\(\int x \sqrt {b x+c x^2} \, dx\) [3]

Optimal result

Integrand size = 15, antiderivative size = 81 \[ \int x \sqrt {b x+c x^2} \, dx=-\frac {b (b+2 c x) \sqrt {b x+c x^2}}{8 c^2}+\frac {\left (b x+c x^2\right )^{3/2}}{3 c}+\frac {b^3 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {654, 626, 634, 212} \[ \int x \sqrt {b x+c x^2} \, dx=\frac {b^3 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{5/2}}-\frac {b (b+2 c x) \sqrt {b x+c x^2}}{8 c^2}+\frac {\left (b x+c x^2\right )^{3/2}}{3 c} \]

[In]

Int[x*Sqrt[b*x + c*x^2],x]

[Out]

-1/8*(b*(b + 2*c*x)*Sqrt[b*x + c*x^2])/c^2 + (b*x + c*x^2)^(3/2)/(3*c) + (b^3*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c
*x^2]])/(8*c^(5/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 634

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (b x+c x^2\right )^{3/2}}{3 c}-\frac {b \int \sqrt {b x+c x^2} \, dx}{2 c} \\ & = -\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}+\frac {b^3 \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c^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}+\frac {b^3 \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{8 c^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}+\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.19 \[ \int x \sqrt {b x+c x^2} \, dx=\frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (-3 b^2+2 b c x+8 c^2 x^2\right )+\frac {6 b^3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )}{\sqrt {x} \sqrt {b+c x}}\right )}{24 c^{5/2}} \]

[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) + (6*b^3*ArcTanh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b] + Sqrt
[b + c*x])])/(Sqrt[x]*Sqrt[b + c*x])))/(24*c^(5/2))

Maple [A] (verified)

Time = 2.43 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {\left (-8 c^{2} x^{2}-2 b c x +3 b^{2}\right ) x \left (c x +b \right )}{24 c^{2} \sqrt {x \left (c x +b \right )}}+\frac {b^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {5}{2}}}\)	\(73\)
default	\(\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\)	\(79\)
pseudoelliptic	\(\frac {8 c^{\frac {5}{2}} \sqrt {x \left (c x +b \right )}\, x^{2}+2 b \,c^{\frac {3}{2}} x \sqrt {x \left (c x +b \right )}+3 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right ) b^{3}-3 b^{2} \sqrt {c}\, \sqrt {x \left (c x +b \right )}}{24 c^{\frac {5}{2}}}\)	\(79\)

[In]

int(x*(c*x^2+b*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/24*(-8*c^2*x^2-2*b*c*x+3*b^2)*x*(c*x+b)/c^2/(x*(c*x+b))^(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))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.83 \[ \int x \sqrt {b x+c x^2} \, dx=\left [\frac {3 \, b^{3} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (8 \, c^{3} x^{2} + 2 \, b c^{2} x - 3 \, b^{2} c\right )} \sqrt {c x^{2} + b x}}{48 \, c^{3}}, -\frac {3 \, b^{3} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (8 \, c^{3} x^{2} + 2 \, b c^{2} x - 3 \, b^{2} c\right )} \sqrt {c x^{2} + b x}}{24 \, c^{3}}\right ] \]

[In]

integrate(x*(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.44 \[ \int x \sqrt {b x+c x^2} \, dx=\begin {cases} \frac {b^{3} \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{16 c^{2}} + \sqrt {b x + c x^{2}} \left (- \frac {b^{2}}{8 c^{2}} + \frac {b x}{12 c} + \frac {x^{2}}{3}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (b x\right )^{\frac {5}{2}}}{5 b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((b**3*Piecewise((log(b + 2*sqrt(c)*sqrt(b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(b**2/c, 0)), ((b/(2*c) +
x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True))/(16*c**2) + sqrt(b*x + c*x**2)*(-b**2/(8*c**2) + b*x/(12*
c) + x**2/3), Ne(c, 0)), (2*(b*x)**(5/2)/(5*b**2), Ne(b, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05 \[ \int x \sqrt {b x+c x^2} \, dx=-\frac {\sqrt {c x^{2} + b x} b x}{4 \, c} + \frac {b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {5}{2}}} - \frac {\sqrt {c x^{2} + b x} b^{2}}{8 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{3 \, c} \]

[In]

integrate(x*(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

-1/4*sqrt(c*x^2 + b*x)*b*x/c + 1/16*b^3*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 1/8*sqrt(c*x^2
+ b*x)*b^2/c^2 + 1/3*(c*x^2 + b*x)^(3/2)/c

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88 \[ \int x \sqrt {b x+c x^2} \, dx=\frac {1}{24} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, x + \frac {b}{c}\right )} x - \frac {3 \, b^{2}}{c^{2}}\right )} - \frac {b^{3} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{16 \, c^{\frac {5}{2}}} \]

[In]

integrate(x*(c*x^2+b*x)^(1/2),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*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt
(c) + b))/c^(5/2)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.85 \[ \int x \sqrt {b x+c x^2} \, dx=\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2} \]

[In]

int(x*(b*x + c*x^2)^(1/2),x)

[Out]

(b^3*log((b + 2*c*x)/c^(1/2) + 2*(b*x + c*x^2)^(1/2)))/(16*c^(5/2)) + ((b*x + c*x^2)^(1/2)*(8*c^2*x^2 - 3*b^2
+ 2*b*c*x))/(24*c^2)