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

Optimal. Leaf size=72 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 \sqrt{c}}+\frac{3}{4} b \sqrt{b x+c x^2}+\frac{\left (b x+c x^2\right )^{3/2}}{2 x} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.086668, antiderivative size = 72, 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 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 \sqrt{c}}+\frac{3}{4} b \sqrt{b x+c x^2}+\frac{\left (b x+c x^2\right )^{3/2}}{2 x} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/x^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.50966, size = 63, normalized size = 0.88 \[ \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 \sqrt{c}} + \frac{3 b \sqrt{b x + c x^{2}}}{4} + \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0775002, size = 71, normalized size = 0.99 \[ \frac{1}{4} \sqrt{x (b+c x)} \left (\frac{3 b^2 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{c} \sqrt{x} \sqrt{b+c x}}+5 b+2 c x\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 99, normalized size = 1.4 \[ 2\,{\frac{ \left ( c{x}^{2}+bx \right ) ^{5/2}}{b{x}^{2}}}-2\,{\frac{c \left ( c{x}^{2}+bx \right ) ^{3/2}}{b}}-{\frac{3\,cx}{2}\sqrt{c{x}^{2}+bx}}-{\frac{3\,b}{4}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x^2,x)

[Out]

2/b/x^2*(c*x^2+b*x)^(5/2)-2*c/b*(c*x^2+b*x)^(3/2)-3/2*c*(c*x^2+b*x)^(1/2)*x-3/4*
b*(c*x^2+b*x)^(1/2)+3/8/c^(1/2)*b^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))

_______________________________________________________________________________________

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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.242955, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \, \sqrt{c x^{2} + b x}{\left (2 \, c x + 5 \, b\right )} \sqrt{c}}{8 \, \sqrt{c}}, \frac{3 \, b^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) + \sqrt{c x^{2} + b x}{\left (2 \, c x + 5 \, b\right )} \sqrt{-c}}{4 \, \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222239, size = 81, normalized size = 1.12 \[ -\frac{3 \, b^{2}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, \sqrt{c}} + \frac{1}{4} \, \sqrt{c x^{2} + b x}{\left (2 \, c x + 5 \, b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/8*b^2*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/sqrt(c) + 1/4*s
qrt(c*x^2 + b*x)*(2*c*x + 5*b)