\(\int x \sqrt {c x^2} (a+b x) \, dx\) [758]

Optimal result

Integrand size = 16, antiderivative size = 35 \[ \int x \sqrt {c x^2} (a+b x) \, dx=\frac {1}{3} a x^2 \sqrt {c x^2}+\frac {1}{4} b x^3 \sqrt {c x^2} \]

[Out]

1/3*a*x^2*(c*x^2)^(1/2)+1/4*b*x^3*(c*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {15, 45} \[ \int x \sqrt {c x^2} (a+b x) \, dx=\frac {1}{3} a x^2 \sqrt {c x^2}+\frac {1}{4} b x^3 \sqrt {c x^2} \]

[In]

Int[x*Sqrt[c*x^2]*(a + b*x),x]

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c x^2} \int x^2 (a+b x) \, dx}{x} \\ & = \frac {\sqrt {c x^2} \int \left (a x^2+b x^3\right ) \, dx}{x} \\ & = \frac {1}{3} a x^2 \sqrt {c x^2}+\frac {1}{4} b x^3 \sqrt {c x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.69 \[ \int x \sqrt {c x^2} (a+b x) \, dx=\frac {1}{12} x^2 \sqrt {c x^2} (4 a+3 b x) \]

[In]

Integrate[x*Sqrt[c*x^2]*(a + b*x),x]

[Out]

(x^2*Sqrt[c*x^2]*(4*a + 3*b*x))/12

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.60

method	result	size
gosper	\(\frac {x^{2} \left (3 b x +4 a \right ) \sqrt {c \,x^{2}}}{12}\)	\(21\)
default	\(\frac {x^{2} \left (3 b x +4 a \right ) \sqrt {c \,x^{2}}}{12}\)	\(21\)
risch	\(\frac {a \,x^{2} \sqrt {c \,x^{2}}}{3}+\frac {b \,x^{3} \sqrt {c \,x^{2}}}{4}\)	\(28\)
trager	\(\frac {\left (3 b \,x^{3}+4 a \,x^{2}+3 b \,x^{2}+4 a x +3 b x +4 a +3 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{12 x}\)	\(49\)

[In]

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

[Out]

1/12*x^2*(3*b*x+4*a)*(c*x^2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63 \[ \int x \sqrt {c x^2} (a+b x) \, dx=\frac {1}{12} \, {\left (3 \, b x^{3} + 4 \, a x^{2}\right )} \sqrt {c x^{2}} \]

[In]

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

[Out]

1/12*(3*b*x^3 + 4*a*x^2)*sqrt(c*x^2)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int x \sqrt {c x^2} (a+b x) \, dx=\frac {a x^{2} \sqrt {c x^{2}}}{3} + \frac {b x^{3} \sqrt {c x^{2}}}{4} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80 \[ \int x \sqrt {c x^2} (a+b x) \, dx=\frac {\left (c x^{2}\right )^{\frac {3}{2}} b x}{4 \, c} + \frac {\left (c x^{2}\right )^{\frac {3}{2}} a}{3 \, c} \]

[In]

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

[Out]

1/4*(c*x^2)^(3/2)*b*x/c + 1/3*(c*x^2)^(3/2)*a/c

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63 \[ \int x \sqrt {c x^2} (a+b x) \, dx=\frac {1}{12} \, {\left (3 \, b x^{4} \mathrm {sgn}\left (x\right ) + 4 \, a x^{3} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \]

[In]

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

[Out]

1/12*(3*b*x^4*sgn(x) + 4*a*x^3*sgn(x))*sqrt(c)

Mupad [F(-1)]

Timed out. \[ \int x \sqrt {c x^2} (a+b x) \, dx=\int x\,\sqrt {c\,x^2}\,\left (a+b\,x\right ) \,d x \]

[In]

int(x*(c*x^2)^(1/2)*(a + b*x),x)

[Out]

int(x*(c*x^2)^(1/2)*(a + b*x), x)