[next] [prev] [prev-tail] [tail] [up]

3.2204 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=264 \[ -\frac{c^{5/2} g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e^2}-\frac{2 c^2 g \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 e^2 (d+e x)^7 (2 c d-b e)}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^5}+\frac{2 c g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3} \]

[Out]

(-2*c^2*g*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*(d + e*x)) + (2*c*g*(d
*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(d + e*x)^3) - (2*g*(d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*e^2*(d + e*x)^5) - (2*(e*f - d*g)*(d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(7*e^2*(2*c*d - b*e)*(d + e*x)^7) - (c^(5/2
)*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])
])/e^2

_______________________________________________________________________________________

Rubi [A]  time = 1.0937, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{c^{5/2} g \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e^2}-\frac{2 c^2 g \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 e^2 (d+e x)^7 (2 c d-b e)}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^5}+\frac{2 c g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3} \]

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)^7,x]

[Out]

(-2*c^2*g*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*(d + e*x)) + (2*c*g*(d
*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(d + e*x)^3) - (2*g*(d*(c*d -
b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*e^2*(d + e*x)^5) - (2*(e*f - d*g)*(d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(7*e^2*(2*c*d - b*e)*(d + e*x)^7) - (c^(5/2
)*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])
])/e^2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 106.919, size = 241, normalized size = 0.91 \[ - \frac{c^{\frac{5}{2}} g \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{e^{2}} - \frac{2 c^{2} g \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \left (d + e x\right )} + \frac{2 c g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{3}} - \frac{2 g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{5 e^{2} \left (d + e x\right )^{5}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{7 e^{2} \left (d + e x\right )^{7} \left (b e - 2 c d\right )} \]

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)**7,x)

[Out]

-c**(5/2)*g*atan(-e*(-b - 2*c*x)/(2*sqrt(c)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b
*e + c*d))))/e**2 - 2*c**2*g*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(e**
2*(d + e*x)) + 2*c*g*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(3*e**2*(
d + e*x)**3) - 2*g*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(5*e**2*(d
+ e*x)**5) - 2*(d*g - e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(7*
e**2*(d + e*x)**7*(b*e - 2*c*d))

_______________________________________________________________________________________

Mathematica [C]  time = 1.8299, size = 266, normalized size = 1.01 \[ ((d+e x) (c (d-e x)-b e))^{5/2} \left (-\frac{2 \left (-c^2 (d+e x)^3 (161 b e g-337 c d g+15 c e f)+c (d+e x)^2 (2 c d-b e) (77 b e g-199 c d g+45 c e f)+3 (d+e x) (b e-2 c d)^2 (-7 b e g+29 c d g-15 c e f)+15 (2 c d-b e)^3 (e f-d g)\right )}{105 e^2 (d+e x)^6 (2 c d-b e) (b e-c d+c e x)^2}-\frac{i c^{5/2} g \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{e^2 (d+e x)^{5/2} (c (d-e x)-b e)^{5/2}}\right ) \]

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)^7,x]

[Out]

((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((-2*(15*(2*c*d - b*e)^3*(e*f - d*g) +
3*(-2*c*d + b*e)^2*(-15*c*e*f + 29*c*d*g - 7*b*e*g)*(d + e*x) + c*(2*c*d - b*e)*
(45*c*e*f - 199*c*d*g + 77*b*e*g)*(d + e*x)^2 - c^2*(15*c*e*f - 337*c*d*g + 161*
b*e*g)*(d + e*x)^3))/(105*e^2*(2*c*d - b*e)*(d + e*x)^6*(-(c*d) + b*e + c*e*x)^2
) - (I*c^(5/2)*g*Log[((-I)*e*(b + 2*c*x))/Sqrt[c] + 2*Sqrt[d + e*x]*Sqrt[-(b*e)
+ c*(d - e*x)]])/(e^2*(d + e*x)^(5/2)*(-(b*e) + c*(d - e*x))^(5/2)))

_______________________________________________________________________________________

Maple [B]  time = 0.026, size = 1905, normalized size = 7.2 \[ \text{result too large to display} \]

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)^7,x)

[Out]

-32/5*g/e*c^3/(-b*e^2+2*c*d*e)^4/(d/e+x)^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d
/e+x))^(7/2)-256/15*g*e*c^4/(-b*e^2+2*c*d*e)^5/(d/e+x)^2*(-c*(d/e+x)^2*e^2+(-b*e
^2+2*c*d*e)*(d/e+x))^(7/2)+2*g*e^7*c^3/(-b*e^2+2*c*d*e)^5*b^4*(-c*(d/e+x)^2*e^2+
(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)+4/15*g/e^5*c/(-b*e^2+2*c*d*e)^2/(d/e+x)^5*(-c*(d
/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(7/2)-16/15*g/e^3*c^2/(-b*e^2+2*c*d*e)^3/(
d/e+x)^4*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(7/2)+4*g*e^7*c^4/(-b*e^2+2
*c*d*e)^5*b^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x-12*g*e^6*c^4/(
-b*e^2+2*c*d*e)^5*b^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*d+g*e^9*
c^3/(-b*e^2+2*c*d*e)^5*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))-64/3*g*e^4*c
^6/(-b*e^2+2*c*d*e)^5*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x-32/3
*g*e^4*c^5/(-b*e^2+2*c*d*e)^5*d*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2
)*b-32*g*e^4*c^7/(-b*e^2+2*c*d*e)^5*d^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+
x))^(1/2)*x-16*g*e^4*c^6/(-b*e^2+2*c*d*e)^5*d^3*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*
e)*(d/e+x))^(1/2)*b-32*g*e^4*c^8/(-b*e^2+2*c*d*e)^5*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))+24*g*e^5*c^5/(-b*e^2+2*c*d*e)^5*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2
*c*d*e)*(d/e+x))^(1/2)*d^2+32/3*g*e^5*c^5/(-b*e^2+2*c*d*e)^5*b*(-c*(d/e+x)^2*e^2
+(-b*e^2+2*c*d*e)*(d/e+x))^(3/2)*x-2/7*(-d*g+e*f)/e^8/(-b*e^2+2*c*d*e)/(d/e+x)^7
*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(7/2)-2/5*g/e^7/(-b*e^2+2*c*d*e)/(d
/e+x)^6*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(7/2)-24*g*e^6*c^5/(-b*e^2+2
*c*d*e)^5*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*d+40*g*e^7*c^5
/(-b*e^2+2*c*d*e)^5*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^2-80*g*e^6*c^
6/(-b*e^2+2*c*d*e)^5*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+48*g*e^5*c
^6/(-b*e^2+2*c*d*e)^5*b*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(1/2)*x*d^2+
80*g*e^5*c^7/(-b*e^2+2*c*d*e)^5*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^4-1
0*g*e^8*c^4/(-b*e^2+2*c*d*e)^5*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-25
6/15*g*e^3*c^5/(-b*e^2+2*c*d*e)^5*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/e+x))^(5
/2)+16/3*g*e^5*c^4/(-b*e^2+2*c*d*e)^5*b^2*(-c*(d/e+x)^2*e^2+(-b*e^2+2*c*d*e)*(d/
e+x))^(3/2)

_______________________________________________________________________________________

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)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 3.35751, 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)^7,x, algorithm="fricas")

[Out]

[1/210*(105*((2*c^3*d*e^4 - b*c^2*e^5)*g*x^4 + 4*(2*c^3*d^2*e^3 - b*c^2*d*e^4)*g
*x^3 + 6*(2*c^3*d^3*e^2 - b*c^2*d^2*e^3)*g*x^2 + 4*(2*c^3*d^4*e - b*c^2*d^3*e^2)
*g*x + (2*c^3*d^5 - b*c^2*d^4*e)*g)*sqrt(-c)*log(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 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2
*c*e*x + b*e)*sqrt(-c)) + 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((15*c^3*
e^4*f - (337*c^3*d*e^3 - 161*b*c^2*e^4)*g)*x^3 - (45*(c^3*d*e^3 - b*c^2*e^4)*f +
 (613*c^3*d^2*e^2 - 130*b*c^2*d*e^3 - 77*b^2*c*e^4)*g)*x^2 - 15*(c^3*d^3*e - 3*b
*c^2*d^2*e^2 + 3*b^2*c*d*e^3 - b^3*e^4)*f - (167*c^3*d^4 - 60*b*c^2*d^3*e + 4*b^
2*c*d^2*e^2 - 6*b^3*d*e^3)*g + (45*(c^3*d^2*e^2 - 2*b*c^2*d*e^3 + b^2*c*e^4)*f -
 (563*c^3*d^3*e - 209*b*c^2*d^2*e^2 + 17*b^2*c*d*e^3 - 21*b^3*e^4)*g)*x))/(2*c*d
^5*e^2 - b*d^4*e^3 + (2*c*d*e^6 - b*e^7)*x^4 + 4*(2*c*d^2*e^5 - b*d*e^6)*x^3 + 6
*(2*c*d^3*e^4 - b*d^2*e^5)*x^2 + 4*(2*c*d^4*e^3 - b*d^3*e^4)*x), -1/105*(105*((2
*c^3*d*e^4 - b*c^2*e^5)*g*x^4 + 4*(2*c^3*d^2*e^3 - b*c^2*d*e^4)*g*x^3 + 6*(2*c^3
*d^3*e^2 - b*c^2*d^2*e^3)*g*x^2 + 4*(2*c^3*d^4*e - b*c^2*d^3*e^2)*g*x + (2*c^3*d
^5 - b*c^2*d^4*e)*g)*sqrt(c)*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))) - 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((1
5*c^3*e^4*f - (337*c^3*d*e^3 - 161*b*c^2*e^4)*g)*x^3 - (45*(c^3*d*e^3 - b*c^2*e^
4)*f + (613*c^3*d^2*e^2 - 130*b*c^2*d*e^3 - 77*b^2*c*e^4)*g)*x^2 - 15*(c^3*d^3*e
 - 3*b*c^2*d^2*e^2 + 3*b^2*c*d*e^3 - b^3*e^4)*f - (167*c^3*d^4 - 60*b*c^2*d^3*e
+ 4*b^2*c*d^2*e^2 - 6*b^3*d*e^3)*g + (45*(c^3*d^2*e^2 - 2*b*c^2*d*e^3 + b^2*c*e^
4)*f - (563*c^3*d^3*e - 209*b*c^2*d^2*e^2 + 17*b^2*c*d*e^3 - 21*b^3*e^4)*g)*x))/
(2*c*d^5*e^2 - b*d^4*e^3 + (2*c*d*e^6 - b*e^7)*x^4 + 4*(2*c*d^2*e^5 - b*d*e^6)*x
^3 + 6*(2*c*d^3*e^4 - b*d^2*e^5)*x^2 + 4*(2*c*d^4*e^3 - b*d^3*e^4)*x)]

_______________________________________________________________________________________

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)**7,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 2.05285, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

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)^7,x, algorithm="giac")

[Out]

sage0*x

[next] [prev] [prev-tail] [front] [up]