3.2236 \(\int \frac{(f+g x) \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=223 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-5 c d g+c e f)}{e^2 \sqrt{d+e x} (2 c d-b e)}+\frac{(2 b e g-5 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2 \sqrt{2 c d-b e}} \]

[Out]

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

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Rubi [A]  time = 0.814991, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^{5/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-5 c d g+c e f)}{e^2 \sqrt{d+e x} (2 c d-b e)}+\frac{(2 b e g-5 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2 \sqrt{2 c d-b e}} \]

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)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 91.2422, size = 202, normalized size = 0.91 \[ - \frac{\left (2 b e g - 5 c d g + c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{e^{2} \sqrt{b e - 2 c d}} + \frac{\left (2 b e g - 5 c d g + c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \sqrt{d + e x} \left (b e - 2 c d\right )} - \frac{\left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{e^{2} \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )} \]

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)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.370625, size = 134, normalized size = 0.6 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (\frac{(2 b e g-5 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{\sqrt{2 c d-b e} \sqrt{c (d-e x)-b e}}+\frac{d g-e f}{d+e x}+2 g\right )}{e^2 \sqrt{d+e x}} \]

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)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.032, size = 359, normalized size = 1.6 \[{\frac{1}{{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( -2\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) xb{e}^{2}g+5\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) xcdeg-\arctan \left ({1\sqrt{-cex-be+cd}{\frac{1}{\sqrt{be-2\,cd}}}} \right ) xc{e}^{2}f+2\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}xeg-2\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bdeg+5\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) c{d}^{2}g-\arctan \left ({1\sqrt{-cex-be+cd}{\frac{1}{\sqrt{be-2\,cd}}}} \right ) cdef+3\,\sqrt{be-2\,cd}\sqrt{-cex-be+cd}dg-\sqrt{be-2\,cd}\sqrt{-cex-be+cd}ef \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-cex-be+cd}}}{\frac{1}{\sqrt{be-2\,cd}}}} \]

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)^(5/2),x)

[Out]

(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(3/2)*(-2*arctan((-c*e*x-b*e+c*d)
^(1/2)/(b*e-2*c*d)^(1/2))*x*b*e^2*g+5*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^
(1/2))*x*c*d*e*g-arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c*e^2*f+2*(b
*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*x*e*g-2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*
e-2*c*d)^(1/2))*b*d*e*g+5*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c*d^2
*g-arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c*d*e*f+3*(b*e-2*c*d)^(1/2)*
(-c*e*x-b*e+c*d)^(1/2)*d*g-(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)*e*f)/(-c*e*x
-b*e+c*d)^(1/2)/e^2/(b*e-2*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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.299477, size = 1, normalized size = 0. \[ \left [\frac{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c e f -{\left (5 \, c d - 2 \, b e\right )} g\right )} \sqrt{e x + d} \log \left (\frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c d - b e\right )} \sqrt{e x + d} -{\left (c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \,{\left (c d e - b e^{2}\right )} x\right )} \sqrt{2 \, c d - b e}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \,{\left (2 \, c e^{2} g x^{2} +{\left (c d e - b e^{2}\right )} f - 3 \,{\left (c d^{2} - b d e\right )} g -{\left (c e^{2} f -{\left (c d e + 2 \, b e^{2}\right )} g\right )} x\right )} \sqrt{2 \, c d - b e}}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{2 \, c d - b e} \sqrt{e x + d} e^{2}}, -\frac{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (c e f -{\left (5 \, c d - 2 \, b e\right )} g\right )} \sqrt{e x + d} \arctan \left (\frac{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{-2 \, c d + b e} \sqrt{e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) +{\left (2 \, c e^{2} g x^{2} +{\left (c d e - b e^{2}\right )} f - 3 \,{\left (c d^{2} - b d e\right )} g -{\left (c e^{2} f -{\left (c d e + 2 \, b e^{2}\right )} g\right )} x\right )} \sqrt{-2 \, c d + b e}}{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{-2 \, c d + b e} \sqrt{e x + d} e^{2}}\right ] \]

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)^(5/2),x, algorithm="fricas")

[Out]

[1/2*(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(c*e*f - (5*c*d - 2*b*e)*g)*sqr
t(e*x + d)*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*(2*c*e^2*g*x^2 + (c*d*e - b*e^2)*f - 3*(c*d^2
- b*d*e)*g - (c*e^2*f - (c*d*e + 2*b*e^2)*g)*x)*sqrt(2*c*d - b*e))/(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)*e^2), -(sqrt(-c*
e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(c*e*f - (5*c*d - 2*b*e)*g)*sqrt(e*x + d)*arc
tan(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*e^2*g*x^2 + (c*d*e - b*e^2)*f - 3*
(c*d^2 - b*d*e)*g - (c*e^2*f - (c*d*e + 2*b*e^2)*g)*x)*sqrt(-2*c*d + b*e))/(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)*e^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]

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)**(5/2),x)

[Out]

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**(5/2), x)

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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)^(5/2),x, algorithm="giac")

[Out]

Timed out