3.2203 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=352 \[ -\frac{c^{3/2} (5 b e g-12 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 e^2}-\frac{c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (5 b e g-12 c d g+2 c e f)}{e^2 (2 c d-b e)}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^6 (2 c d-b e)}+\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (5 b e g-12 c d g+2 c e f)}{15 e^2 (d+e x)^4 (2 c d-b e)}-\frac{2 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (5 b e g-12 c d g+2 c e f)}{3 e^2 (d+e x)^2 (2 c d-b e)} \]

[Out]

-((c^2*(2*c*e*f - 12*c*d*g + 5*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])
/(e^2*(2*c*d - b*e))) - (2*c*(2*c*e*f - 12*c*d*g + 5*b*e*g)*(d*(c*d - b*e) - b*e
^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^2) + (2*(2*c*e*f - 12*c*
d*g + 5*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(15*e^2*(2*c*d - b*e
)*(d + e*x)^4) - (2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(5*
e^2*(2*c*d - b*e)*(d + e*x)^6) - (c^(3/2)*(2*c*e*f - 12*c*d*g + 5*b*e*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])])/(2*e^2)

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Rubi [A]  time = 1.21307, antiderivative size = 352, 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 \[ -\frac{c^{3/2} (5 b e g-12 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 e^2}-\frac{c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (5 b e g-12 c d g+2 c e f)}{e^2 (2 c d-b e)}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^6 (2 c d-b e)}+\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (5 b e g-12 c d g+2 c e f)}{15 e^2 (d+e x)^4 (2 c d-b e)}-\frac{2 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (5 b e g-12 c d g+2 c e f)}{3 e^2 (d+e x)^2 (2 c d-b e)} \]

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)^6,x]

[Out]

-((c^2*(2*c*e*f - 12*c*d*g + 5*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])
/(e^2*(2*c*d - b*e))) - (2*c*(2*c*e*f - 12*c*d*g + 5*b*e*g)*(d*(c*d - b*e) - b*e
^2*x - c*e^2*x^2)^(3/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^2) + (2*(2*c*e*f - 12*c*
d*g + 5*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(15*e^2*(2*c*d - b*e
)*(d + e*x)^4) - (2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(5*
e^2*(2*c*d - b*e)*(d + e*x)^6) - (c^(3/2)*(2*c*e*f - 12*c*d*g + 5*b*e*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])])/(2*e^2)

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Rubi in Sympy [A]  time = 130.787, size = 340, normalized size = 0.97 \[ - \frac{2 c^{\frac{3}{2}} \left (\frac{5 b e g}{4} - 3 c d g + \frac{c e f}{2}\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{e^{2}} + \frac{2 c^{2} \left (\frac{5 b e g}{2} - 6 c d g + c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \left (b e - 2 c d\right )} + \frac{2 c \left (5 b e g - 12 c d g + 2 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{2} \left (b e - 2 c d\right )} - \frac{4 \left (\frac{5 b e g}{2} - 6 c d g + c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{15 e^{2} \left (d + e x\right )^{4} \left (b e - 2 c d\right )} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{5 e^{2} \left (d + e x\right )^{6} \left (b e - 2 c d\right )} \]

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)**6,x)

[Out]

-2*c**(3/2)*(5*b*e*g/4 - 3*c*d*g + c*e*f/2)*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*(5*b*e*g/2 - 6*c*d*g
+ c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(e**2*(b*e - 2*c*d)) + 2
*c*(5*b*e*g - 12*c*d*g + 2*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3
/2)/(3*e**2*(d + e*x)**2*(b*e - 2*c*d)) - 4*(5*b*e*g/2 - 6*c*d*g + c*e*f)*(-b*e*
*2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(15*e**2*(d + e*x)**4*(b*e - 2*c*d))
- 2*(d*g - e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(5*e**2*(d +
e*x)**6*(b*e - 2*c*d))

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Mathematica [C]  time = 2.34665, size = 244, normalized size = 0.69 \[ -\frac{((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{2 \left (2 c (d+e x)^2 (35 b e g-93 c d g+23 c e f)+2 (d+e x) (2 c d-b e) (-5 b e g+21 c d g-11 c e f)+6 (b e-2 c d)^2 (e f-d g)-15 c^2 g (d+e x)^3\right )}{(d+e x)^5 (b e-c d+c e x)^2}+\frac{15 i c^{3/2} (5 b e g+2 c (e f-6 d g)) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{(d+e x)^{5/2} (c (d-e x)-b e)^{5/2}}\right )}{30 e^2} \]

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)^6,x]

[Out]

-(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((2*(6*(-2*c*d + b*e)^2*(e*f - d*g) +
2*(2*c*d - b*e)*(-11*c*e*f + 21*c*d*g - 5*b*e*g)*(d + e*x) + 2*c*(23*c*e*f - 93
*c*d*g + 35*b*e*g)*(d + e*x)^2 - 15*c^2*g*(d + e*x)^3))/((d + e*x)^5*(-(c*d) + b
*e + c*e*x)^2) + ((15*I)*c^(3/2)*(5*b*e*g + 2*c*(e*f - 6*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))))/(30*e^2)

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Maple [B]  time = 0.034, size = 5440, normalized size = 15.5 \[ \text{output too large to display} \]

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)^6,x)

[Out]

result too large to display

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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)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.55119, 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)^6,x, algorithm="fricas")

[Out]

[1/60*(15*(2*c^2*d^3*e*f + (2*c^2*e^4*f - (12*c^2*d*e^3 - 5*b*c*e^4)*g)*x^3 + 3*
(2*c^2*d*e^3*f - (12*c^2*d^2*e^2 - 5*b*c*d*e^3)*g)*x^2 - (12*c^2*d^4 - 5*b*c*d^3
*e)*g + 3*(2*c^2*d^2*e^2*f - (12*c^2*d^3*e - 5*b*c*d^2*e^2)*g)*x)*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*(15*c^2*e^3*g*x^3 - (4
6*c^2*e^3*f - 7*(33*c^2*d*e^2 - 10*b*c*e^3)*g)*x^2 - 2*(13*c^2*d^2*e - b*c*d*e^2
+ 3*b^2*e^3)*f + (141*c^2*d^3 - 32*b*c*d^2*e - 4*b^2*d*e^2)*g - (2*(24*c^2*d*e^
2 + 11*b*c*e^3)*f - (333*c^2*d^2*e - 78*b*c*d*e^2 - 10*b^2*e^3)*g)*x)*sqrt(-c*e^
2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2
), -1/30*(15*(2*c^2*d^3*e*f + (2*c^2*e^4*f - (12*c^2*d*e^3 - 5*b*c*e^4)*g)*x^3 +
3*(2*c^2*d*e^3*f - (12*c^2*d^2*e^2 - 5*b*c*d*e^3)*g)*x^2 - (12*c^2*d^4 - 5*b*c*
d^3*e)*g + 3*(2*c^2*d^2*e^2*f - (12*c^2*d^3*e - 5*b*c*d^2*e^2)*g)*x)*sqrt(c)*arc
tan(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*(15*c^2*e^3*g*x^3 - (46*c^2*e^3*f - 7*(33*c^2*d*e^2 - 10*b*c*e^3)*g)*x^2 - 2*(
13*c^2*d^2*e - b*c*d*e^2 + 3*b^2*e^3)*f + (141*c^2*d^3 - 32*b*c*d^2*e - 4*b^2*d*
e^2)*g - (2*(24*c^2*d*e^2 + 11*b*c*e^3)*f - (333*c^2*d^2*e - 78*b*c*d*e^2 - 10*b
^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(e^5*x^3 + 3*d*e^4*x^2
+ 3*d^2*e^3*x + d^3*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)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.66553, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

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)^6,x, algorithm="giac")

[Out]

sage0*x