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

Optimal. Leaf size=126 \[ \frac{e^2 \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{d^{3/2} (c d-b e)^{3/2}}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (c d-b e)} \]

[Out]

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

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Rubi [A]  time = 0.272606, antiderivative size = 126, 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 \[ \frac{e^2 \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{d^{3/2} (c d-b e)^{3/2}}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (c d-b e)} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 35.5989, size = 110, normalized size = 0.87 \[ - \frac{e^{2} \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{d^{\frac{3}{2}} \left (b e - c d\right )^{\frac{3}{2}}} - \frac{2 \left (b \left (b e - c d\right ) + c x \left (b e - 2 c d\right )\right )}{b^{2} d \left (b e - c d\right ) \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.262756, size = 129, normalized size = 1.02 \[ -\frac{2 \left (\sqrt{d} \sqrt{b e-c d} \left (b^2 e-b c d+b c e x-2 c^2 d x\right )+b^2 e^2 \sqrt{x} \sqrt{b+c x} \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )\right )}{b^2 d^{3/2} \sqrt{x (b+c x)} (b e-c d)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.013, size = 403, normalized size = 3.2 \[ -2\,{\frac{e}{d \left ( be-cd \right ) }{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}}-2\,{\frac{exc}{d \left ( be-cd \right ) b}{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}}+4\,{\frac{{c}^{2}x}{ \left ( be-cd \right ){b}^{2}}{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}}+2\,{\frac{c}{ \left ( be-cd \right ) b}{\frac{1}{\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}}+{\frac{e}{d \left ( be-cd \right ) }\ln \left ({1 \left ( -2\,{\frac{d \left ( be-cd \right ) }{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230054, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{c x^{2} + b x} b^{2} e^{2} \log \left (-\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} - \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right ) + 2 \,{\left (b c d - b^{2} e +{\left (2 \, c^{2} d - b c e\right )} x\right )} \sqrt{c d^{2} - b d e}}{{\left (b^{2} c d^{2} - b^{3} d e\right )} \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x}}, -\frac{2 \,{\left (\sqrt{c x^{2} + b x} b^{2} e^{2} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) +{\left (b c d - b^{2} e +{\left (2 \, c^{2} d - b c e\right )} x\right )} \sqrt{-c d^{2} + b d e}\right )}}{{\left (b^{2} c d^{2} - b^{3} d e\right )} \sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-(sqrt(c*x^2 + b*x)*b^2*e^2*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*(b*c*d - b^2*e + (2*c^2*d -
b*c*e)*x)*sqrt(c*d^2 - b*d*e))/((b^2*c*d^2 - b^3*d*e)*sqrt(c*d^2 - b*d*e)*sqrt(
c*x^2 + b*x)), -2*(sqrt(c*x^2 + b*x)*b^2*e^2*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c
*x^2 + b*x)/((c*d - b*e)*x)) + (b*c*d - b^2*e + (2*c^2*d - b*c*e)*x)*sqrt(-c*d^2
+ b*d*e))/((b^2*c*d^2 - b^3*d*e)*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. \[ \int \frac{1}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.2285, size = 225, normalized size = 1.79 \[ -\frac{2 \,{\left (\frac{{\left (2 \, c^{2} d^{2} - b c d e\right )} x}{b^{2} c d^{3} - b^{3} d^{2} e} + \frac{b c d^{2} - b^{2} d e}{b^{2} c d^{3} - b^{3} d^{2} e}\right )}}{\sqrt{c x^{2} + b x}} - \frac{2 \, \arctan \left (\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e}}\right ) e^{2}}{{\left (c d^{2} - b d e\right )} \sqrt{-c d^{2} + b d e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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