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

Optimal. Leaf size=64 \[ -\frac{2 \left (b x+c x^2\right )^{3/2}}{x^2}+3 c \sqrt{b x+c x^2}+3 b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0848387, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{2 \left (b x+c x^2\right )^{3/2}}{x^2}+3 c \sqrt{b x+c x^2}+3 b \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.15333, size = 60, normalized size = 0.94 \[ 3 b \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )} + 3 c \sqrt{b x + c x^{2}} - \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0681781, size = 76, normalized size = 1.19 \[ \frac{\sqrt{b+c x} \left (\sqrt{b+c x} (c x-2 b)+3 b \sqrt{c} \sqrt{x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )\right )}{\sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b + c*x]*((-2*b + c*x)*Sqrt[b + c*x] + 3*b*Sqrt[c]*Sqrt[x]*Log[c*Sqrt[x] +
Sqrt[c]*Sqrt[b + c*x]]))/Sqrt[x*(b + c*x)]

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Maple [B]  time = 0.006, size = 124, normalized size = 1.9 \[ -2\,{\frac{ \left ( c{x}^{2}+bx \right ) ^{5/2}}{b{x}^{3}}}+8\,{\frac{c \left ( c{x}^{2}+bx \right ) ^{5/2}}{{b}^{2}{x}^{2}}}-8\,{\frac{{c}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}{{b}^{2}}}-6\,{\frac{{c}^{2}\sqrt{c{x}^{2}+bx}x}{b}}-3\,c\sqrt{c{x}^{2}+bx}+{\frac{3\,b}{2}\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x^3,x)

[Out]

-2/b/x^3*(c*x^2+b*x)^(5/2)+8*c/b^2/x^2*(c*x^2+b*x)^(5/2)-8*c^2/b^2*(c*x^2+b*x)^(
3/2)-6*c^2/b*(c*x^2+b*x)^(1/2)*x-3*c*(c*x^2+b*x)^(1/2)+3/2*c^(1/2)*b*ln((1/2*b+c
*x)/c^(1/2)+(c*x^2+b*x)^(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((c*x^2 + b*x)^(3/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231591, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b \sqrt{c} x \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \, \sqrt{c x^{2} + b x}{\left (c x - 2 \, b\right )}}{2 \, x}, \frac{3 \, b \sqrt{-c} x \arctan \left (\frac{\sqrt{c x^{2} + b x}}{\sqrt{-c} x}\right ) + \sqrt{c x^{2} + b x}{\left (c x - 2 \, b\right )}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.225969, size = 103, normalized size = 1.61 \[ -\frac{3}{2} \, b \sqrt{c}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right ) + \sqrt{c x^{2} + b x} c + \frac{2 \, b^{2}}{\sqrt{c} x - \sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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