3.2273 \(\int \frac{f+g x}{\sqrt{d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.840659, 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 \[ -\frac{e f-d g}{e^2 \sqrt{d+e x} (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{\sqrt{d+e x} (-2 b e g+c d g+3 c e f)}{e^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{(-2 b e g+c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2 (2 c d-b e)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 91.5786, size = 206, normalized size = 0.92 \[ - \frac{\sqrt{d + e x} \left (2 b e g - c d g - 3 c e f\right )}{e^{2} \left (b e - 2 c d\right )^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{\left (2 b e g - c d g - 3 c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{e^{2} \left (b e - 2 c d\right )^{\frac{5}{2}}} - \frac{d g - e f}{e^{2} \sqrt{d + e x} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)/(e*x+d)**(1/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 1.32117, size = 188, normalized size = 0.84 \[ \frac{(d+e x)^{3/2} \left (\frac{(c (d-e x)-b e) \left (b e (e (f-2 g x)-3 d g)+c \left (3 d^2 g+d e (f+g x)+3 e^2 f x\right )\right )}{(d+e x) (b e-2 c d)^2}-\frac{(c (d-e x)-b e)^{3/2} (-2 b e g+c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(2 c d-b e)^{5/2}}\right )}{e^2 ((d+e x) (c (d-e x)-b e))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.042, size = 479, normalized size = 2.2 \[{\frac{1}{ \left ( cex+be-cd \right ){e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( 2\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) \sqrt{-cex-be+cd}xb{e}^{2}g-\arctan \left ({1\sqrt{-cex-be+cd}{\frac{1}{\sqrt{be-2\,cd}}}} \right ) \sqrt{-cex-be+cd}xcdeg-3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) \sqrt{-cex-be+cd}xc{e}^{2}f+2\,\sqrt{be-2\,cd}xb{e}^{2}g-\sqrt{be-2\,cd}xcdeg-3\,\sqrt{be-2\,cd}xc{e}^{2}f+2\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) \sqrt{-cex-be+cd}bdeg-\arctan \left ({1\sqrt{-cex-be+cd}{\frac{1}{\sqrt{be-2\,cd}}}} \right ) \sqrt{-cex-be+cd}c{d}^{2}g-3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) \sqrt{-cex-be+cd}cdef+3\,\sqrt{be-2\,cd}bdeg-\sqrt{be-2\,cd}b{e}^{2}f-3\,\sqrt{be-2\,cd}c{d}^{2}g-\sqrt{be-2\,cd}cdef \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}} \left ( be-2\,cd \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)/(e*x+d)^(1/2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)

[Out]

1/(e*x+d)^(3/2)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(2*arctan((-c*e*x-b*e+c*d
)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x*b*e^2*g-arctan((-c*e*x-b*e+c
*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x*c*d*e*g-3*arctan((-c*e*x-b
*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*x*c*e^2*f+2*(b*e-2*c*d)^
(1/2)*x*b*e^2*g-(b*e-2*c*d)^(1/2)*x*c*d*e*g-3*(b*e-2*c*d)^(1/2)*x*c*e^2*f+2*arct
an((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*b*d*e*g-arct
an((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c*d^2*g-3*ar
ctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*(-c*e*x-b*e+c*d)^(1/2)*c*d*e*f+3*
(b*e-2*c*d)^(1/2)*b*d*e*g-(b*e-2*c*d)^(1/2)*b*e^2*f-3*(b*e-2*c*d)^(1/2)*c*d^2*g-
(b*e-2*c*d)^(1/2)*c*d*e*f)/(c*e*x+b*e-c*d)/e^2/(b*e-2*c*d)^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.303313, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[1/2*(2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*((c*d*e + b
*e^2)*f + 3*(c*d^2 - b*d*e)*g + (3*c*e^2*f + (c*d*e - 2*b*e^2)*g)*x)*sqrt(e*x +
d) + ((3*c^2*e^4*f + (c^2*d*e^3 - 2*b*c*e^4)*g)*x^3 + (3*(c^2*d*e^3 + b*c*e^4)*f
+ (c^2*d^2*e^2 - b*c*d*e^3 - 2*b^2*e^4)*g)*x^2 - 3*(c^2*d^3*e - b*c*d^2*e^2)*f
- (c^2*d^4 - 3*b*c*d^3*e + 2*b^2*d^2*e^2)*g - (3*(c^2*d^2*e^2 - 2*b*c*d*e^3)*f +
(c^2*d^3*e - 4*b*c*d^2*e^2 + 4*b^2*d*e^3)*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)))/((4*c^3*d^
5*e^2 - 8*b*c^2*d^4*e^3 + 5*b^2*c*d^3*e^4 - b^3*d^2*e^5 - (4*c^3*d^2*e^5 - 4*b*c
^2*d*e^6 + b^2*c*e^7)*x^3 - (4*c^3*d^3*e^4 - 3*b^2*c*d*e^6 + b^3*e^7)*x^2 + (4*c
^3*d^4*e^3 - 12*b*c^2*d^3*e^4 + 9*b^2*c*d^2*e^5 - 2*b^3*d*e^6)*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)*((c*d*e + b
*e^2)*f + 3*(c*d^2 - b*d*e)*g + (3*c*e^2*f + (c*d*e - 2*b*e^2)*g)*x)*sqrt(e*x +
d) - ((3*c^2*e^4*f + (c^2*d*e^3 - 2*b*c*e^4)*g)*x^3 + (3*(c^2*d*e^3 + b*c*e^4)*f
+ (c^2*d^2*e^2 - b*c*d*e^3 - 2*b^2*e^4)*g)*x^2 - 3*(c^2*d^3*e - b*c*d^2*e^2)*f
- (c^2*d^4 - 3*b*c*d^3*e + 2*b^2*d^2*e^2)*g - (3*(c^2*d^2*e^2 - 2*b*c*d*e^3)*f +
(c^2*d^3*e - 4*b*c*d^2*e^2 + 4*b^2*d*e^3)*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^3*d^5*e^2 - 8*b*c^2*d^4*e^3 + 5*b^2*c*d^3*e^4 - b^3*d^2*e^5 -
(4*c^3*d^2*e^5 - 4*b*c^2*d*e^6 + b^2*c*e^7)*x^3 - (4*c^3*d^3*e^4 - 3*b^2*c*d*e^6
+ b^3*e^7)*x^2 + (4*c^3*d^4*e^3 - 12*b*c^2*d^3*e^4 + 9*b^2*c*d^2*e^5 - 2*b^3*d*
e^6)*x)*sqrt(-2*c*d + b*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{3}{2}} \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)/(e*x+d)**(1/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Integral((f + g*x)/((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*sqrt(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.620051, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

sage0*x