3.2238 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{9/2}} \, dx\)
Optimal. Leaf size=307 \[ \frac{c^2 (-2 b e g+3 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 (2 c d-b e)^{5/2}}+\frac{c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+3 c d g+c e f)}{8 e^2 (d+e x)^{3/2} (2 c d-b e)^2}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+3 c d g+c e f)}{4 e^2 (d+e x)^{5/2} (2 c d-b e)} \]
[Out]
-((c*e*f + 3*c*d*g - 2*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^2*
(2*c*d - b*e)*(d + e*x)^(5/2)) + (c*(c*e*f + 3*c*d*g - 2*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)^2*(d + e*x)^(3/2)) - ((e*f - d*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)^(9
/2)) + (c^2*(c*e*f + 3*c*d*g - 2*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)^(5/2))
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Rubi [A] time = 1.19082, antiderivative size = 307, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.109
\[ \frac{c^2 (-2 b e g+3 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{8 e^2 (2 c d-b e)^{5/2}}+\frac{c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+3 c d g+c e f)}{8 e^2 (d+e x)^{3/2} (2 c d-b e)^2}-\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{9/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+3 c d g+c e f)}{4 e^2 (d+e x)^{5/2} (2 c d-b e)} \]
Antiderivative was successfully verified.
[In] Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(9/2),x]
[Out]
-((c*e*f + 3*c*d*g - 2*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*e^2*
(2*c*d - b*e)*(d + e*x)^(5/2)) + (c*(c*e*f + 3*c*d*g - 2*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)^2*(d + e*x)^(3/2)) - ((e*f - d*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)^(9
/2)) + (c^2*(c*e*f + 3*c*d*g - 2*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)^(5/2))
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Rubi in Sympy [A] time = 119.872, size = 282, normalized size = 0.92 \[ \frac{c^{2} \left (2 b e g - 3 c d g - c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{8 e^{2} \left (b e - 2 c d\right )^{\frac{5}{2}}} - \frac{c \left (2 b e g - 3 c d g - c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{8 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right )^{2}} - \frac{\left (2 b e g - 3 c d g - c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{4 e^{2} \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )} - \frac{\left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right )} \]
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)**(1/2)/(e*x+d)**(9/2),x)
[Out]
c**2*(2*b*e*g - 3*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)**(5/2)) - c*(2*b*e
*g - 3*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)**2) - (2*b*e*g - 3*c*d*g - c*e*f)*sqrt(-b*e**2*x - c
*e**2*x**2 + d*(-b*e + c*d))/(4*e**2*(d + e*x)**(5/2)*(b*e - 2*c*d)) - (d*g - e*
f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(3/2)/(3*e**2*(d + e*x)**(9/2)*(b
*e - 2*c*d))
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Mathematica [A] time = 1.14149, size = 209, normalized size = 0.68 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{3 c^2 (-2 b e g+3 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(2 c d-b e)^{5/2} \sqrt{c (d-e x)-b e}}+\frac{3 c (-2 b e g+3 c d g+c e f)}{(d+e x) (b e-2 c d)^2}+\frac{2 (6 b e g-13 c d g+c e f)}{(d+e x)^2 (2 c d-b e)}+\frac{8 (d g-e f)}{(d+e x)^3}\right )}{24 e^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(9/2),x]
[Out]
(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((8*(-(e*f) + d*g))/(d + e*x)^3 + (2*(c*
e*f - 13*c*d*g + 6*b*e*g))/((2*c*d - b*e)*(d + e*x)^2) + (3*c*(c*e*f + 3*c*d*g -
2*b*e*g))/((-2*c*d + b*e)^2*(d + e*x)) + (3*c^2*(c*e*f + 3*c*d*g - 2*b*e*g)*Arc
Tanh[Sqrt[c*d - b*e - c*e*x]/Sqrt[2*c*d - b*e]])/((2*c*d - b*e)^(5/2)*Sqrt[-(b*e
) + c*(d - e*x)])))/(24*e^2*Sqrt[d + e*x])
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Maple [B] time = 0.046, size = 1033, normalized size = 3.4 \[ \text{result too large to display} \]
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)^(1/2)/(e*x+d)^(9/2),x)
[Out]
1/24*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(-8*b^2*e^3*f*(b*e-2*c*d)^(1/2)*(-c*
e*x-b*e+c*d)^(1/2)-11*c^2*d^3*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-3*arcta
n((-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-9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*
c*d)^(1/2))*c^3*d^4*g-2*x*b*c*e^3*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)-34*
x*c^2*d^2*e*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+10*x*c^2*d*e^2*f*(b*e-2*c
*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+12*b*c*d^2*e*g*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*
d)^(1/2)+30*b*c*d*e^2*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^2*b*c^2*d*e^3*g+18*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-6*x^2*b*c*e^3*g*(b*e-2*c*d)^(1/2)*(
-c*e*x-b*e+c*d)^(1/2)+9*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)-25*c^2*d^2*e*f*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+6*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-9*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-27*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-27*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+6*arctan((-
c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^3*e*g+38*x*b*c*d*e^2*g*(b*e-2*c*
d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2))/(e*x+d)^(7/2)/(b*e-2*c*d)^(5/2)/e^2/(-c*e*x-b*e
+c*d)^(1/2)
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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(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^(9/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.316965, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^(9/2),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 + (3*c^2*d*e^2
- 2*b*c*e^3)*g)*x^2 - (25*c^2*d^2*e - 30*b*c*d*e^2 + 8*b^2*e^3)*f - (11*c^2*d^3
- 12*b*c*d^2*e + 4*b^2*d*e^2)*g + 2*((5*c^2*d*e^2 - b*c*e^3)*f - (17*c^2*d^2*e
- 19*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 + (3*c^3*d*e^4 - 2*b*c^2*e^5)*g)*x^4 + 4*(c^3*d*e^4*f + (3*c^3*d
^2*e^3 - 2*b*c^2*d*e^4)*g)*x^3 + 6*(c^3*d^2*e^3*f + (3*c^3*d^3*e^2 - 2*b*c^2*d^2
*e^3)*g)*x^2 + (3*c^3*d^5 - 2*b*c^2*d^4*e)*g + 4*(c^3*d^3*e^2*f + (3*c^3*d^4*e -
2*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)*s
qrt(2*c*d - b*e))/(e^2*x^2 + 2*d*e*x + d^2)))/((4*c^2*d^6*e^2 - 4*b*c*d^5*e^3 +
b^2*d^4*e^4 + (4*c^2*d^2*e^6 - 4*b*c*d*e^7 + b^2*e^8)*x^4 + 4*(4*c^2*d^3*e^5 - 4
*b*c*d^2*e^6 + b^2*d*e^7)*x^3 + 6*(4*c^2*d^4*e^4 - 4*b*c*d^3*e^5 + b^2*d^2*e^6)*
x^2 + 4*(4*c^2*d^5*e^3 - 4*b*c*d^4*e^4 + b^2*d^3*e^5)*x)*sqrt(2*c*d - b*e)), 1/2
4*(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(3*(c^2*e^3*f + (3*c^2*d*e^2 - 2*b
*c*e^3)*g)*x^2 - (25*c^2*d^2*e - 30*b*c*d*e^2 + 8*b^2*e^3)*f - (11*c^2*d^3 - 12*
b*c*d^2*e + 4*b^2*d*e^2)*g + 2*((5*c^2*d*e^2 - b*c*e^3)*f - (17*c^2*d^2*e - 19*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 + (3*c^3*d*e^4 - 2*b*c^2*e^5)*g)*x^4 + 4*(c^3*d*e^4*f + (3*c^3*d^2*e^
3 - 2*b*c^2*d*e^4)*g)*x^3 + 6*(c^3*d^2*e^3*f + (3*c^3*d^3*e^2 - 2*b*c^2*d^2*e^3)
*g)*x^2 + (3*c^3*d^5 - 2*b*c^2*d^4*e)*g + 4*(c^3*d^3*e^2*f + (3*c^3*d^4*e - 2*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)))/((4*c^2*d^6*e^2 -
4*b*c*d^5*e^3 + b^2*d^4*e^4 + (4*c^2*d^2*e^6 - 4*b*c*d*e^7 + b^2*e^8)*x^4 + 4*(4
*c^2*d^3*e^5 - 4*b*c*d^2*e^6 + b^2*d*e^7)*x^3 + 6*(4*c^2*d^4*e^4 - 4*b*c*d^3*e^5
+ b^2*d^2*e^6)*x^2 + 4*(4*c^2*d^5*e^3 - 4*b*c*d^4*e^4 + b^2*d^3*e^5)*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)**(1/2)/(e*x+d)**(9/2),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(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^(9/2),x, algorithm="giac")
[Out]
Timed out