3.2260 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx\)

Optimal. Leaf size=383 \[ \frac{5 c^3 (-8 b e g+15 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{64 e^2 (2 c d-b e)^{3/2}}-\frac{5 c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+15 c d g+c e f)}{64 e^2 (d+e x)^{3/2} (2 c d-b e)}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (d+e x)^{15/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+15 c d g+c e f)}{24 e^2 (d+e x)^{11/2} (2 c d-b e)}+\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+15 c d g+c e f)}{96 e^2 (d+e x)^{7/2} (2 c d-b e)} \]

[Out]

(-5*c^2*(c*e*f + 15*c*d*g - 8*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/
(64*e^2*(2*c*d - b*e)*(d + e*x)^(3/2)) + (5*c*(c*e*f + 15*c*d*g - 8*b*e*g)*(d*(c
*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(96*e^2*(2*c*d - b*e)*(d + e*x)^(7/2)) -
((c*e*f + 15*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(24*
e^2*(2*c*d - b*e)*(d + e*x)^(11/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(7/2))/(4*e^2*(2*c*d - b*e)*(d + e*x)^(15/2)) + (5*c^3*(c*e*f + 15*c*d*
g - 8*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])])/(64*e^2*(2*c*d - b*e)^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 1.30597, antiderivative size = 383, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{5 c^3 (-8 b e g+15 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{64 e^2 (2 c d-b e)^{3/2}}-\frac{5 c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-8 b e g+15 c d g+c e f)}{64 e^2 (d+e x)^{3/2} (2 c d-b e)}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4 e^2 (d+e x)^{15/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-8 b e g+15 c d g+c e f)}{24 e^2 (d+e x)^{11/2} (2 c d-b e)}+\frac{5 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-8 b e g+15 c d g+c e f)}{96 e^2 (d+e x)^{7/2} (2 c d-b e)} \]

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)^(15/2),x]

[Out]

(-5*c^2*(c*e*f + 15*c*d*g - 8*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/
(64*e^2*(2*c*d - b*e)*(d + e*x)^(3/2)) + (5*c*(c*e*f + 15*c*d*g - 8*b*e*g)*(d*(c
*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(96*e^2*(2*c*d - b*e)*(d + e*x)^(7/2)) -
((c*e*f + 15*c*d*g - 8*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(24*
e^2*(2*c*d - b*e)*(d + e*x)^(11/2)) - ((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(7/2))/(4*e^2*(2*c*d - b*e)*(d + e*x)^(15/2)) + (5*c^3*(c*e*f + 15*c*d*
g - 8*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])])/(64*e^2*(2*c*d - b*e)^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 158.965, size = 357, normalized size = 0.93 \[ - \frac{5 c^{3} \left (8 b e g - 15 c d g - c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{64 e^{2} \left (b e - 2 c d\right )^{\frac{3}{2}}} - \frac{5 c^{2} \left (8 b e g - 15 c d g - c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{64 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right )} + \frac{5 c \left (8 b e g - 15 c d g - c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{96 e^{2} \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right )} - \frac{\left (8 b e g - 15 c d g - c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{24 e^{2} \left (d + e x\right )^{\frac{11}{2}} \left (b e - 2 c d\right )} - \frac{\left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{4 e^{2} \left (d + e x\right )^{\frac{15}{2}} \left (b e - 2 c d\right )} \]

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)**(15/2),x)

[Out]

-5*c**3*(8*b*e*g - 15*c*d*g - 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)))/(64*e**2*(b*e - 2*c*d)**(3/2)) - 5*c
**2*(8*b*e*g - 15*c*d*g - c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/
(64*e**2*(d + e*x)**(3/2)*(b*e - 2*c*d)) + 5*c*(8*b*e*g - 15*c*d*g - c*e*f)*(-b*
e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(96*e**2*(d + e*x)**(7/2)*(b*e - 2
*c*d)) - (8*b*e*g - 15*c*d*g - c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))
**(5/2)/(24*e**2*(d + e*x)**(11/2)*(b*e - 2*c*d)) - (d*g - e*f)*(-b*e**2*x - c*e
**2*x**2 + d*(-b*e + c*d))**(7/2)/(4*e**2*(d + e*x)**(15/2)*(b*e - 2*c*d))

_______________________________________________________________________________________

Mathematica [A]  time = 2.28369, size = 274, normalized size = 0.72 \[ \frac{((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{15 c^3 (-8 b e g+15 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(2 c d-b e)^{3/2} (c (d-e x)-b e)^{5/2}}-\frac{-3 c^2 (d+e x)^3 (88 b e g-181 c d g+5 c e f)+2 c (d+e x)^2 (2 c d-b e) (104 b e g-267 c d g+59 c e f)+8 (d+e x) (b e-2 c d)^2 (-8 b e g+33 c d g-17 c e f)+48 (2 c d-b e)^3 (e f-d g)}{(d+e x)^4 (2 c d-b e) (b e-c d+c e x)^2}\right )}{192 e^2 (d+e x)^{5/2}} \]

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)^(15/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*(-((48*(2*c*d - b*e)^3*(e*f - d*g) + 8
*(-2*c*d + b*e)^2*(-17*c*e*f + 33*c*d*g - 8*b*e*g)*(d + e*x) + 2*c*(2*c*d - b*e)
*(59*c*e*f - 267*c*d*g + 104*b*e*g)*(d + e*x)^2 - 3*c^2*(5*c*e*f - 181*c*d*g + 8
8*b*e*g)*(d + e*x)^3)/((2*c*d - b*e)*(d + e*x)^4*(-(c*d) + b*e + c*e*x)^2)) + (1
5*c^3*(c*e*f + 15*c*d*g - 8*b*e*g)*ArcTanh[Sqrt[c*d - b*e - c*e*x]/Sqrt[2*c*d -
b*e]])/((2*c*d - b*e)^(3/2)*(-(b*e) + c*(d - e*x))^(5/2))))/(192*e^2*(d + e*x)^(
5/2))

_______________________________________________________________________________________

Maple [B]  time = 0.05, size = 1517, normalized size = 4. \[ \text{result too large to display} \]

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)^(15/2),x)

[Out]

-1/192*(15*x^3*c^3*e^4*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+64*x*b^3*e^4*g
*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+16*b^3*d*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*
(b*e-2*c*d)^(1/2)-61*c^3*d^3*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+120*ar
ctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^4*b*c^3*e^5*g-225*arctan((-c*e*
x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^4*c^4*d*e^4*g-900*arctan((-c*e*x-b*e+c*d)^
(1/2)/(b*e-2*c*d)^(1/2))*x^3*c^4*d^2*e^3*g-60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e
-2*c*d)^(1/2))*x^3*c^4*d*e^4*f-1350*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1
/2))*x^2*c^4*d^3*e^2*g-90*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*c
^4*d^2*e^3*f-900*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^4*d^4*e*g-
60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^4*d^3*e^2*f+120*arctan((
-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^3*d^4*e*g-147*c^3*d^4*g*(-c*e*x-b*e
+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2)
)*x^4*c^4*e^5*f-15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^4*e*f-
549*x*c^3*d^3*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+117*x*c^3*d^2*e^2*f*(
-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-24*b^2*c*d^2*e^2*g*(-c*e*x-b*e+c*d)^(1/2
)*(b*e-2*c*d)^(1/2)-152*b^2*c*d*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+5
0*b*c^2*d^3*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+150*b*c^2*d^2*e^2*f*(-c
*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-543*x^3*c^3*d*e^3*g*(-c*e*x-b*e+c*d)^(1/2)
*(b*e-2*c*d)^(1/2)+480*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^3*b*c^
3*d*e^4*g+720*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x^2*b*c^3*d^2*e^3
*g+480*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c^3*d^3*e^2*g+264*x^
3*b*c^2*e^4*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+208*x^2*b^2*c*e^4*g*(-c*e
*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+118*x^2*b*c^2*e^4*f*(-c*e*x-b*e+c*d)^(1/2)*(
b*e-2*c*d)^(1/2)-561*x^2*c^3*d^2*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-
191*x^2*c^3*d*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+136*x*b^2*c*e^4*f*(
-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+48*b^3*e^4*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e
-2*c*d)^(1/2)-225*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^4*d^5*g+204
*x*b*c^2*d^2*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-308*x*b*c^2*d*e^3*f*
(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-158*x^2*b*c^2*d*e^3*g*(-c*e*x-b*e+c*d)^
(1/2)*(b*e-2*c*d)^(1/2)-104*x*b^2*c*d*e^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(
1/2))*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(b*e-2*c*d)^(3/2)/e^2/(-c*e*x-b*e+c
*d)^(1/2)/(e*x+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((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(g*x + f)/(e*x + d)^(15/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.333308, 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)^(15/2),x, algorithm="fricas")

[Out]

[1/384*(2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(3*(5*c^3*e^4*f - (181*c^3*
d*e^3 - 88*b*c^2*e^4)*g)*x^3 - ((191*c^3*d*e^3 - 118*b*c^2*e^4)*f + (561*c^3*d^2
*e^2 + 158*b*c^2*d*e^3 - 208*b^2*c*e^4)*g)*x^2 - (61*c^3*d^3*e - 150*b*c^2*d^2*e
^2 + 152*b^2*c*d*e^3 - 48*b^3*e^4)*f - (147*c^3*d^4 - 50*b*c^2*d^3*e + 24*b^2*c*
d^2*e^2 - 16*b^3*d*e^3)*g + ((117*c^3*d^2*e^2 - 308*b*c^2*d*e^3 + 136*b^2*c*e^4)
*f - (549*c^3*d^3*e - 204*b*c^2*d^2*e^2 + 104*b^2*c*d*e^3 - 64*b^3*e^4)*g)*x)*sq
rt(2*c*d - b*e)*sqrt(e*x + d) + 15*(c^4*d^5*e*f + (c^4*e^6*f + (15*c^4*d*e^5 - 8
*b*c^3*e^6)*g)*x^5 + 5*(c^4*d*e^5*f + (15*c^4*d^2*e^4 - 8*b*c^3*d*e^5)*g)*x^4 +
10*(c^4*d^2*e^4*f + (15*c^4*d^3*e^3 - 8*b*c^3*d^2*e^4)*g)*x^3 + 10*(c^4*d^3*e^3*
f + (15*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3)*g)*x^2 + (15*c^4*d^6 - 8*b*c^3*d^5*e)*g +
5*(c^4*d^4*e^2*f + (15*c^4*d^5*e - 8*b*c^3*d^4*e^2)*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)))/
((2*c*d^6*e^2 - b*d^5*e^3 + (2*c*d*e^7 - b*e^8)*x^5 + 5*(2*c*d^2*e^6 - b*d*e^7)*
x^4 + 10*(2*c*d^3*e^5 - b*d^2*e^6)*x^3 + 10*(2*c*d^4*e^4 - b*d^3*e^5)*x^2 + 5*(2
*c*d^5*e^3 - b*d^4*e^4)*x)*sqrt(2*c*d - b*e)), 1/192*(sqrt(-c*e^2*x^2 - b*e^2*x
+ c*d^2 - b*d*e)*(3*(5*c^3*e^4*f - (181*c^3*d*e^3 - 88*b*c^2*e^4)*g)*x^3 - ((191
*c^3*d*e^3 - 118*b*c^2*e^4)*f + (561*c^3*d^2*e^2 + 158*b*c^2*d*e^3 - 208*b^2*c*e
^4)*g)*x^2 - (61*c^3*d^3*e - 150*b*c^2*d^2*e^2 + 152*b^2*c*d*e^3 - 48*b^3*e^4)*f
- (147*c^3*d^4 - 50*b*c^2*d^3*e + 24*b^2*c*d^2*e^2 - 16*b^3*d*e^3)*g + ((117*c^
3*d^2*e^2 - 308*b*c^2*d*e^3 + 136*b^2*c*e^4)*f - (549*c^3*d^3*e - 204*b*c^2*d^2*
e^2 + 104*b^2*c*d*e^3 - 64*b^3*e^4)*g)*x)*sqrt(-2*c*d + b*e)*sqrt(e*x + d) - 15*
(c^4*d^5*e*f + (c^4*e^6*f + (15*c^4*d*e^5 - 8*b*c^3*e^6)*g)*x^5 + 5*(c^4*d*e^5*f
+ (15*c^4*d^2*e^4 - 8*b*c^3*d*e^5)*g)*x^4 + 10*(c^4*d^2*e^4*f + (15*c^4*d^3*e^3
- 8*b*c^3*d^2*e^4)*g)*x^3 + 10*(c^4*d^3*e^3*f + (15*c^4*d^4*e^2 - 8*b*c^3*d^3*e
^3)*g)*x^2 + (15*c^4*d^6 - 8*b*c^3*d^5*e)*g + 5*(c^4*d^4*e^2*f + (15*c^4*d^5*e -
8*b*c^3*d^4*e^2)*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)))/((2*c*d^6*e^2
- b*d^5*e^3 + (2*c*d*e^7 - b*e^8)*x^5 + 5*(2*c*d^2*e^6 - b*d*e^7)*x^4 + 10*(2*c
*d^3*e^5 - b*d^2*e^6)*x^3 + 10*(2*c*d^4*e^4 - b*d^3*e^5)*x^2 + 5*(2*c*d^5*e^3 -
b*d^4*e^4)*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)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**(15/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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)^(15/2),x, algorithm="giac")

[Out]

Timed out