\(\int \frac {x^3 (a+b x)}{\sqrt {c x^2}} \, dx\) [780]

Optimal result

Integrand size = 18, antiderivative size = 35 \[ \int \frac {x^3 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {a x^4}{3 \sqrt {c x^2}}+\frac {b x^5}{4 \sqrt {c x^2}} \]

[Out]

1/3*a*x^4/(c*x^2)^(1/2)+1/4*b*x^5/(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} \[ \int \frac {x^3 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {a x^4}{3 \sqrt {c x^2}}+\frac {b x^5}{4 \sqrt {c x^2}} \]

[In]

Int[(x^3*(a + b*x))/Sqrt[c*x^2],x]

[Out]

(a*x^4)/(3*Sqrt[c*x^2]) + (b*x^5)/(4*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^2 (a+b x) \, dx}{\sqrt {c x^2}} \\ & = \frac {x \int \left (a x^2+b x^3\right ) \, dx}{\sqrt {c x^2}} \\ & = \frac {a x^4}{3 \sqrt {c x^2}}+\frac {b x^5}{4 \sqrt {c x^2}} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

(x^4*(4*a + 3*b*x))/(12*Sqrt[c*x^2])

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.60

method	result	size
gosper	\(\frac {x^{4} \left (3 b x +4 a \right )}{12 \sqrt {c \,x^{2}}}\)	\(21\)
default	\(\frac {x^{4} \left (3 b x +4 a \right )}{12 \sqrt {c \,x^{2}}}\)	\(21\)
risch	\(\frac {a \,x^{4}}{3 \sqrt {c \,x^{2}}}+\frac {b \,x^{5}}{4 \sqrt {c \,x^{2}}}\)	\(28\)
trager	\(\frac {\left (3 b \,x^{3}+4 a \,x^{2}+3 b \,x^{2}+4 a x +3 b x +4 a +3 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{12 c x}\)	\(52\)

[In]

int(x^3*(b*x+a)/(c*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.71 \[ \int \frac {x^3 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {{\left (3 \, b x^{3} + 4 \, a x^{2}\right )} \sqrt {c x^{2}}}{12 \, c} \]

[In]

integrate(x^3*(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)/c

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {a x^{4}}{3 \sqrt {c x^{2}}} + \frac {b x^{5}}{4 \sqrt {c x^{2}}} \]

[In]

integrate(x**3*(b*x+a)/(c*x**2)**(1/2),x)

[Out]

a*x**4/(3*sqrt(c*x**2)) + b*x**5/(4*sqrt(c*x**2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {\sqrt {c x^{2}} b x^{3}}{4 \, c} + \frac {\sqrt {c x^{2}} a x^{2}}{3 \, c} \]

[In]

integrate(x^3*(b*x+a)/(c*x^2)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63 \[ \int \frac {x^3 (a+b x)}{\sqrt {c x^2}} \, dx=\frac {3 \, b x^{4} + 4 \, a x^{3}}{12 \, \sqrt {c} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(x^3*(b*x+a)/(c*x^2)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 (a+b x)}{\sqrt {c x^2}} \, dx=\int \frac {x^3\,\left (a+b\,x\right )}{\sqrt {c\,x^2}} \,d x \]

[In]

int((x^3*(a + b*x))/(c*x^2)^(1/2),x)

[Out]

int((x^3*(a + b*x))/(c*x^2)^(1/2), x)