3.2248 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=307 \[ -\frac{c^2 (-6 b e g+11 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 (2 c d-b e)^{3/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (d+e x)^{11/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-6 b e g+11 c d g+c e f)}{12 e^2 (d+e x)^{7/2} (2 c d-b e)}+\frac{c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+11 c d g+c e f)}{8 e^2 (d+e x)^{3/2} (2 c d-b e)} \]

[Out]

(c*(c*e*f + 11*c*d*g - 6*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(8*e^
2*(2*c*d - b*e)*(d + e*x)^(3/2)) - ((c*e*f + 11*c*d*g - 6*b*e*g)*(d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2)^(3/2))/(12*e^2*(2*c*d - b*e)*(d + e*x)^(7/2)) - ((e*f - d
*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^
(11/2)) - (c^2*(c*e*f + 11*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)^(3/2))

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Rubi [A]  time = 1.06678, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{c^2 (-6 b e g+11 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 (2 c d-b e)^{3/2}}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (d+e x)^{11/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-6 b e g+11 c d g+c e f)}{12 e^2 (d+e x)^{7/2} (2 c d-b e)}+\frac{c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+11 c d g+c e f)}{8 e^2 (d+e x)^{3/2} (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^(11/2),x]

[Out]

(c*(c*e*f + 11*c*d*g - 6*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(8*e^
2*(2*c*d - b*e)*(d + e*x)^(3/2)) - ((c*e*f + 11*c*d*g - 6*b*e*g)*(d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2)^(3/2))/(12*e^2*(2*c*d - b*e)*(d + e*x)^(7/2)) - ((e*f - d
*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3*e^2*(2*c*d - b*e)*(d + e*x)^
(11/2)) - (c^2*(c*e*f + 11*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)^(3/2))

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Rubi in Sympy [A]  time = 119.25, size = 280, normalized size = 0.91 \[ \frac{c^{2} \left (6 b e g - 11 c d g - c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{8 e^{2} \left (b e - 2 c d\right )^{\frac{3}{2}}} + \frac{c \left (6 b e g - 11 c d g - c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{8 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right )} - \frac{\left (6 b e g - 11 c d g - c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{12 e^{2} \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right )} - \frac{\left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{3 e^{2} \left (d + e x\right )^{\frac{11}{2}} \left (b e - 2 c d\right )} \]

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)**(3/2)/(e*x+d)**(11/2),x)

[Out]

c**2*(6*b*e*g - 11*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)))/(8*e**2*(b*e - 2*c*d)**(3/2)) + c*(6*b*
e*g - 11*c*d*g - c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(8*e**2*(
d + e*x)**(3/2)*(b*e - 2*c*d)) - (6*b*e*g - 11*c*d*g - c*e*f)*(-b*e**2*x - c*e**
2*x**2 + d*(-b*e + c*d))**(3/2)/(12*e**2*(d + e*x)**(7/2)*(b*e - 2*c*d)) - (d*g
- e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(3*e**2*(d + e*x)**(11/
2)*(b*e - 2*c*d))

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Mathematica [A]  time = 2.02098, size = 234, normalized size = 0.76 \[ -\frac{((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac{3 c^2 (-6 b e g+11 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(2 c d-b e)^{3/2} (c (d-e x)-b e)^{3/2}}+\frac{3 c (d+e x)^2 (10 b e g-21 c d g+c e f)+2 (d+e x) (2 c d-b e) (-6 b e g+19 c d g-7 c e f)+8 (b e-2 c d)^2 (e f-d g)}{(d+e x)^3 (2 c d-b e) (c (d-e x)-b e)}\right )}{24 e^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^(11/2),x]

[Out]

-(((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((8*(-2*c*d + b*e)^2*(e*f - d*g) + 2*
(2*c*d - b*e)*(-7*c*e*f + 19*c*d*g - 6*b*e*g)*(d + e*x) + 3*c*(c*e*f - 21*c*d*g
+ 10*b*e*g)*(d + e*x)^2)/((2*c*d - b*e)*(d + e*x)^3*(-(b*e) + c*(d - e*x))) + (3
*c^2*(c*e*f + 11*c*d*g - 6*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))^(3/2))))/(24*e^2*(d + e*x)^(3/
2))

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Maple [B]  time = 0.046, size = 999, normalized size = 3.3 \[ \text{result too large to display} \]

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)^(3/2)/(e*x+d)^(11/2),x)

[Out]

1/24*(8*b^2*e^3*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-19*c^2*d^3*g*(b*e-2*c
*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/
2))*x^3*c^3*e^4*f-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*e*f
-33*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^4*g+14*x*b*c*e^3*f*(b
*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-50*x*c^2*d^2*e*g*(b*e-2*c*d)^(1/2)*(-c*e*
x-b*e+c*d)^(1/2)-22*x*c^2*d*e^2*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-18*b*
c*d*e^2*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+54*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+54*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+30*x^2*b*c*e^3*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*
d)^(1/2)-63*x^2*c^2*d*e^2*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+3*x^2*c^2*e
^3*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+12*x*b^2*e^3*g*(b*e-2*c*d)^(1/2)*(
-c*e*x-b*e+c*d)^(1/2)+4*b^2*d*e^2*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+7*c
^2*d^2*e*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+18*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-33*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-99*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-9*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-99*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^3*d^3*e*g-9*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+18*arctan((-c*e*x-b*e+
c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^3*e*g-2*x*b*c*d*e^2*g*(b*e-2*c*d)^(1/2)*(-
c*e*x-b*e+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)^(7/2)

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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((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/(e*x + d)^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312122, 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)^(3/2)*(g*x + f)/(e*x + d)^(11/2),x, algorithm="fricas")

[Out]

[-1/48*(2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(3*(c^2*e^3*f - (21*c^2*d*e
^2 - 10*b*c*e^3)*g)*x^2 + (7*c^2*d^2*e - 18*b*c*d*e^2 + 8*b^2*e^3)*f - (19*c^2*d
^3 - 4*b^2*d*e^2)*g - 2*((11*c^2*d*e^2 - 7*b*c*e^3)*f + (25*c^2*d^2*e + b*c*d*e^
2 - 6*b^2*e^3)*g)*x)*sqrt(2*c*d - b*e)*sqrt(e*x + d) - 3*(c^3*d^4*e*f + (c^3*e^5
*f + (11*c^3*d*e^4 - 6*b*c^2*e^5)*g)*x^4 + 4*(c^3*d*e^4*f + (11*c^3*d^2*e^3 - 6*
b*c^2*d*e^4)*g)*x^3 + 6*(c^3*d^2*e^3*f + (11*c^3*d^3*e^2 - 6*b*c^2*d^2*e^3)*g)*x
^2 + (11*c^3*d^5 - 6*b*c^2*d^4*e)*g + 4*(c^3*d^3*e^2*f + (11*c^3*d^4*e - 6*b*c^2
*d^3*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^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)*sqrt(2*c*d - b*e)), -1/24*(sqrt(-c*e^2*x^2 - b*e^2*
x + c*d^2 - b*d*e)*(3*(c^2*e^3*f - (21*c^2*d*e^2 - 10*b*c*e^3)*g)*x^2 + (7*c^2*d
^2*e - 18*b*c*d*e^2 + 8*b^2*e^3)*f - (19*c^2*d^3 - 4*b^2*d*e^2)*g - 2*((11*c^2*d
*e^2 - 7*b*c*e^3)*f + (25*c^2*d^2*e + b*c*d*e^2 - 6*b^2*e^3)*g)*x)*sqrt(-2*c*d +
b*e)*sqrt(e*x + d) - 3*(c^3*d^4*e*f + (c^3*e^5*f + (11*c^3*d*e^4 - 6*b*c^2*e^5)
*g)*x^4 + 4*(c^3*d*e^4*f + (11*c^3*d^2*e^3 - 6*b*c^2*d*e^4)*g)*x^3 + 6*(c^3*d^2*
e^3*f + (11*c^3*d^3*e^2 - 6*b*c^2*d^2*e^3)*g)*x^2 + (11*c^3*d^5 - 6*b*c^2*d^4*e)
*g + 4*(c^3*d^3*e^2*f + (11*c^3*d^4*e - 6*b*c^2*d^3*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^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)*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)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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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)^(3/2)*(g*x + f)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

Timed out