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

Optimal. Leaf size=78 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{3/2}}+\frac{b (b+2 c x) \sqrt{b x+c x^2}}{8 c}+\frac{1}{3} \left (b x+c x^2\right )^{3/2} \]

[Out]

(b*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(8*c) + (b*x + c*x^2)^(3/2)/3 - (b^3*ArcTanh[(
Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(3/2))

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Rubi [A]  time = 0.0782406, antiderivative size = 78, 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 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{3/2}}+\frac{b (b+2 c x) \sqrt{b x+c x^2}}{8 c}+\frac{1}{3} \left (b x+c x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/x,x]

[Out]

(b*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(8*c) + (b*x + c*x^2)^(3/2)/3 - (b^3*ArcTanh[(
Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(3/2))

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Rubi in Sympy [A]  time = 8.25874, size = 66, normalized size = 0.85 \[ - \frac{b^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 c^{\frac{3}{2}}} + \frac{b \left (b + 2 c x\right ) \sqrt{b x + c x^{2}}}{8 c} + \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**3*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(8*c**(3/2)) + b*(b + 2*c*x)*sqrt(b*x
+ c*x**2)/(8*c) + (b*x + c*x**2)**(3/2)/3

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Mathematica [A]  time = 0.0935506, size = 89, normalized size = 1.14 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (3 b^2+14 b c x+8 c^2 x^2\right )-\frac{3 b^3 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{24 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(3*b^2 + 14*b*c*x + 8*c^2*x^2) - (3*b^3*Log[c*Sqrt[x
] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(24*c^(3/2))

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Maple [A]  time = 0.007, size = 81, normalized size = 1. \[{\frac{1}{3} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{bx}{4}\sqrt{c{x}^{2}+bx}}+{\frac{{b}^{2}}{8\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x,x)

[Out]

1/3*(c*x^2+b*x)^(3/2)+1/4*b*(c*x^2+b*x)^(1/2)*x+1/8/c*(c*x^2+b*x)^(1/2)*b^2-1/16
*b^3/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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231982, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{3} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (8 \, c^{2} x^{2} + 14 \, b c x + 3 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{48 \, c^{\frac{3}{2}}}, -\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (8 \, c^{2} x^{2} + 14 \, b c x + 3 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{24 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(3*b^3*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c) + 2*(8*c^2*x^2 + 1
4*b*c*x + 3*b^2)*sqrt(c*x^2 + b*x)*sqrt(c))/c^(3/2), -1/24*(3*b^3*arctan(sqrt(c*
x^2 + b*x)*sqrt(-c)/(c*x)) - (8*c^2*x^2 + 14*b*c*x + 3*b^2)*sqrt(c*x^2 + b*x)*sq
rt(-c))/(sqrt(-c)*c)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x*(b + c*x))**(3/2)/x, x)

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GIAC/XCAS [A]  time = 0.222696, size = 97, normalized size = 1.24 \[ \frac{b^{3}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{3}{2}}} + \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \, c x + 7 \, b\right )} x + \frac{3 \, b^{2}}{c}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*b^3*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(3/2) + 1/24*
sqrt(c*x^2 + b*x)*(2*(4*c*x + 7*b)*x + 3*b^2/c)