3.3 \(\int x \sqrt{b x+c x^2} \, dx\)

Optimal. Leaf size=81 \[ \frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{5/2}}-\frac{b (b+2 c x) \sqrt{b x+c x^2}}{8 c^2}+\frac{\left (b x+c x^2\right )^{3/2}}{3 c} \]

[Out]

-(b*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(8*c^2) + (b*x + c*x^2)^(3/2)/(3*c) + (b^3*Ar
cTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0740085, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{5/2}}-\frac{b (b+2 c x) \sqrt{b x+c x^2}}{8 c^2}+\frac{\left (b x+c x^2\right )^{3/2}}{3 c} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[b*x + c*x^2],x]

[Out]

-(b*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(8*c^2) + (b*x + c*x^2)^(3/2)/(3*c) + (b^3*Ar
cTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.54215, size = 70, normalized size = 0.86 \[ \frac{b^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 c^{\frac{5}{2}}} - \frac{b \left (b + 2 c x\right ) \sqrt{b x + c x^{2}}}{8 c^{2}} + \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**3*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(8*c**(5/2)) - b*(b + 2*c*x)*sqrt(b*x +
c*x**2)/(8*c**2) + (b*x + c*x**2)**(3/2)/(3*c)

_______________________________________________________________________________________

Mathematica [A]  time = 0.094282, size = 89, normalized size = 1.1 \[ \frac{\sqrt{x (b+c x)} \left (\frac{3 b^3 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (-3 b^2+2 b c x+8 c^2 x^2\right )\right )}{24 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[b*x + c*x^2],x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 87, normalized size = 1.1 \[{\frac{1}{3\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{bx}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^2+b*x)^(1/2),x)

[Out]

1/3*(c*x^2+b*x)^(3/2)/c-1/4*b/c*x*(c*x^2+b*x)^(1/2)-1/8*b^2/c^2*(c*x^2+b*x)^(1/2
)+1/16*b^3/c^(5/2)*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(sqrt(c*x^2 + b*x)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.232512, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{3} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (8 \, c^{2} x^{2} + 2 \, b c x - 3 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{c}}{48 \, c^{\frac{5}{2}}}, \frac{3 \, b^{3} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (8 \, c^{2} x^{2} + 2 \, b c x - 3 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-c}}{24 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)*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 + 2
*b*c*x - 3*b^2)*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2), 1/24*(3*b^3*arctan(sqrt(c*x^
2 + b*x)*sqrt(-c)/(c*x)) + (8*c^2*x^2 + 2*b*c*x - 3*b^2)*sqrt(c*x^2 + b*x)*sqrt(
-c))/(sqrt(-c)*c^2)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*sqrt(x*(b + c*x)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21969, size = 99, normalized size = 1.22 \[ \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \, x + \frac{b}{c}\right )} x - \frac{3 \, b^{2}}{c^{2}}\right )} - \frac{b^{3}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*sqrt(c*x^2 + b*x)*(2*(4*x + b/c)*x - 3*b^2/c^2) - 1/16*b^3*ln(abs(-2*(sqrt(
c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(5/2)