3.2245 \(\int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=250 \[ \frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac{2 (2 c d-b e) (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 (2 c d-b e)^{3/2} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e^2 (d+e x)^{5/2}} \]

[Out]

(2*(2*c*d - b*e)*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*Sqr
t[d + e*x]) + (2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2
*(d + e*x)^(3/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*c*e^2*(
d + e*x)^(5/2)) - (2*(2*c*d - b*e)^(3/2)*(e*f - d*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 = 0.989854, antiderivative size = 250, 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 \[ \frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac{2 (2 c d-b e) (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt{d+e x}}-\frac{2 (2 c d-b e)^{3/2} (e f-d g) \tanh ^{-1}\left (\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt{d+e x} \sqrt{2 c d-b e}}\right )}{e^2}-\frac{2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c e^2 (d+e x)^{5/2}} \]

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)^(5/2),x]

[Out]

(2*(2*c*d - b*e)*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e^2*Sqr
t[d + e*x]) + (2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e^2
*(d + e*x)^(3/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*c*e^2*(
d + e*x)^(5/2)) - (2*(2*c*d - b*e)^(3/2)*(e*f - d*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 = 104.821, size = 223, normalized size = 0.89 \[ - \frac{2 \left (b e - 2 c d\right )^{\frac{3}{2}} \left (d g - e f\right ) \operatorname{atan}{\left (\frac{\sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{\sqrt{d + e x} \sqrt{b e - 2 c d}} \right )}}{e^{2}} + \frac{2 \left (b e - 2 c d\right ) \left (d g - e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \sqrt{d + e x}} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{3 e^{2} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 g \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{5 c e^{2} \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.939411, size = 196, normalized size = 0.78 \[ \frac{2 ((d+e x) (c (d-e x)-b e))^{3/2} \left (\frac{-3 b^2 e^2 g-2 b c e (-13 d g+10 e f+3 e g x)+c^2 \left (-38 d^2 g+d e (35 f+11 g x)-e^2 x (5 f+3 g x)\right )}{c (c (d-e x)-b e)}+\frac{15 (2 c d-b e)^{3/2} (d g-e f) \tanh ^{-1}\left (\frac{\sqrt{-b e+c d-c e x}}{\sqrt{2 c d-b e}}\right )}{(c (d-e x)-b e)^{3/2}}\right )}{15 e^2 (d+e x)^{3/2}} \]

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)^(5/2),x]

[Out]

(2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((-3*b^2*e^2*g - 2*b*c*e*(10*e*f - 1
3*d*g + 3*e*g*x) + c^2*(-38*d^2*g - e^2*x*(5*f + 3*g*x) + d*e*(35*f + 11*g*x)))/
(c*(-(b*e) + c*(d - e*x))) + (15*(2*c*d - b*e)^(3/2)*(-(e*f) + d*g)*ArcTanh[Sqrt
[c*d - b*e - c*e*x]/Sqrt[2*c*d - b*e]])/(-(b*e) + c*(d - e*x))^(3/2)))/(15*e^2*(
d + e*x)^(3/2))

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Maple [B]  time = 0.031, size = 601, normalized size = 2.4 \[ -{\frac{2}{15\,c{e}^{2}}\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}} \left ( 3\,{x}^{2}{c}^{2}{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+15\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{2}cd{e}^{2}g-15\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){b}^{2}c{e}^{3}f-60\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) b{c}^{2}{d}^{2}eg+60\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ) b{c}^{2}d{e}^{2}f+60\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{3}{d}^{3}g-60\,\arctan \left ({\frac{\sqrt{-cex-be+cd}}{\sqrt{be-2\,cd}}} \right ){c}^{3}{d}^{2}ef+6\,xbc{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-11\,x{c}^{2}deg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+5\,x{c}^{2}{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+3\,{b}^{2}{e}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-26\,bcdeg\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+20\,bc{e}^{2}f\sqrt{-cex-be+cd}\sqrt{be-2\,cd}+38\,{c}^{2}{d}^{2}g\sqrt{-cex-be+cd}\sqrt{be-2\,cd}-35\,{c}^{2}def\sqrt{-cex-be+cd}\sqrt{be-2\,cd} \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{-cex-be+cd}}}{\frac{1}{\sqrt{be-2\,cd}}}} \]

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)^(5/2),x)

[Out]

-2/15*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*(3*x^2*c^2*e^2*g*(-c*e*x-b*e+c*d)^(
1/2)*(b*e-2*c*d)^(1/2)+15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c
*d*e^2*g-15*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*e^3*f-60*arct
an((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d^2*e*g+60*arctan((-c*e*x-b*e
+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c^2*d*e^2*f+60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b
*e-2*c*d)^(1/2))*c^3*d^3*g-60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c
^3*d^2*e*f+6*x*b*c*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-11*x*c^2*d*e*g
*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+5*x*c^2*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*(
b*e-2*c*d)^(1/2)+3*b^2*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-26*b*c*d*e
*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+20*b*c*e^2*f*(-c*e*x-b*e+c*d)^(1/2)*
(b*e-2*c*d)^(1/2)+38*c^2*d^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-35*c^2*d
*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2))/(e*x+d)^(1/2)/(-c*e*x-b*e+c*d)^(1
/2)/c/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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302526, 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)^(5/2),x, algorithm="fricas")

[Out]

[1/15*(6*c^3*e^4*g*x^4 + 2*(5*c^3*e^4*f - (11*c^3*d*e^3 - 9*b*c^2*e^4)*g)*x^3 +
15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*((2*c^2*d*e - b*
c*e^2)*f - (2*c^2*d^2 - b*c*d*e)*g)*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)) - 2*(5*(7*c^3*d*e^3 - 5*b
*c^2*e^4)*f - (35*c^3*d^2*e^2 - 34*b*c^2*d*e^3 + 9*b^2*c*e^4)*g)*x^2 + 10*(7*c^3
*d^3*e - 11*b*c^2*d^2*e^2 + 4*b^2*c*d*e^3)*f - 2*(38*c^3*d^4 - 64*b*c^2*d^3*e +
29*b^2*c*d^2*e^2 - 3*b^3*d*e^3)*g - 2*(5*(c^3*d^2*e^2 + 6*b*c^2*d*e^3 - 4*b^2*c*
e^4)*f - (11*c^3*d^3*e + 21*b*c^2*d^2*e^2 - 20*b^2*c*d*e^3 + 3*b^3*e^4)*g)*x)/(s
qrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)*c*e^2), 2/15*(3*c^3*e^4*
g*x^4 + (5*c^3*e^4*f - (11*c^3*d*e^3 - 9*b*c^2*e^4)*g)*x^3 - 15*sqrt(-c*e^2*x^2
- b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b*e)*((2*c^2*d*e - b*c*e^2)*f - (2*c^2*
d^2 - b*c*d*e)*g)*sqrt(e*x + d)*arctan(-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))) - (5*(7*c^3*d*e^3 - 5*b*c^2*e^4)*f - (35*c^3*d^2*e^2 - 34*b*c^2*d*e^3
+ 9*b^2*c*e^4)*g)*x^2 + 5*(7*c^3*d^3*e - 11*b*c^2*d^2*e^2 + 4*b^2*c*d*e^3)*f -
(38*c^3*d^4 - 64*b*c^2*d^3*e + 29*b^2*c*d^2*e^2 - 3*b^3*d*e^3)*g - (5*(c^3*d^2*e
^2 + 6*b*c^2*d*e^3 - 4*b^2*c*e^4)*f - (11*c^3*d^3*e + 21*b*c^2*d^2*e^2 - 20*b^2*
c*d*e^3 + 3*b^3*e^4)*g)*x)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x
+ d)*c*e^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]

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)**(3/2)/(e*x+d)**(5/2),x)

[Out]

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

[Out]

Timed out