3.2246 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=288 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^{7/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (2 b e g-7 c d g+3 c e f)}{3 e^2 (d+e x)^{3/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-7 c d g+3 c e f)}{e^2 \sqrt{d+e x}}+\frac{\sqrt{2 c d-b e} (2 b e g-7 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2} \]

[Out]

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

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Rubi [A]  time = 1.08266, antiderivative size = 288, 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 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^{7/2} (2 c d-b e)}-\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (2 b e g-7 c d g+3 c e f)}{3 e^2 (d+e x)^{3/2} (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (2 b e g-7 c d g+3 c e f)}{e^2 \sqrt{d+e x}}+\frac{\sqrt{2 c d-b e} (2 b e g-7 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2} \]

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)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 117.898, size = 270, normalized size = 0.94 \[ \frac{\sqrt{b e - 2 c d} \left (2 b e g - 7 c d g + 3 c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{e^{2}} - \frac{2 \left (b e g - \frac{7 c d g}{2} + \frac{3 c e f}{2}\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \sqrt{d + e x}} + \frac{\left (2 b e g - 7 c d g + 3 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - 2 c d\right )} - \frac{\left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{e^{2} \left (d + e x\right )^{\frac{7}{2}} \left (b e - 2 c d\right )} \]

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)**(7/2),x)

[Out]

sqrt(b*e - 2*c*d)*(2*b*e*g - 7*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 - 2*(b*e*g - 7*c*d*g
/2 + 3*c*e*f/2)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(e**2*sqrt(d + e*
x)) + (2*b*e*g - 7*c*d*g + 3*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**
(3/2)/(3*e**2*(d + e*x)**(3/2)*(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)/(e**2*(d + e*x)**(7/2)*(b*e - 2*c*d))

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Mathematica [A]  time = 1.10429, size = 195, normalized size = 0.68 \[ \frac{((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac{b e (-11 d g+3 e f-8 e g x)+2 c \left (13 d^2 g+d e (9 g x-6 f)-e^2 x (3 f+g x)\right )}{(d+e x) (c (d-e x)-b e)}+\frac{3 \sqrt{2 c d-b e} (2 b e g-7 c d g+3 c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(c (d-e x)-b e)^{3/2}}\right )}{3 e^2 (d+e x)^{3/2}} \]

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)^(7/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((b*e*(3*e*f - 11*d*g - 8*e*g*x) + 2*c
*(13*d^2*g - e^2*x*(3*f + g*x) + d*e*(-6*f + 9*g*x)))/((d + e*x)*(-(b*e) + c*(d
- e*x))) + (3*Sqrt[2*c*d - b*e]*(3*c*e*f - 7*c*d*g + 2*b*e*g)*ArcTanh[Sqrt[c*d -
b*e - c*e*x]/Sqrt[2*c*d - b*e]])/(-(b*e) + c*(d - e*x))^(3/2)))/(3*e^2*(d + e*x
)^(3/2))

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Maple [B]  time = 0.038, size = 695, normalized size = 2.4 \[{\frac{1}{3\,{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( 6\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{b}^{2}{e}^{3}g-33\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) xbcd{e}^{2}g+9\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) xbc{e}^{3}f+42\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}{d}^{2}eg-18\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}d{e}^{2}f-2\,{x}^{2}c{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+6\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{2}d{e}^{2}g-33\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bc{d}^{2}eg+9\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bcd{e}^{2}f+42\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}{d}^{3}g-18\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}{d}^{2}ef-8\,xb{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+18\,xcdeg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-6\,xc{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-11\,bdeg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+3\,b{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+26\,c{d}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-12\,cdef\sqrt{-cex-be+cd}\sqrt{be-2\,cd} \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-cex-be+cd}}}{\frac{1}{\sqrt{be-2\,cd}}}} \]

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)^(3/2)/(e*x+d)^(7/2),x)

[Out]

1/3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e
-2*c*d)^(1/2))*x*b^2*e^3*g-33*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x
*b*c*d*e^2*g+9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*b*c*e^3*f+42*a
rctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^2*d^2*e*g-18*arctan((-c*e*x-
b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^2*d*e^2*f-2*x^2*c*e^2*g*(-c*e*x-b*e+c*d)^(
1/2)*(b*e-2*c*d)^(1/2)+6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*d*
e^2*g-33*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d^2*e*g+9*arctan((
-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d*e^2*f+42*arctan((-c*e*x-b*e+c*d)^
(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^3*g-18*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^
(1/2))*c^2*d^2*e*f-8*x*b*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+18*x*c*d
*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-6*x*c*e^2*f*(-c*e*x-b*e+c*d)^(1/2)
*(b*e-2*c*d)^(1/2)-11*b*d*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+3*b*e^2*f
*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+26*c*d^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e
-2*c*d)^(1/2)-12*c*d*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2))/(e*x+d)^(3/2)
/(-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((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.307287, size = 1, normalized size = 0. \[ \left [\frac{4 \, c^{2} e^{3} g x^{3} + 3 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (3 \, c e f -{\left (7 \, c d - 2 \, b e\right )} g\right )} \sqrt{2 \, c d - b e} \sqrt{e x + d} \log \left (-\frac{c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \,{\left (c d e - b e^{2}\right )} x - 2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{2 \, c d - b e} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 4 \,{\left (3 \, c^{2} e^{3} f - 5 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} - 6 \,{\left (4 \, c^{2} d^{2} e - 5 \, b c d e^{2} + b^{2} e^{3}\right )} f + 2 \,{\left (26 \, c^{2} d^{3} - 37 \, b c d^{2} e + 11 \, b^{2} d e^{2}\right )} g + 2 \,{\left (3 \,{\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} f -{\left (8 \, c^{2} d^{2} e + 15 \, b c d e^{2} - 8 \, b^{2} e^{3}\right )} g\right )} x}{6 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} e^{2}}, \frac{2 \, c^{2} e^{3} g x^{3} + 3 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (3 \, c e f -{\left (7 \, c d - 2 \, b e\right )} g\right )} \sqrt{-2 \, c d + b e} \sqrt{e x + d} \arctan \left (-\frac{\sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c d - b e\right )} \sqrt{e x + d}}{{\left (c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e\right )} \sqrt{-2 \, c d + b e}}\right ) + 2 \,{\left (3 \, c^{2} e^{3} f - 5 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} - 3 \,{\left (4 \, c^{2} d^{2} e - 5 \, b c d e^{2} + b^{2} e^{3}\right )} f +{\left (26 \, c^{2} d^{3} - 37 \, b c d^{2} e + 11 \, b^{2} d e^{2}\right )} g +{\left (3 \,{\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} f -{\left (8 \, c^{2} d^{2} e + 15 \, b c d e^{2} - 8 \, b^{2} e^{3}\right )} g\right )} x}{3 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} e^{2}}\right ] \]

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)^(7/2),x, algorithm="fricas")

[Out]

[1/6*(4*c^2*e^3*g*x^3 + 3*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(3*c*e*f -
(7*c*d - 2*b*e)*g)*sqrt(2*c*d - b*e)*sqrt(e*x + d)*log(-(c*e^2*x^2 - 3*c*d^2 + 2
*b*d*e - 2*(c*d*e - b*e^2)*x - 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*x^2 + 2*d*e*x + d^2)) + 4*(3*c^2*e^3*f - 5*(2*
c^2*d*e^2 - b*c*e^3)*g)*x^2 - 6*(4*c^2*d^2*e - 5*b*c*d*e^2 + b^2*e^3)*f + 2*(26*
c^2*d^3 - 37*b*c*d^2*e + 11*b^2*d*e^2)*g + 2*(3*(2*c^2*d*e^2 + b*c*e^3)*f - (8*c
^2*d^2*e + 15*b*c*d*e^2 - 8*b^2*e^3)*g)*x)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 -
b*d*e)*sqrt(e*x + d)*e^2), 1/3*(2*c^2*e^3*g*x^3 + 3*sqrt(-c*e^2*x^2 - b*e^2*x +
c*d^2 - b*d*e)*(3*c*e*f - (7*c*d - 2*b*e)*g)*sqrt(-2*c*d + b*e)*sqrt(e*x + d)*ar
ctan(-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 + b*e^2*x - c*d^2 + b*d*e)*sqrt(-2*c*d + b*e))) + 2*(3*c^2*e^3*f - 5*(2
*c^2*d*e^2 - b*c*e^3)*g)*x^2 - 3*(4*c^2*d^2*e - 5*b*c*d*e^2 + b^2*e^3)*f + (26*c
^2*d^3 - 37*b*c*d^2*e + 11*b^2*d*e^2)*g + (3*(2*c^2*d*e^2 + b*c*e^3)*f - (8*c^2*
d^2*e + 15*b*c*d*e^2 - 8*b^2*e^3)*g)*x)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d
*e)*sqrt(e*x + d)*e^2)]

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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)**(7/2),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)^(7/2),x, algorithm="giac")

[Out]

Timed out