378, 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
\[ -\frac{e f-d g}{2 e^2 \sqrt{d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{5 c \sqrt{d+e x} (-4 b e g+c d g+7 c e f)}{4 e^2 (2 c d-b e)^4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{5 (-4 b e g+c d g+7 c e f)}{12 e^2 \sqrt{d+e x} (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{\sqrt{d+e x} (-4 b e g+c d g+7 c e f)}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{5 c (-4 b e g+c d g+7 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{9/2}} \]
Antiderivative was successfully verified.
[In] Int[(f + g*x)/(Sqrt[d + e*x]*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]
[Out]
-(e*f - d*g)/(2*e^2*(2*c*d - b*e)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2)^(3/2)) + ((7*c*e*f + c*d*g - 4*b*e*g)*Sqrt[d + e*x])/(6*e^2*(2*c*d - b*e)^
2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (5*(7*c*e*f + c*d*g - 4*b*e*g))
/(12*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*(7*c*e*f + c*d*g - 4*b*e*g)*Sqrt[d + e*x])/(4*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*(7*c*e*f + c*d*g - 4*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])])/(
4*e^2*(2*c*d - b*e)^(9/2))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 163.807, size = 359, normalized size = 0.95 \[ - \frac{5 c \sqrt{d + e x} \left (4 b e g - c d g - 7 c e f\right )}{4 e^{2} \left (b e - 2 c d\right )^{4} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{5 c \left (4 b e g - c d g - 7 c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{4 e^{2} \left (b e - 2 c d\right )^{\frac{9}{2}}} - \frac{\sqrt{d + e x} \left (4 b e g - c d g - 7 c e f\right )}{6 e^{2} \left (b e - 2 c d\right )^{2} \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} - \frac{d g - e f}{2 e^{2} \sqrt{d + e x} \left (b e - 2 c d\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} - \frac{5 \left (4 b e g - c d g - 7 c e f\right )}{12 e^{2} \sqrt{d + e x} \left (b e - 2 c d\right )^{3} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)/(e*x+d)**(1/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)
[Out]
-5*c*sqrt(d + e*x)*(4*b*e*g - c*d*g - 7*c*e*f)/(4*e**2*(b*e - 2*c*d)**4*sqrt(-b*
e**2*x - c*e**2*x**2 + d*(-b*e + c*d))) - 5*c*(4*b*e*g - c*d*g - 7*c*e*f)*atan(s
qrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(sqrt(d + e*x)*sqrt(b*e - 2*c*d)))
/(4*e**2*(b*e - 2*c*d)**(9/2)) - sqrt(d + e*x)*(4*b*e*g - c*d*g - 7*c*e*f)/(6*e*
*2*(b*e - 2*c*d)**2*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)) - (d*g -
e*f)/(2*e**2*sqrt(d + e*x)*(b*e - 2*c*d)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*
d))**(3/2)) - 5*(4*b*e*g - c*d*g - 7*c*e*f)/(12*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)))
_______________________________________________________________________________________
Mathematica [A] time = 2.90443, size = 292, normalized size = 0.77 \[ \frac{(d+e x)^{5/2} \left (\frac{(c (d-e x)-b e) \left (8 c (d+e x)^2 (2 c d-b e) (-b e g+c d g+c e f)-24 c (d+e x)^2 (b e-c d+c e x) (-2 b e g+c d g+3 c e f)-3 (d+e x) (b e-c d+c e x)^2 (-4 b e g-3 c d g+11 c e f)+6 (2 c d-b e) (d g-e f) (b e-c d+c e x)^2\right )}{(d+e x)^2 (b e-2 c d)^4}-\frac{15 c (c (d-e x)-b e)^{5/2} (-4 b e g+c d g+7 c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(2 c d-b e)^{9/2}}\right )}{12 e^2 ((d+e x) (c (d-e x)-b e))^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(f + g*x)/(Sqrt[d + e*x]*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2)),x]
[Out]
((d + e*x)^(5/2)*(((-(b*e) + c*(d - e*x))*(8*c*(2*c*d - b*e)*(c*e*f + c*d*g - b*
e*g)*(d + e*x)^2 - 24*c*(3*c*e*f + c*d*g - 2*b*e*g)*(d + e*x)^2*(-(c*d) + b*e +
c*e*x) + 6*(2*c*d - b*e)*(-(e*f) + d*g)*(-(c*d) + b*e + c*e*x)^2 - 3*(11*c*e*f -
3*c*d*g - 4*b*e*g)*(d + e*x)*(-(c*d) + b*e + c*e*x)^2))/((-2*c*d + b*e)^4*(d +
e*x)^2) - (15*c*(7*c*e*f + c*d*g - 4*b*e*g)*(-(b*e) + c*(d - e*x))^(5/2)*ArcTanh
[Sqrt[c*d - b*e - c*e*x]/Sqrt[2*c*d - b*e]])/(2*c*d - b*e)^(9/2)))/(12*e^2*((d +
e*x)*(-(b*e) + c*(d - e*x)))^(5/2))
_______________________________________________________________________________________
Maple [B] time = 0.054, size = 1528, normalized size = 4. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
[Out]
1/12*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(120*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*(-c*e*x-b*e+c*d)^(1/2)-90*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*(-c*e*x-b*e+c*d)^(1/2)-210*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*(-c*e*x-b*e+c*d)^(1/2
)-105*(b*e-2*c*d)^(1/2)*x^3*c^3*e^4*f+12*(b*e-2*c*d)^(1/2)*x*b^3*e^4*g+6*(b*e-2*
c*d)^(1/2)*b^3*d*e^3*g+43*(b*e-2*c*d)^(1/2)*c^3*d^3*e*f-15*arctan((-c*e*x-b*e+c*
d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^3*d^2*e^2*g*(-c*e*x-b*e+c*d)^(1/2)-105*arctan(
(-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c^3*d*e^3*f*(-c*e*x-b*e+c*d)^(1/2)
+15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^3*d^3*e*g*(-c*e*x-b*e+c
*d)^(1/2)+105*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*(
-c*e*x-b*e+c*d)^(1/2)+60*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*(-c*e*x-b*e+c*d)^(1/2)-75*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1
/2))*b*c^2*d^3*e*g*(-c*e*x-b*e+c*d)^(1/2)-105*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*(-c*e*x-b*e+c*d)^(1/2)+109*(b*e-2*c*d)^(1/2)*x*b^
2*c*d*e^3*g-120*(b*e-2*c*d)^(1/2)*x*b*c^2*d^2*e^2*g-196*(b*e-2*c*d)^(1/2)*x*b*c^
2*d*e^3*f+6*(b*e-2*c*d)^(1/2)*b^3*e^4*f+61*(b*e-2*c*d)^(1/2)*c^3*d^4*g-21*(b*e-2
*c*d)^(1/2)*x*b^2*c*e^4*f+23*(b*e-2*c*d)^(1/2)*x*c^3*d^3*e*g+161*(b*e-2*c*d)^(1/
2)*x*c^3*d^2*e^2*f-35*(b*e-2*c*d)^(1/2)*x^2*c^3*d*e^3*f-140*(b*e-2*c*d)^(1/2)*x^
2*b*c^2*e^4*f-5*(b*e-2*c*d)^(1/2)*x^2*c^3*d^2*e^2*g+60*(b*e-2*c*d)^(1/2)*x^3*b*c
^2*e^4*g-15*(b*e-2*c*d)^(1/2)*x^3*c^3*d*e^3*g+80*(b*e-2*c*d)^(1/2)*x^2*b^2*c*e^4
*g-105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^3*e^4*f*(-c*e*x-b*
e+c*d)^(1/2)+105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*e*f*(-
c*e*x-b*e+c*d)^(1/2)+65*(b*e-2*c*d)^(1/2)*b^2*c*d^2*e^2*g-57*(b*e-2*c*d)^(1/2)*b
^2*c*d*e^3*f-132*(b*e-2*c*d)^(1/2)*b*c^2*d^3*e*g+16*(b*e-2*c*d)^(1/2)*b*c^2*d^2*
e^2*f+15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^4*g*(-c*e*x-b*e+
c*d)^(1/2)+60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*b*c^2*e^4*g*(
-c*e*x-b*e+c*d)^(1/2)-15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^
3*d*e^3*g*(-c*e*x-b*e+c*d)^(1/2)+60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1
/2))*x^2*b^2*c*e^4*g*(-c*e*x-b*e+c*d)^(1/2)-105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b
*e-2*c*d)^(1/2))*x^2*b*c^2*e^4*f*(-c*e*x-b*e+c*d)^(1/2)+45*arctan((-c*e*x-b*e+c*
d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c^2*d*e^3*g*(-c*e*x-b*e+c*d)^(1/2))/(e*x+d)^(5
/2)/(c*e*x+b*e-c*d)^2/e^2/(b*e-2*c*d)^(9/2)
_______________________________________________________________________________________
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)^(5/2)*sqrt(e*x + d)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.406731, 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)^(5/2)*sqrt(e*x + d)),x, algorithm="fricas")
[Out]
[-1/24*(2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(15*(7*c^3*e^4*f + (c^3*d*e
^3 - 4*b*c^2*e^4)*g)*x^3 + 5*(7*(c^3*d*e^3 + 4*b*c^2*e^4)*f + (c^3*d^2*e^2 - 16*
b^2*c*e^4)*g)*x^2 - (43*c^3*d^3*e + 16*b*c^2*d^2*e^2 - 57*b^2*c*d*e^3 + 6*b^3*e^
4)*f - (61*c^3*d^4 - 132*b*c^2*d^3*e + 65*b^2*c*d^2*e^2 + 6*b^3*d*e^3)*g - (7*(2
3*c^3*d^2*e^2 - 28*b*c^2*d*e^3 - 3*b^2*c*e^4)*f + (23*c^3*d^3*e - 120*b*c^2*d^2*
e^2 + 109*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 + (c^4*d*e^5 - 4*b*c^3*e^6)*g)*x^5 + (7*(c^4*d*e^5 + 2*b*c^3*e^6)*f
+ (c^4*d^2*e^4 - 2*b*c^3*d*e^5 - 8*b^2*c^2*e^6)*g)*x^4 - (7*(2*c^4*d^2*e^4 - 4*
b*c^3*d*e^5 - b^2*c^2*e^6)*f + (2*c^4*d^3*e^3 - 12*b*c^3*d^2*e^4 + 15*b^2*c^2*d*
e^5 + 4*b^3*c*e^6)*g)*x^3 - (7*(2*c^4*d^3*e^3 - 3*b^2*c^2*d*e^5)*f + (2*c^4*d^4*
e^2 - 8*b*c^3*d^3*e^3 - 3*b^2*c^2*d^2*e^4 + 12*b^3*c*d*e^5)*g)*x^2 + 7*(c^4*d^5*
e - 2*b*c^3*d^4*e^2 + b^2*c^2*d^3*e^3)*f + (c^4*d^6 - 6*b*c^3*d^5*e + 9*b^2*c^2*
d^4*e^2 - 4*b^3*c*d^3*e^3)*g + (7*(c^4*d^4*e^2 - 4*b*c^3*d^3*e^3 + 3*b^2*c^2*d^2
*e^4)*f + (c^4*d^5*e - 8*b*c^3*d^4*e^2 + 19*b^2*c^2*d^3*e^3 - 12*b^3*c*d^2*e^4)*
g)*x)*log((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^6*d^9*e^2 - 64*b*c^5*d^8*e^3 + 104*b^2*c^4*d^7
*e^4 - 88*b^3*c^3*d^6*e^5 + 41*b^4*c^2*d^5*e^6 - 10*b^5*c*d^4*e^7 + b^6*d^3*e^8
+ (16*c^6*d^4*e^7 - 32*b*c^5*d^3*e^8 + 24*b^2*c^4*d^2*e^9 - 8*b^3*c^3*d*e^10 + b
^4*c^2*e^11)*x^5 + (16*c^6*d^5*e^6 - 40*b^2*c^4*d^3*e^8 + 40*b^3*c^3*d^2*e^9 - 1
5*b^4*c^2*d*e^10 + 2*b^5*c*e^11)*x^4 - (32*c^6*d^6*e^5 - 128*b*c^5*d^5*e^6 + 160
*b^2*c^4*d^4*e^7 - 80*b^3*c^3*d^3*e^8 + 10*b^4*c^2*d^2*e^9 + 4*b^5*c*d*e^10 - b^
6*e^11)*x^3 - (32*c^6*d^7*e^4 - 64*b*c^5*d^6*e^5 + 80*b^3*c^3*d^4*e^7 - 70*b^4*c
^2*d^3*e^8 + 24*b^5*c*d^2*e^9 - 3*b^6*d*e^10)*x^2 + (16*c^6*d^8*e^3 - 96*b*c^5*d
^7*e^4 + 200*b^2*c^4*d^6*e^5 - 200*b^3*c^3*d^5*e^6 + 105*b^4*c^2*d^4*e^7 - 28*b^
5*c*d^3*e^8 + 3*b^6*d^2*e^9)*x)*sqrt(2*c*d - b*e)), -1/12*(sqrt(-c*e^2*x^2 - b*e
^2*x + c*d^2 - b*d*e)*(15*(7*c^3*e^4*f + (c^3*d*e^3 - 4*b*c^2*e^4)*g)*x^3 + 5*(7
*(c^3*d*e^3 + 4*b*c^2*e^4)*f + (c^3*d^2*e^2 - 16*b^2*c*e^4)*g)*x^2 - (43*c^3*d^3
*e + 16*b*c^2*d^2*e^2 - 57*b^2*c*d*e^3 + 6*b^3*e^4)*f - (61*c^3*d^4 - 132*b*c^2*
d^3*e + 65*b^2*c*d^2*e^2 + 6*b^3*d*e^3)*g - (7*(23*c^3*d^2*e^2 - 28*b*c^2*d*e^3
- 3*b^2*c*e^4)*f + (23*c^3*d^3*e - 120*b*c^2*d^2*e^2 + 109*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 + (c^4*d*e^5 - 4*
b*c^3*e^6)*g)*x^5 + (7*(c^4*d*e^5 + 2*b*c^3*e^6)*f + (c^4*d^2*e^4 - 2*b*c^3*d*e^
5 - 8*b^2*c^2*e^6)*g)*x^4 - (7*(2*c^4*d^2*e^4 - 4*b*c^3*d*e^5 - b^2*c^2*e^6)*f +
(2*c^4*d^3*e^3 - 12*b*c^3*d^2*e^4 + 15*b^2*c^2*d*e^5 + 4*b^3*c*e^6)*g)*x^3 - (7
*(2*c^4*d^3*e^3 - 3*b^2*c^2*d*e^5)*f + (2*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3 - 3*b^2*
c^2*d^2*e^4 + 12*b^3*c*d*e^5)*g)*x^2 + 7*(c^4*d^5*e - 2*b*c^3*d^4*e^2 + b^2*c^2*
d^3*e^3)*f + (c^4*d^6 - 6*b*c^3*d^5*e + 9*b^2*c^2*d^4*e^2 - 4*b^3*c*d^3*e^3)*g +
(7*(c^4*d^4*e^2 - 4*b*c^3*d^3*e^3 + 3*b^2*c^2*d^2*e^4)*f + (c^4*d^5*e - 8*b*c^3
*d^4*e^2 + 19*b^2*c^2*d^3*e^3 - 12*b^3*c*d^2*e^4)*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^6*d^9*e^2 - 64*b*c^5*d^8*e^3 + 104*b^2*c^4*d^7*e^4 - 8
8*b^3*c^3*d^6*e^5 + 41*b^4*c^2*d^5*e^6 - 10*b^5*c*d^4*e^7 + b^6*d^3*e^8 + (16*c^
6*d^4*e^7 - 32*b*c^5*d^3*e^8 + 24*b^2*c^4*d^2*e^9 - 8*b^3*c^3*d*e^10 + b^4*c^2*e
^11)*x^5 + (16*c^6*d^5*e^6 - 40*b^2*c^4*d^3*e^8 + 40*b^3*c^3*d^2*e^9 - 15*b^4*c^
2*d*e^10 + 2*b^5*c*e^11)*x^4 - (32*c^6*d^6*e^5 - 128*b*c^5*d^5*e^6 + 160*b^2*c^4
*d^4*e^7 - 80*b^3*c^3*d^3*e^8 + 10*b^4*c^2*d^2*e^9 + 4*b^5*c*d*e^10 - b^6*e^11)*
x^3 - (32*c^6*d^7*e^4 - 64*b*c^5*d^6*e^5 + 80*b^3*c^3*d^4*e^7 - 70*b^4*c^2*d^3*e
^8 + 24*b^5*c*d^2*e^9 - 3*b^6*d*e^10)*x^2 + (16*c^6*d^8*e^3 - 96*b*c^5*d^7*e^4 +
200*b^2*c^4*d^6*e^5 - 200*b^3*c^3*d^5*e^6 + 105*b^4*c^2*d^4*e^7 - 28*b^5*c*d^3*
e^8 + 3*b^6*d^2*e^9)*x)*sqrt(-2*c*d + b*e))]
_______________________________________________________________________________________
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)**(1/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.732723, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]
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)^(5/2)*sqrt(e*x + d)),x, algorithm="giac")
[Out]
sage0*x