\(\int \frac {x^2 (a+b x)}{\sqrt {c x^2}} \, dx\) [781]

Optimal result

Integrand size = 18, antiderivative size = 35 \[ \int \frac {x^2 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {a x^3}{2 \sqrt {c x^2}}+\frac {b x^4}{3 \sqrt {c x^2}} \]

[Out]

1/2*a*x^3/(c*x^2)^(1/2)+1/3*b*x^4/(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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \[ \int \frac {x^2 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {a x^3}{2 \sqrt {c x^2}}+\frac {b x^4}{3 \sqrt {c x^2}} \]

[In]

Int[(x^2*(a + b*x))/Sqrt[c*x^2],x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.71 \[ \int \frac {x^2 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {x \sqrt {c x^2} (3 a+2 b x)}{6 c} \]

[In]

Integrate[(x^2*(a + b*x))/Sqrt[c*x^2],x]

[Out]

(x*Sqrt[c*x^2]*(3*a + 2*b*x))/(6*c)

Maple [A] (verified)

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

method	result	size
gosper	\(\frac {x^{3} \left (2 b x +3 a \right )}{6 \sqrt {c \,x^{2}}}\)	\(21\)
default	\(\frac {x^{3} \left (2 b x +3 a \right )}{6 \sqrt {c \,x^{2}}}\)	\(21\)
risch	\(\frac {a \,x^{3}}{2 \sqrt {c \,x^{2}}}+\frac {b \,x^{4}}{3 \sqrt {c \,x^{2}}}\)	\(28\)
trager	\(\frac {\left (2 b \,x^{2}+3 a x +2 b x +3 a +2 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{6 c x}\)	\(40\)

[In]

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

[Out]

1/6*x^3*(2*b*x+3*a)/(c*x^2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.66 \[ \int \frac {x^2 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {{\left (2 \, b x^{2} + 3 \, a x\right )} \sqrt {c x^{2}}}{6 \, c} \]

[In]

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

[Out]

1/6*(2*b*x^2 + 3*a*x)*sqrt(c*x^2)/c

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {x^2 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {a x^{3}}{2 \sqrt {c x^{2}}} + \frac {b x^{4}}{3 \sqrt {c x^{2}}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.74 \[ \int \frac {x^2 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {\sqrt {c x^{2}} b x^{2}}{3 \, c} + \frac {a x^{2}}{2 \, \sqrt {c}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63 \[ \int \frac {x^2 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {2 \, b x^{3} + 3 \, a x^{2}}{6 \, \sqrt {c} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

1/6*(2*b*x^3 + 3*a*x^2)/(sqrt(c)*sgn(x))

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.66 \[ \int \frac {x^2 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {2\,b\,\sqrt {x^6}+3\,a\,x\,\sqrt {x^2}}{6\,\sqrt {c}} \]

[In]

int((x^2*(a + b*x))/(c*x^2)^(1/2),x)

[Out]

(2*b*(x^6)^(1/2) + 3*a*x*(x^2)^(1/2))/(6*c^(1/2))