3.327 \(\int \frac{1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=207 \[ -\frac{e \sqrt{b x+c x^2} \left (3 b^2 e^2-4 b c d e+4 c^2 d^2\right )}{b^2 d^2 (d+e x) (c d-b e)^2}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x) (c d-b e)}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{5/2} (c d-b e)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.665674, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{e \sqrt{b x+c x^2} \left (3 b^2 e^2-4 b c d e+4 c^2 d^2\right )}{b^2 d^2 (d+e x) (c d-b e)^2}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x) (c d-b e)}+\frac{3 e^2 (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{5/2} (c d-b e)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^2*(b*x + c*x^2)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 63.5692, size = 187, normalized size = 0.9 \[ - \frac{3 e^{2} \left (b e - 2 c d\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{2 d^{\frac{5}{2}} \left (b e - c d\right )^{\frac{5}{2}}} - \frac{2 \left (b \left (b e - c d\right ) + c x \left (b e - 2 c d\right )\right )}{b^{2} d \left (d + e x\right ) \left (b e - c d\right ) \sqrt{b x + c x^{2}}} - \frac{e \sqrt{b x + c x^{2}} \left (3 b^{2} e^{2} - 4 b c d e + 4 c^{2} d^{2}\right )}{b^{2} d^{2} \left (d + e x\right ) \left (b e - c d\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*e**2*(b*e - 2*c*d)*atan((-b*d + x*(b*e - 2*c*d))/(2*sqrt(d)*sqrt(b*e - c*d)*s
qrt(b*x + c*x**2)))/(2*d**(5/2)*(b*e - c*d)**(5/2)) - 2*(b*(b*e - c*d) + c*x*(b*
e - 2*c*d))/(b**2*d*(d + e*x)*(b*e - c*d)*sqrt(b*x + c*x**2)) - e*sqrt(b*x + c*x
**2)*(3*b**2*e**2 - 4*b*c*d*e + 4*c**2*d**2)/(b**2*d**2*(d + e*x)*(b*e - c*d)**2
)

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Mathematica [A]  time = 0.771777, size = 164, normalized size = 0.79 \[ \frac{x^{3/2} \left (\frac{(b+c x)^2 \left (\frac{-\frac{2 c^3 x}{(b+c x) (c d-b e)^2}-\frac{2}{d^2}}{b^2}-\frac{e^3 x}{d^2 (d+e x) (c d-b e)^2}\right )}{\sqrt{x}}+\frac{3 e^2 (b+c x)^{3/2} (2 c d-b e) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{5/2} (b e-c d)^{5/2}}\right )}{(x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^2*(b*x + c*x^2)^(3/2)),x]

[Out]

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

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Maple [B]  time = 0.016, size = 893, normalized size = 4.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/d/(b*e-c*d)/(d/e+x)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)-
3*e^2/d^2/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*
b+9*e/d/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c-
3*e^2/d^2/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*
x*c+12*e/d/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/
2)*x*c^2-12/(b*e-c*d)^2/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^
(1/2)*x*c^3-6/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^
(1/2)*c^2+3/2*e^2/d^2/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^
2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d
/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b-3*e/d/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1
/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(
d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c-8*c^2/d/(b*e-c
*d)/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x-4*c/d/(b*e-c
*d)/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(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(1/((c*x^2 + b*x)^(3/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238152, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left (2 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3} +{\left (2 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} x\right )} \sqrt{c x^{2} + b x} \log \left (-\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} - \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right ) + 2 \,{\left (2 \, b c^{2} d^{3} - 4 \, b^{2} c d^{2} e + 2 \, b^{3} d e^{2} +{\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3}\right )} x^{2} +{\left (4 \, c^{3} d^{3} - 2 \, b c^{2} d^{2} e - 2 \, b^{2} c d e^{2} + 3 \, b^{3} e^{3}\right )} x\right )} \sqrt{c d^{2} - b d e}}{2 \,{\left (b^{2} c^{2} d^{5} - 2 \, b^{3} c d^{4} e + b^{4} d^{3} e^{2} +{\left (b^{2} c^{2} d^{4} e - 2 \, b^{3} c d^{3} e^{2} + b^{4} d^{2} e^{3}\right )} x\right )} \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x}}, -\frac{3 \,{\left (2 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3} +{\left (2 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} x\right )} \sqrt{c x^{2} + b x} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) +{\left (2 \, b c^{2} d^{3} - 4 \, b^{2} c d^{2} e + 2 \, b^{3} d e^{2} +{\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3}\right )} x^{2} +{\left (4 \, c^{3} d^{3} - 2 \, b c^{2} d^{2} e - 2 \, b^{2} c d e^{2} + 3 \, b^{3} e^{3}\right )} x\right )} \sqrt{-c d^{2} + b d e}}{{\left (b^{2} c^{2} d^{5} - 2 \, b^{3} c d^{4} e + b^{4} d^{3} e^{2} +{\left (b^{2} c^{2} d^{4} e - 2 \, b^{3} c d^{3} e^{2} + b^{4} d^{2} e^{3}\right )} x\right )} \sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^(3/2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

[-1/2*(3*(2*b^2*c*d^2*e^2 - b^3*d*e^3 + (2*b^2*c*d*e^3 - b^3*e^4)*x)*sqrt(c*x^2
+ b*x)*log(-(2*(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) - sqrt(c*d^2 - b*d*e)*(b*d + (2
*c*d - b*e)*x))/(e*x + d)) + 2*(2*b*c^2*d^3 - 4*b^2*c*d^2*e + 2*b^3*d*e^2 + (4*c
^3*d^2*e - 4*b*c^2*d*e^2 + 3*b^2*c*e^3)*x^2 + (4*c^3*d^3 - 2*b*c^2*d^2*e - 2*b^2
*c*d*e^2 + 3*b^3*e^3)*x)*sqrt(c*d^2 - b*d*e))/((b^2*c^2*d^5 - 2*b^3*c*d^4*e + b^
4*d^3*e^2 + (b^2*c^2*d^4*e - 2*b^3*c*d^3*e^2 + b^4*d^2*e^3)*x)*sqrt(c*d^2 - b*d*
e)*sqrt(c*x^2 + b*x)), -(3*(2*b^2*c*d^2*e^2 - b^3*d*e^3 + (2*b^2*c*d*e^3 - b^3*e
^4)*x)*sqrt(c*x^2 + b*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d -
b*e)*x)) + (2*b*c^2*d^3 - 4*b^2*c*d^2*e + 2*b^3*d*e^2 + (4*c^3*d^2*e - 4*b*c^2*d
*e^2 + 3*b^2*c*e^3)*x^2 + (4*c^3*d^3 - 2*b*c^2*d^2*e - 2*b^2*c*d*e^2 + 3*b^3*e^3
)*x)*sqrt(-c*d^2 + b*d*e))/((b^2*c^2*d^5 - 2*b^3*c*d^4*e + b^4*d^3*e^2 + (b^2*c^
2*d^4*e - 2*b^3*c*d^3*e^2 + b^4*d^2*e^3)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*
x))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((x*(b + c*x))**(3/2)*(d + e*x)**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^(3/2)*(e*x + d)^2),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + b*x)^(3/2)*(e*x + d)^2), x)