3.2247 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=305 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (d+e x)^{9/2} (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-9 c d g+c e f)}{4 e^2 (d+e x)^{5/2} (2 c d-b e)}+\frac{3 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-9 c d g+c e f)}{4 e^2 \sqrt{d+e x} (2 c d-b e)}-\frac{3 c (4 b e g-9 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 \sqrt{2 c d-b e}} \]

[Out]

(3*c*(c*e*f - 9*c*d*g + 4*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)*Sqrt[d + e*x]) + ((c*e*f - 9*c*d*g + 4*b*e*g)*(d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2)^(3/2))/(4*e^2*(2*c*d - b*e)*(d + e*x)^(5/2)) - ((e*f - d*g)
*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(2*e^2*(2*c*d - b*e)*(d + e*x)^(9/
2)) - (3*c*(c*e*f - 9*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*Sqrt[2*c*d - b*e])

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Rubi [A]  time = 1.07087, antiderivative size = 305, 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 \[ -\frac{(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{2 e^2 (d+e x)^{9/2} (2 c d-b e)}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (4 b e g-9 c d g+c e f)}{4 e^2 (d+e x)^{5/2} (2 c d-b e)}+\frac{3 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (4 b e g-9 c d g+c e f)}{4 e^2 \sqrt{d+e x} (2 c d-b e)}-\frac{3 c (4 b e g-9 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{4 e^2 \sqrt{2 c d-b e}} \]

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)^(9/2),x]

[Out]

(3*c*(c*e*f - 9*c*d*g + 4*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)*Sqrt[d + e*x]) + ((c*e*f - 9*c*d*g + 4*b*e*g)*(d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2)^(3/2))/(4*e^2*(2*c*d - b*e)*(d + e*x)^(5/2)) - ((e*f - d*g)
*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(2*e^2*(2*c*d - b*e)*(d + e*x)^(9/
2)) - (3*c*(c*e*f - 9*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*Sqrt[2*c*d - b*e])

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Rubi in Sympy [A]  time = 117.793, size = 282, normalized size = 0.92 \[ \frac{3 c \left (4 b e g - 9 c d g + c e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{4 e^{2} \sqrt{b e - 2 c d}} - \frac{3 c \left (4 b e g - 9 c d g + c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{4 e^{2} \sqrt{d + e x} \left (b e - 2 c d\right )} - \frac{\left (4 b e g - 9 c d g + c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{4 e^{2} \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )} - \frac{\left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{2 e^{2} \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right )} \]

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)**(3/2)/(e*x+d)**(9/2),x)

[Out]

3*c*(4*b*e*g - 9*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)))/(4*e**2*sqrt(b*e - 2*c*d)) - 3*c*(4*b*e*g
- 9*c*d*g + c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(4*e**2*sqrt(
d + e*x)*(b*e - 2*c*d)) - (4*b*e*g - 9*c*d*g + c*e*f)*(-b*e**2*x - c*e**2*x**2 +
d*(-b*e + c*d))**(3/2)/(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))**(5/2)/(2*e**2*(d + e*x)**(9/2)*(b*e -
2*c*d))

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Mathematica [A]  time = 1.06702, size = 191, normalized size = 0.63 \[ \frac{((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac{c \left (17 d^2 g-d e (f-29 g x)+e^2 x (8 g x-5 f)\right )-2 b e (d g+e (f+2 g x))}{(d+e x)^2 (b e-c d+c e x)}-\frac{3 c (4 b e g-9 c d g+c e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{\sqrt{2 c d-b e} (c (d-e x)-b e)^{3/2}}\right )}{4 e^2 (d+e x)^{3/2}} \]

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)^(9/2),x]

[Out]

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((-2*b*e*(d*g + e*(f + 2*g*x)) + c*(17
*d^2*g - d*e*(f - 29*g*x) + e^2*x*(-5*f + 8*g*x)))/((d + e*x)^2*(-(c*d) + b*e +
c*e*x)) - (3*c*(c*e*f - 9*c*d*g + 4*b*e*g)*ArcTanh[Sqrt[c*d - b*e - c*e*x]/Sqrt[
2*c*d - b*e]])/(Sqrt[2*c*d - b*e]*(-(b*e) + c*(d - e*x))^(3/2))))/(4*e^2*(d + e*
x)^(3/2))

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Maple [B]  time = 0.039, size = 665, normalized size = 2.2 \[{\frac{1}{4\,{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( 12\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}bc{e}^{3}g-27\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}{c}^{2}d{e}^{2}g+3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){x}^{2}{c}^{2}{e}^{3}f+24\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) xbcd{e}^{2}g-54\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}{d}^{2}eg+6\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) x{c}^{2}d{e}^{2}f-8\,{x}^{2}c{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+12\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) bc{d}^{2}eg-27\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}{d}^{3}g+3\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{2}{d}^{2}ef+4\,xb{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-29\,xcdeg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+5\,xc{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+2\,bdeg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+2\,b{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-17\,c{d}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+cdef\sqrt{-cex-be+cd}\sqrt{be-2\,cd} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{-cex-be+cd}}}{\frac{1}{\sqrt{be-2\,cd}}}} \]

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)^(9/2),x)

[Out]

1/4*(-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))*x^2*b*c*e^3*g-27*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2)
)*x^2*c^2*d*e^2*g+3*arctan((-c*e*x-b*e+c*d)^(1/2)/(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))*x*b*c*d*e^2*g-54*arctan((
-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*x*c^2*d^2*e*g+6*arctan((-c*e*x-b*e+c*d)
^(1/2)/(b*e-2*c*d)^(1/2))*x*c^2*d*e^2*f-8*x^2*c*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*
e-2*c*d)^(1/2)+12*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d^2*e*g-2
7*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^3*g+3*arctan((-c*e*x-b*
e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*f+4*x*b*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(
b*e-2*c*d)^(1/2)-29*x*c*d*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+5*x*c*e^2
*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+2*b*d*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*
e-2*c*d)^(1/2)+2*b*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-17*c*d^2*g*(-c
*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+c*d*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)
^(1/2))/(e*x+d)^(5/2)/(-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)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.330602, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^(9/2),x, algorithm="fricas")

[Out]

[1/8*(3*(c^2*d*e*f - (9*c^2*d^2 - 4*b*c*d*e)*g + (c^2*e^2*f - (9*c^2*d*e - 4*b*c
*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)*log(-(2*sqr
t(-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*(8*c^2*e^3*g*x^3 - (5*c^2*e^3*f - (21*c^2*d*e^2 + 4*b*c*e^3)*g)*x^
2 + (c^2*d^2*e + b*c*d*e^2 - 2*b^2*e^3)*f - (17*c^2*d^3 - 19*b*c*d^2*e + 2*b^2*d
*e^2)*g + ((4*c^2*d*e^2 - 7*b*c*e^3)*f - (12*c^2*d^2*e - 31*b*c*d*e^2 + 4*b^2*e^
3)*g)*x)*sqrt(2*c*d - b*e))/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e^3*x +
d*e^2)*sqrt(2*c*d - b*e)*sqrt(e*x + d)), 1/4*(3*(c^2*d*e*f - (9*c^2*d^2 - 4*b*c
*d*e)*g + (c^2*e^2*f - (9*c^2*d*e - 4*b*c*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x +
c*d^2 - b*d*e)*sqrt(e*x + d)*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)) + (8*c^2
*e^3*g*x^3 - (5*c^2*e^3*f - (21*c^2*d*e^2 + 4*b*c*e^3)*g)*x^2 + (c^2*d^2*e + b*c
*d*e^2 - 2*b^2*e^3)*f - (17*c^2*d^3 - 19*b*c*d^2*e + 2*b^2*d*e^2)*g + ((4*c^2*d*
e^2 - 7*b*c*e^3)*f - (12*c^2*d^2*e - 31*b*c*d*e^2 + 4*b^2*e^3)*g)*x)*sqrt(-2*c*d
+ b*e))/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e^3*x + d*e^2)*sqrt(-2*c*d
+ b*e)*sqrt(e*x + d))]

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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)**(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((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/(e*x + d)^(9/2),x, algorithm="giac")

[Out]

Timed out