3.321 \(\int \frac{1}{(d+e x)^3 \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=180 \[ \frac{\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{5/2} (c d-b e)^{5/2}}-\frac{3 e \sqrt{b x+c x^2} (2 c d-b e)}{4 d^2 (d+e x) (c d-b e)^2}-\frac{e \sqrt{b x+c x^2}}{2 d (d+e x)^2 (c d-b e)} \]

[Out]

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

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Rubi [A]  time = 0.505719, antiderivative size = 180, 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 \[ \frac{\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{5/2} (c d-b e)^{5/2}}-\frac{3 e \sqrt{b x+c x^2} (2 c d-b e)}{4 d^2 (d+e x) (c d-b e)^2}-\frac{e \sqrt{b x+c x^2}}{2 d (d+e x)^2 (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 53.7226, size = 155, normalized size = 0.86 \[ \frac{e \sqrt{b x + c x^{2}}}{2 d \left (d + e x\right )^{2} \left (b e - c d\right )} + \frac{3 e \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{4 d^{2} \left (d + e x\right ) \left (b e - c d\right )^{2}} + \frac{\left (\frac{3 b^{2} e^{2}}{8} - b c d e + c^{2} d^{2}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{d^{\frac{5}{2}} \left (b e - c d\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*sqrt(b*x + c*x**2)/(2*d*(d + e*x)**2*(b*e - c*d)) + 3*e*(b*e - 2*c*d)*sqrt(b*x
+ c*x**2)/(4*d**2*(d + e*x)*(b*e - c*d)**2) + (3*b**2*e**2/8 - b*c*d*e + c**2*d
**2)*atan((-b*d + x*(b*e - 2*c*d))/(2*sqrt(d)*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)
))/(d**(5/2)*(b*e - c*d)**(5/2))

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Mathematica [A]  time = 0.452623, size = 164, normalized size = 0.91 \[ \frac{\sqrt{x} \left (\frac{\sqrt{b+c x} \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b e-c d}}-\frac{\sqrt{d} e \sqrt{x} (b+c x) (2 c d (4 d+3 e x)-b e (5 d+3 e x))}{(d+e x)^2}\right )}{4 d^{5/2} \sqrt{x (b+c x)} (c d-b e)^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(-((Sqrt[d]*e*Sqrt[x]*(b + c*x)*(2*c*d*(4*d + 3*e*x) - b*e*(5*d + 3*e*x
)))/(d + e*x)^2) + ((8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*Sqrt[b + c*x]*ArcTan[(Sq
rt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[-(c*d) + b*e]))/(4*d^(5
/2)*(c*d - b*e)^2*Sqrt[x*(b + c*x)])

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Maple [B]  time = 0.015, size = 798, normalized size = 4.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/e/d/(b*e-c*d)/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^
(1/2)+3/4*e/d^2/(b*e-c*d)^2/(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)*b-3/2/d/(b*e-c*d)^2/(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)*c-3/8*e/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^2+3/2/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))*b*c-
3/2/e/(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^2-1/2/e*c/d/(b*e-c*d)/(-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))

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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/(sqrt(c*x^2 + b*x)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239576, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (8 \, c d^{2} e - 5 \, b d e^{2} + 3 \,{\left (2 \, c d e^{2} - b e^{3}\right )} x\right )} \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x} -{\left (8 \, c^{2} d^{4} - 8 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} +{\left (8 \, c^{2} d^{2} e^{2} - 8 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} x^{2} + 2 \,{\left (8 \, c^{2} d^{3} e - 8 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\right )} x\right )} \log \left (\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} + \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right )}{8 \,{\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} +{\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )} \sqrt{c d^{2} - b d e}}, -\frac{{\left (8 \, c d^{2} e - 5 \, b d e^{2} + 3 \,{\left (2 \, c d e^{2} - b e^{3}\right )} x\right )} \sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x} +{\left (8 \, c^{2} d^{4} - 8 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} +{\left (8 \, c^{2} d^{2} e^{2} - 8 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} x^{2} + 2 \,{\left (8 \, c^{2} d^{3} e - 8 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\right )} x\right )} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right )}{4 \,{\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} +{\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )} \sqrt{-c d^{2} + b d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*(2*(8*c*d^2*e - 5*b*d*e^2 + 3*(2*c*d*e^2 - b*e^3)*x)*sqrt(c*d^2 - b*d*e)*s
qrt(c*x^2 + b*x) - (8*c^2*d^4 - 8*b*c*d^3*e + 3*b^2*d^2*e^2 + (8*c^2*d^2*e^2 - 8
*b*c*d*e^3 + 3*b^2*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + 3*b^2*d*e^3)*x)*l
og((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)))/((c^2*d^6 - 2*b*c*d^5*e + b^2*d^4*e^2 + (c^2*d^4*e^2 - 2*b*c*
d^3*e^3 + b^2*d^2*e^4)*x^2 + 2*(c^2*d^5*e - 2*b*c*d^4*e^2 + b^2*d^3*e^3)*x)*sqrt
(c*d^2 - b*d*e)), -1/4*((8*c*d^2*e - 5*b*d*e^2 + 3*(2*c*d*e^2 - b*e^3)*x)*sqrt(-
c*d^2 + b*d*e)*sqrt(c*x^2 + b*x) + (8*c^2*d^4 - 8*b*c*d^3*e + 3*b^2*d^2*e^2 + (8
*c^2*d^2*e^2 - 8*b*c*d*e^3 + 3*b^2*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + 3
*b^2*d*e^3)*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)))/
((c^2*d^6 - 2*b*c*d^5*e + b^2*d^4*e^2 + (c^2*d^4*e^2 - 2*b*c*d^3*e^3 + b^2*d^2*e
^4)*x^2 + 2*(c^2*d^5*e - 2*b*c*d^4*e^2 + b^2*d^3*e^3)*x)*sqrt(-c*d^2 + b*d*e))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.580919, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x