3.2274 \(\int \frac{f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=308 \[ -\frac{e f-d g}{2 e^2 (d+e x)^{3/2} (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{3 c \sqrt{d+e x} (-4 b e g+3 c d g+5 c e f)}{4 e^2 (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{-4 b e g+3 c d g+5 c e f}{4 e^2 \sqrt{d+e x} (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{3 c (-4 b e g+3 c d g+5 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 1.08291, antiderivative size = 308, 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 \[ -\frac{e f-d g}{2 e^2 (d+e x)^{3/2} (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{3 c \sqrt{d+e x} (-4 b e g+3 c d g+5 c e f)}{4 e^2 (2 c d-b e)^3 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{-4 b e g+3 c d g+5 c e f}{4 e^2 \sqrt{d+e x} (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{3 c (-4 b e g+3 c d g+5 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 (2 c d-b e)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 125.394, size = 291, normalized size = 0.94 \[ \frac{3 c \sqrt{d + e x} \left (4 b e g - 3 c d g - 5 c e f\right )}{4 e^{2} \left (b e - 2 c d\right )^{3} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{3 c \left (4 b e g - 3 c d g - 5 c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{4 e^{2} \left (b e - 2 c d\right )^{\frac{7}{2}}} + \frac{4 b e g - 3 c d g - 5 c e f}{4 e^{2} \sqrt{d + e x} \left (b e - 2 c d\right )^{2} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} - \frac{d g - e f}{2 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*c*sqrt(d + e*x)*(4*b*e*g - 3*c*d*g - 5*c*e*f)/(4*e**2*(b*e - 2*c*d)**3*sqrt(-b
*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))) + 3*c*(4*b*e*g - 3*c*d*g - 5*c*e*f)*ata
n(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
)))/(4*e**2*(b*e - 2*c*d)**(7/2)) + (4*b*e*g - 3*c*d*g - 5*c*e*f)/(4*e**2*sqrt(d
+ e*x)*(b*e - 2*c*d)**2*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))) - (d*g
- e*f)/(2*e**2*(d + e*x)**(3/2)*(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 = 1.97501, size = 243, normalized size = 0.79 \[ \frac{(d+e x)^{3/2} \left (\frac{(c (d-e x)-b e) \left (8 c (d+e x)^2 (-b e g+c d g+c e f)+(d+e x) (b e-c d+c e x) (-4 b e g+c d g+7 c e f)+2 (2 c d-b e) (d g-e f) (c (d-e x)-b e)\right )}{(d+e x)^2 (2 c d-b e)^3}-\frac{3 c (c (d-e x)-b e)^{3/2} (-4 b e g+3 c d g+5 c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(2 c d-b e)^{7/2}}\right )}{4 e^2 ((d+e x) (c (d-e x)-b e))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(f + g*x)/((d + e*x)^(3/2)*(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))*(8*c*(c*e*f + c*d*g - b*e*g)*(d + e*x)
^2 + (7*c*e*f + c*d*g - 4*b*e*g)*(d + e*x)*(-(c*d) + b*e + c*e*x) + 2*(2*c*d - b
*e)*(-(e*f) + d*g)*(-(b*e) + c*(d - e*x))))/((2*c*d - b*e)^3*(d + e*x)^2) - (3*c
*(5*c*e*f + 3*c*d*g - 4*b*e*g)*(-(b*e) + c*(d - e*x))^(3/2)*ArcTanh[Sqrt[c*d - b
*e - c*e*x]/Sqrt[2*c*d - b*e]])/(2*c*d - b*e)^(7/2)))/(4*e^2*((d + e*x)*(-(b*e)
+ c*(d - e*x)))^(3/2))

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Maple [B]  time = 0.046, size = 824, normalized size = 2.7 \[ -{\frac{1}{ \left ( 4\,cex+4\,be-4\,cd \right ){e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( 12\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) \sqrt{-cex-be+cd}{x}^{2}bc{e}^{3}g-9\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) \sqrt{-cex-be+cd}{x}^{2}{c}^{2}d{e}^{2}g-15\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) \sqrt{-cex-be+cd}{x}^{2}{c}^{2}{e}^{3}f+12\,\sqrt{be-2\,cd}{x}^{2}bc{e}^{3}g-9\,\sqrt{be-2\,cd}{x}^{2}{c}^{2}d{e}^{2}g-15\,\sqrt{be-2\,cd}{x}^{2}{c}^{2}{e}^{3}f+24\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) \sqrt{-cex-be+cd}xbcd{e}^{2}g-18\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) \sqrt{-cex-be+cd}x{c}^{2}{d}^{2}eg-30\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) \sqrt{-cex-be+cd}x{c}^{2}d{e}^{2}f+4\,\sqrt{be-2\,cd}x{b}^{2}{e}^{3}g+13\,\sqrt{be-2\,cd}xbcd{e}^{2}g-5\,\sqrt{be-2\,cd}xbc{e}^{3}f-12\,\sqrt{be-2\,cd}x{c}^{2}{d}^{2}eg-20\,\sqrt{be-2\,cd}x{c}^{2}d{e}^{2}f+12\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) \sqrt{-cex-be+cd}bc{d}^{2}eg-9\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) \sqrt{-cex-be+cd}{c}^{2}{d}^{3}g-15\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) \sqrt{-cex-be+cd}{c}^{2}{d}^{2}ef+2\,\sqrt{be-2\,cd}{b}^{2}d{e}^{2}g+2\,\sqrt{be-2\,cd}{b}^{2}{e}^{3}f+9\,\sqrt{be-2\,cd}bc{d}^{2}eg-13\,\sqrt{be-2\,cd}bcd{e}^{2}f-11\,\sqrt{be-2\,cd}{c}^{2}{d}^{3}g+3\,\sqrt{be-2\,cd}{c}^{2}{d}^{2}ef \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}} \left ( be-2\,cd \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/(e*x+d)^(5/2)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(12*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^2*b*c*e^3*g-9*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^2*c^2*d*e^2*g-15
*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^2*c^2
*e^3*f+12*(b*e-2*c*d)^(1/2)*x^2*b*c*e^3*g-9*(b*e-2*c*d)^(1/2)*x^2*c^2*d*e^2*g-15
*(b*e-2*c*d)^(1/2)*x^2*c^2*e^3*f+24*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*c*d*e^2*g-18*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^2*d^2*e*g-30*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^2*d*e^2*f+4*(b*e-2*c*d)^(1/2)*
x*b^2*e^3*g+13*(b*e-2*c*d)^(1/2)*x*b*c*d*e^2*g-5*(b*e-2*c*d)^(1/2)*x*b*c*e^3*f-1
2*(b*e-2*c*d)^(1/2)*x*c^2*d^2*e*g-20*(b*e-2*c*d)^(1/2)*x*c^2*d*e^2*f+12*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)*b*c*d^2*e*g-9*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^2*d^3*g-
15*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)*c^2*d
^2*e*f+2*(b*e-2*c*d)^(1/2)*b^2*d*e^2*g+2*(b*e-2*c*d)^(1/2)*b^2*e^3*f+9*(b*e-2*c*
d)^(1/2)*b*c*d^2*e*g-13*(b*e-2*c*d)^(1/2)*b*c*d*e^2*f-11*(b*e-2*c*d)^(1/2)*c^2*d
^3*g+3*(b*e-2*c*d)^(1/2)*c^2*d^2*e*f)/(c*e*x+b*e-c*d)/e^2/(b*e-2*c*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((g*x + f)/((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.318196, 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)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[1/8*(2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(3*(5*c^2*e^3*f + (3*c^2*d*e^
2 - 4*b*c*e^3)*g)*x^2 - (3*c^2*d^2*e - 13*b*c*d*e^2 + 2*b^2*e^3)*f + (11*c^2*d^3
- 9*b*c*d^2*e - 2*b^2*d*e^2)*g + (5*(4*c^2*d*e^2 + b*c*e^3)*f + (12*c^2*d^2*e -
13*b*c*d*e^2 - 4*b^2*e^3)*g)*x)*sqrt(2*c*d - b*e)*sqrt(e*x + d) - 3*((5*c^3*e^5
*f + (3*c^3*d*e^4 - 4*b*c^2*e^5)*g)*x^4 + (5*(2*c^3*d*e^4 + b*c^2*e^5)*f + (6*c^
3*d^2*e^3 - 5*b*c^2*d*e^4 - 4*b^2*c*e^5)*g)*x^3 + 3*(5*b*c^2*d*e^4*f + (3*b*c^2*
d^2*e^3 - 4*b^2*c*d*e^4)*g)*x^2 - 5*(c^3*d^4*e - b*c^2*d^3*e^2)*f - (3*c^3*d^5 -
7*b*c^2*d^4*e + 4*b^2*c*d^3*e^2)*g - (5*(2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3)*f + (
6*c^3*d^4*e - 17*b*c^2*d^3*e^2 + 12*b^2*c*d^2*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)))/(
(8*c^4*d^7*e^2 - 20*b*c^3*d^6*e^3 + 18*b^2*c^2*d^5*e^4 - 7*b^3*c*d^4*e^5 + b^4*d
^3*e^6 - (8*c^4*d^3*e^6 - 12*b*c^3*d^2*e^7 + 6*b^2*c^2*d*e^8 - b^3*c*e^9)*x^4 -
(16*c^4*d^4*e^5 - 16*b*c^3*d^3*e^6 + 4*b^3*c*d*e^8 - b^4*e^9)*x^3 - 3*(8*b*c^3*d
^4*e^5 - 12*b^2*c^2*d^3*e^6 + 6*b^3*c*d^2*e^7 - b^4*d*e^8)*x^2 + (16*c^4*d^6*e^3
- 48*b*c^3*d^5*e^4 + 48*b^2*c^2*d^4*e^5 - 20*b^3*c*d^3*e^6 + 3*b^4*d^2*e^7)*x)*
sqrt(2*c*d - b*e)), 1/4*(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(3*(5*c^2*e^
3*f + (3*c^2*d*e^2 - 4*b*c*e^3)*g)*x^2 - (3*c^2*d^2*e - 13*b*c*d*e^2 + 2*b^2*e^3
)*f + (11*c^2*d^3 - 9*b*c*d^2*e - 2*b^2*d*e^2)*g + (5*(4*c^2*d*e^2 + b*c*e^3)*f
+ (12*c^2*d^2*e - 13*b*c*d*e^2 - 4*b^2*e^3)*g)*x)*sqrt(-2*c*d + b*e)*sqrt(e*x +
d) - 3*((5*c^3*e^5*f + (3*c^3*d*e^4 - 4*b*c^2*e^5)*g)*x^4 + (5*(2*c^3*d*e^4 + b*
c^2*e^5)*f + (6*c^3*d^2*e^3 - 5*b*c^2*d*e^4 - 4*b^2*c*e^5)*g)*x^3 + 3*(5*b*c^2*d
*e^4*f + (3*b*c^2*d^2*e^3 - 4*b^2*c*d*e^4)*g)*x^2 - 5*(c^3*d^4*e - b*c^2*d^3*e^2
)*f - (3*c^3*d^5 - 7*b*c^2*d^4*e + 4*b^2*c*d^3*e^2)*g - (5*(2*c^3*d^3*e^2 - 3*b*
c^2*d^2*e^3)*f + (6*c^3*d^4*e - 17*b*c^2*d^3*e^2 + 12*b^2*c*d^2*e^3)*g)*x)*arcta
n(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)))/((8*c^4*d^7*e^2 - 20*b*c^3*d^6*e^3 + 18*b^
2*c^2*d^5*e^4 - 7*b^3*c*d^4*e^5 + b^4*d^3*e^6 - (8*c^4*d^3*e^6 - 12*b*c^3*d^2*e^
7 + 6*b^2*c^2*d*e^8 - b^3*c*e^9)*x^4 - (16*c^4*d^4*e^5 - 16*b*c^3*d^3*e^6 + 4*b^
3*c*d*e^8 - b^4*e^9)*x^3 - 3*(8*b*c^3*d^4*e^5 - 12*b^2*c^2*d^3*e^6 + 6*b^3*c*d^2
*e^7 - b^4*d*e^8)*x^2 + (16*c^4*d^6*e^3 - 48*b*c^3*d^5*e^4 + 48*b^2*c^2*d^4*e^5
- 20*b^3*c*d^3*e^6 + 3*b^4*d^2*e^7)*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)/(e*x+d)**(3/2)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

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

[Out]

[undef, undef, undef, undef, undef, undef, undef, 1]