3.18 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=68 \[ 2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 c \sqrt{b x+c x^2}}{x}-\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.084321, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ 2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 c \sqrt{b x+c x^2}}{x}-\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.30459, size = 61, normalized size = 0.9 \[ 2 c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )} - \frac{2 c \sqrt{b x + c x^{2}}}{x} - \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0562908, size = 80, normalized size = 1.18 \[ -\frac{2 \sqrt{x (b+c x)} \left (\sqrt{b+c x} (b+4 c x)-3 c^{3/2} x^{3/2} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )\right )}{3 x^2 \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(3/2)/x^4,x]

[Out]

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

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Maple [B]  time = 0.007, size = 149, normalized size = 2.2 \[ -{\frac{2}{3\,b{x}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{4\,c}{3\,{b}^{2}{x}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{16\,{c}^{2}}{3\,{b}^{3}{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{16\,{c}^{3}}{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-4\,{\frac{{c}^{3}\sqrt{c{x}^{2}+bx}x}{{b}^{2}}}-2\,{\frac{{c}^{2}\sqrt{c{x}^{2}+bx}}{b}}+{c}^{{\frac{3}{2}}}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x^4,x)

[Out]

-2/3/b/x^4*(c*x^2+b*x)^(5/2)-4/3*c/b^2/x^3*(c*x^2+b*x)^(5/2)+16/3*c^2/b^3/x^2*(c
*x^2+b*x)^(5/2)-16/3*c^3/b^3*(c*x^2+b*x)^(3/2)-4*c^3/b^2*(c*x^2+b*x)^(1/2)*x-2*c
^2/b*(c*x^2+b*x)^(1/2)+c^(3/2)*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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232146, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{\frac{3}{2}} x^{2} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \, \sqrt{c x^{2} + b x}{\left (4 \, c x + b\right )}}{3 \, x^{2}}, \frac{2 \,{\left (3 \, \sqrt{-c} c x^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x}}{\sqrt{-c} x}\right ) - \sqrt{c x^{2} + b x}{\left (4 \, c x + b\right )}\right )}}{3 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(3/2)/x**4,x)

[Out]

Integral((x*(b + c*x))**(3/2)/x**4, x)

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GIAC/XCAS [A]  time = 0.226118, size = 155, normalized size = 2.28 \[ -c^{\frac{3}{2}}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right ) + \frac{2 \,{\left (6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b c + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{2} \sqrt{c} + b^{3}\right )}}{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^4,x, algorithm="giac")

[Out]

-c^(3/2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b)) + 2/3*(6*(sqrt(
c)*x - sqrt(c*x^2 + b*x))^2*b*c + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^2*sqrt(c)
+ b^3)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^3