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

Optimal. Leaf size=117 \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}-\frac{e \sqrt{b x+c x^2}}{d (d+e x) (c d-b e)} \]

[Out]

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

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Rubi [A]  time = 0.223907, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}-\frac{e \sqrt{b x+c x^2}}{d (d+e x) (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.1425, size = 94, normalized size = 0.8 \[ \frac{e \sqrt{b x + c x^{2}}}{d \left (d + e x\right ) \left (b e - c d\right )} + \frac{\left (\frac{b e}{2} - c d\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{d^{\frac{3}{2}} \left (b e - c d\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.151718, size = 122, normalized size = 1.04 \[ \frac{\sqrt{x} \left (\frac{\sqrt{b+c x} (2 c d-b e) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b e-c d}}-\frac{\sqrt{d} e \sqrt{x} (b+c x)}{d+e x}\right )}{d^{3/2} \sqrt{x (b+c x)} (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.013, size = 355, normalized size = 3. \[{\frac{1}{d \left ( be-cd \right ) }\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-1}}-{\frac{b}{2\,d \left ( be-cd \right ) }\ln \left ({1 \left ( -2\,{\frac{d \left ( be-cd \right ) }{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}}+{\frac{c}{e \left ( be-cd \right ) }\ln \left ({1 \left ( -2\,{\frac{d \left ( be-cd \right ) }{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232577, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x} e -{\left (2 \, c d^{2} - b d e +{\left (2 \, c d e - b e^{2}\right )} x\right )} \log \left (\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} + \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right )}{2 \,{\left (c d^{3} - b d^{2} e +{\left (c d^{2} e - b d e^{2}\right )} x\right )} \sqrt{c d^{2} - b d e}}, -\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x} e +{\left (2 \, c d^{2} - b d e +{\left (2 \, c d e - b e^{2}\right )} x\right )} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right )}{{\left (c d^{3} - b d^{2} e +{\left (c d^{2} e - b d e^{2}\right )} x\right )} \sqrt{-c d^{2} + b d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.460399, size = 540, normalized size = 4.62 \[ \frac{{\left (2 \, c d{\rm ln}\left ({\left | 2 \, c d - b e - 2 \, \sqrt{c d^{2} - b d e} \sqrt{c} \right |}\right ) - b e{\rm ln}\left ({\left | 2 \, c d - b e - 2 \, \sqrt{c d^{2} - b d e} \sqrt{c} \right |}\right ) + 2 \, \sqrt{c d^{2} - b d e} \sqrt{c}\right )}{\rm sign}\left (\frac{1}{x e + d}\right )}{2 \,{\left (\sqrt{c d^{2} - b d e} c d^{2} - \sqrt{c d^{2} - b d e} b d e\right )}} - \frac{\sqrt{c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{b e}{x e + d} - \frac{b d e}{{\left (x e + d\right )}^{2}}}}{c d^{2}{\rm sign}\left (\frac{1}{x e + d}\right ) - b d e{\rm sign}\left (\frac{1}{x e + d}\right )} - \frac{{\left (2 \, c d e - b e^{2}\right )}{\rm ln}\left ({\left | 2 \, c d - b e - 2 \, \sqrt{c d^{2} - b d e}{\left (\sqrt{c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{b e}{x e + d} - \frac{b d e}{{\left (x e + d\right )}^{2}}} + \frac{\sqrt{c d^{2} e^{2} - b d e^{3}} e^{\left (-1\right )}}{x e + d}\right )} \right |}\right )}{2 \,{\left (c d^{2} e - b d e^{2}\right )} \sqrt{c d^{2} - b d e}{\rm sign}\left (\frac{1}{x e + d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*c*d*ln(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) - b*e*ln(abs(2*c
*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) + 2*sqrt(c*d^2 - b*d*e)*sqrt(c))*sign
(1/(x*e + d))/(sqrt(c*d^2 - b*d*e)*c*d^2 - sqrt(c*d^2 - b*d*e)*b*d*e) - sqrt(c -
2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2)/(c*d^2
*sign(1/(x*e + d)) - b*d*e*sign(1/(x*e + d))) - 1/2*(2*c*d*e - b*e^2)*ln(abs(2*c
*d - b*e - 2*sqrt(c*d^2 - b*d*e)*(sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 +
b*e/(x*e + d) - b*d*e/(x*e + d)^2) + sqrt(c*d^2*e^2 - b*d*e^3)*e^(-1)/(x*e + d)
)))/((c*d^2*e - b*d*e^2)*sqrt(c*d^2 - b*d*e)*sign(1/(x*e + d)))