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

Optimal. Leaf size=81 \[ \frac{3 c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{5/2}}-\frac{3 c \sqrt{x}}{b^2 \sqrt{b x+c x^2}}-\frac{1}{b \sqrt{x} \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.102254, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 c \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{b^{5/2}}-\frac{3 c \sqrt{x}}{b^2 \sqrt{b x+c x^2}}-\frac{1}{b \sqrt{x} \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.9305, size = 75, normalized size = 0.93 \[ - \frac{1}{b \sqrt{x} \sqrt{b x + c x^{2}}} - \frac{3 c \sqrt{x}}{b^{2} \sqrt{b x + c x^{2}}} + \frac{3 c \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(b*sqrt(x)*sqrt(b*x + c*x**2)) - 3*c*sqrt(x)/(b**2*sqrt(b*x + c*x**2)) + 3*c*
atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/b**(5/2)

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Mathematica [A]  time = 0.0555433, size = 65, normalized size = 0.8 \[ \frac{3 c x \sqrt{b+c x} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} (b+3 c x)}{b^{5/2} \sqrt{x} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 60, normalized size = 0.7 \[{\frac{1}{cx+b}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}xc-3\,cx\sqrt{b}-{b}^{{\frac{3}{2}}} \right ){x}^{-{\frac{3}{2}}}{b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x*(c*x+b))^(1/2)*(3*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)*x*c-3*c*x*b^(1
/2)-b^(3/2))/x^(3/2)/(c*x+b)/b^(5/2)

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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)*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231356, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{c x^{2} + b x}{\left (3 \, c x + b\right )} \sqrt{b} \sqrt{x} - 3 \,{\left (c^{2} x^{3} + b c x^{2}\right )} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right )}{2 \,{\left (b^{2} c x^{3} + b^{3} x^{2}\right )} \sqrt{b}}, -\frac{\sqrt{c x^{2} + b x}{\left (3 \, c x + b\right )} \sqrt{-b} \sqrt{x} - 3 \,{\left (c^{2} x^{3} + b c x^{2}\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )}{{\left (b^{2} c x^{3} + b^{3} x^{2}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.229448, size = 78, normalized size = 0.96 \[ -c{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{3 \, c x + b}{{\left ({\left (c x + b\right )}^{\frac{3}{2}} - \sqrt{c x + b} b\right )} b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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