3.100 \(\int \frac{1}{x^{5/2} \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{5/2}}+\frac{3 c \sqrt{b x+c x^2}}{4 b^2 x^{3/2}}-\frac{\sqrt{b x+c x^2}}{2 b x^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.106695, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{5/2}}+\frac{3 c \sqrt{b x+c x^2}}{4 b^2 x^{3/2}}-\frac{\sqrt{b x+c x^2}}{2 b x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*Sqrt[b*x + c*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.5378, size = 80, normalized size = 0.9 \[ - \frac{\sqrt{b x + c x^{2}}}{2 b x^{\frac{5}{2}}} + \frac{3 c \sqrt{b x + c x^{2}}}{4 b^{2} x^{\frac{3}{2}}} - \frac{3 c^{2} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(c*x**2+b*x)**(1/2),x)

[Out]

-sqrt(b*x + c*x**2)/(2*b*x**(5/2)) + 3*c*sqrt(b*x + c*x**2)/(4*b**2*x**(3/2)) -
3*c**2*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/(4*b**(5/2))

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Mathematica [A]  time = 0.0606173, size = 83, normalized size = 0.93 \[ \frac{\sqrt{b} \left (-2 b^2+b c x+3 c^2 x^2\right )-3 c^2 x^2 \sqrt{b+c x} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )}{4 b^{5/2} x^{3/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*Sqrt[b*x + c*x^2]),x]

[Out]

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

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Maple [A]  time = 0.016, size = 72, normalized size = 0.8 \[ -{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){c}^{2}{x}^{2}-3\,xc\sqrt{cx+b}\sqrt{b}+2\,{b}^{3/2}\sqrt{cx+b} \right ){b}^{-{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(c*x^2+b*x)^(1/2),x)

[Out]

-1/4*(x*(c*x+b))^(1/2)/b^(5/2)*(3*arctanh((c*x+b)^(1/2)/b^(1/2))*c^2*x^2-3*x*c*(
c*x+b)^(1/2)*b^(1/2)+2*b^(3/2)*(c*x+b)^(1/2))/x^(5/2)/(c*x+b)^(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/(sqrt(c*x^2 + b*x)*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231947, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{2} x^{3} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \, \sqrt{c x^{2} + b x}{\left (3 \, c x - 2 \, b\right )} \sqrt{b} \sqrt{x}}{8 \, b^{\frac{5}{2}} x^{3}}, -\frac{3 \, c^{2} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) - \sqrt{c x^{2} + b x}{\left (3 \, c x - 2 \, b\right )} \sqrt{-b} \sqrt{x}}{4 \, \sqrt{-b} b^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*x^(5/2)),x, algorithm="fricas")

[Out]

[1/8*(3*c^2*x^3*log((2*sqrt(c*x^2 + b*x)*b*sqrt(x) - (c*x^2 + 2*b*x)*sqrt(b))/x^
2) + 2*sqrt(c*x^2 + b*x)*(3*c*x - 2*b)*sqrt(b)*sqrt(x))/(b^(5/2)*x^3), -1/4*(3*c
^2*x^3*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) - sqrt(c*x^2 + b*x)*(3*c*x - 2
*b)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b^2*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{\frac{5}{2}} \sqrt{x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Integral(1/(x**(5/2)*sqrt(x*(b + c*x))), x)

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GIAC/XCAS [A]  time = 0.234198, size = 81, normalized size = 0.91 \[ \frac{1}{4} \, c^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{3}{2}} - 5 \, \sqrt{c x + b} b}{b^{2} c^{2} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*x^(5/2)),x, algorithm="giac")

[Out]

1/4*c^2*(3*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^2) + (3*(c*x + b)^(3/2) -
5*sqrt(c*x + b)*b)/(b^2*c^2*x^2))