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

Optimal. Leaf size=117 \[ -\frac{15 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{7/2}}+\frac{15 c^2 \sqrt{x}}{4 b^3 \sqrt{b x+c x^2}}+\frac{5 c}{4 b^2 \sqrt{x} \sqrt{b x+c x^2}}-\frac{1}{2 b x^{3/2} \sqrt{b x+c x^2}} \]

[Out]

-1/(2*b*x^(3/2)*Sqrt[b*x + c*x^2]) + (5*c)/(4*b^2*Sqrt[x]*Sqrt[b*x + c*x^2]) + (
15*c^2*Sqrt[x])/(4*b^3*Sqrt[b*x + c*x^2]) - (15*c^2*ArcTanh[Sqrt[b*x + c*x^2]/(S
qrt[b]*Sqrt[x])])/(4*b^(7/2))

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Rubi [A]  time = 0.144092, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{15 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{7/2}}+\frac{15 c^2 \sqrt{x}}{4 b^3 \sqrt{b x+c x^2}}+\frac{5 c}{4 b^2 \sqrt{x} \sqrt{b x+c x^2}}-\frac{1}{2 b x^{3/2} \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*b*x^(3/2)*Sqrt[b*x + c*x^2]) + (5*c)/(4*b^2*Sqrt[x]*Sqrt[b*x + c*x^2]) + (
15*c^2*Sqrt[x])/(4*b^3*Sqrt[b*x + c*x^2]) - (15*c^2*ArcTanh[Sqrt[b*x + c*x^2]/(S
qrt[b]*Sqrt[x])])/(4*b^(7/2))

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Rubi in Sympy [A]  time = 16.1416, size = 109, normalized size = 0.93 \[ - \frac{1}{2 b x^{\frac{3}{2}} \sqrt{b x + c x^{2}}} + \frac{5 c}{4 b^{2} \sqrt{x} \sqrt{b x + c x^{2}}} + \frac{15 c^{2} \sqrt{x}}{4 b^{3} \sqrt{b x + c x^{2}}} - \frac{15 c^{2} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*b*x**(3/2)*sqrt(b*x + c*x**2)) + 5*c/(4*b**2*sqrt(x)*sqrt(b*x + c*x**2)) +
15*c**2*sqrt(x)/(4*b**3*sqrt(b*x + c*x**2)) - 15*c**2*atanh(sqrt(b*x + c*x**2)/
(sqrt(b)*sqrt(x)))/(4*b**(7/2))

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Mathematica [A]  time = 0.0670099, size = 84, normalized size = 0.72 \[ \frac{\sqrt{b} \left (-2 b^2+5 b c x+15 c^2 x^2\right )-15 c^2 x^2 \sqrt{b+c x} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )}{4 b^{7/2} x^{3/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*(-2*b^2 + 5*b*c*x + 15*c^2*x^2) - 15*c^2*x^2*Sqrt[b + c*x]*ArcTanh[Sqrt
[b + c*x]/Sqrt[b]])/(4*b^(7/2)*x^(3/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.014, size = 76, normalized size = 0.7 \[ -{\frac{1}{4\,cx+4\,b}\sqrt{x \left ( cx+b \right ) } \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{2}{c}^{2}-5\,{b}^{3/2}xc-15\,{x}^{2}{c}^{2}\sqrt{b}+2\,{b}^{5/2} \right ){x}^{-{\frac{5}{2}}}{b}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/x^(5/2)*(x*(c*x+b))^(1/2)*(15*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)*
x^2*c^2-5*b^(3/2)*x*c-15*x^2*c^2*b^(1/2)+2*b^(5/2))/(c*x+b)/b^(7/2)

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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)*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235817, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (15 \, c^{2} x^{2} + 5 \, b c x - 2 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x} + 15 \,{\left (c^{3} x^{4} + b c^{2} x^{3}\right )} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right )}{8 \,{\left (b^{3} c x^{4} + b^{4} x^{3}\right )} \sqrt{b}}, \frac{{\left (15 \, c^{2} x^{2} + 5 \, b c x - 2 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x} - 15 \,{\left (c^{3} x^{4} + b c^{2} x^{3}\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )}{4 \,{\left (b^{3} c x^{4} + b^{4} x^{3}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(2*(15*c^2*x^2 + 5*b*c*x - 2*b^2)*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x) + 15*(c
^3*x^4 + b*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))/((b^3*c*x^4 + b^4*x^3)*sqrt(b)), 1/4*((15*c^2*x^2 + 5*b*c*x - 2*b^2)*sqr
t(c*x^2 + b*x)*sqrt(-b)*sqrt(x) - 15*(c^3*x^4 + b*c^2*x^3)*arctan(sqrt(-b)*sqrt(
x)/sqrt(c*x^2 + b*x)))/((b^3*c*x^4 + b^4*x^3)*sqrt(-b))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.287945, size = 97, normalized size = 0.83 \[ \frac{1}{4} \, c^{2}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{8}{\sqrt{c x + b} b^{3}} + \frac{7 \,{\left (c x + b\right )}^{\frac{3}{2}} - 9 \, \sqrt{c x + b} b}{b^{3} c^{2} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*c^2*(15*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^3) + 8/(sqrt(c*x + b)*b^3
) + (7*(c*x + b)^(3/2) - 9*sqrt(c*x + b)*b)/(b^3*c^2*x^2))