\(\int x (c x^2)^{3/2} (a+b x) \, dx\) [766]

Optimal result

Integrand size = 16, antiderivative size = 37 \[ \int x \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{5} a c x^4 \sqrt {c x^2}+\frac {1}{6} b c x^5 \sqrt {c x^2} \]

[Out]

1/5*a*c*x^4*(c*x^2)^(1/2)+1/6*b*c*x^5*(c*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, 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} \[ \int x \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{5} a c x^4 \sqrt {c x^2}+\frac {1}{6} b c x^5 \sqrt {c x^2} \]

[In]

Int[x*(c*x^2)^(3/2)*(a + b*x),x]

[Out]

(a*c*x^4*Sqrt[c*x^2])/5 + (b*c*x^5*Sqrt[c*x^2])/6

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 {\left (c \sqrt {c x^2}\right ) \int x^4 (a+b x) \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int \left (a x^4+b x^5\right ) \, dx}{x} \\ & = \frac {1}{5} a c x^4 \sqrt {c x^2}+\frac {1}{6} b c x^5 \sqrt {c x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65 \[ \int x \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{30} x^2 \left (c x^2\right )^{3/2} (6 a+5 b x) \]

[In]

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

[Out]

(x^2*(c*x^2)^(3/2)*(6*a + 5*b*x))/30

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.57

method	result	size
gosper	\(\frac {x^{2} \left (5 b x +6 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{30}\)	\(21\)
default	\(\frac {x^{2} \left (5 b x +6 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{30}\)	\(21\)
risch	\(\frac {a c \,x^{4} \sqrt {c \,x^{2}}}{5}+\frac {b c \,x^{5} \sqrt {c \,x^{2}}}{6}\)	\(30\)
trager	\(\frac {c \left (5 b \,x^{5}+6 a \,x^{4}+5 b \,x^{4}+6 a \,x^{3}+5 b \,x^{3}+6 a \,x^{2}+5 b \,x^{2}+6 a x +5 b x +6 a +5 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{30 x}\)	\(74\)

[In]

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

[Out]

1/30*x^2*(5*b*x+6*a)*(c*x^2)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65 \[ \int x \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{30} \, {\left (5 \, b c x^{5} + 6 \, a c x^{4}\right )} \sqrt {c x^{2}} \]

[In]

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

[Out]

1/30*(5*b*c*x^5 + 6*a*c*x^4)*sqrt(c*x^2)

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int x \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {a x^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{5} + \frac {b x^{3} \left (c x^{2}\right )^{\frac {3}{2}}}{6} \]

[In]

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

[Out]

a*x**2*(c*x**2)**(3/2)/5 + b*x**3*(c*x**2)**(3/2)/6

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.76 \[ \int x \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {\left (c x^{2}\right )^{\frac {5}{2}} b x}{6 \, c} + \frac {\left (c x^{2}\right )^{\frac {5}{2}} a}{5 \, c} \]

[In]

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

[Out]

1/6*(c*x^2)^(5/2)*b*x/c + 1/5*(c*x^2)^(5/2)*a/c

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.59 \[ \int x \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{30} \, {\left (5 \, b x^{6} \mathrm {sgn}\left (x\right ) + 6 \, a x^{5} \mathrm {sgn}\left (x\right )\right )} c^{\frac {3}{2}} \]

[In]

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

[Out]

1/30*(5*b*x^6*sgn(x) + 6*a*x^5*sgn(x))*c^(3/2)

Mupad [F(-1)]

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

[In]

int(x*(c*x^2)^(3/2)*(a + b*x),x)

[Out]

int(x*(c*x^2)^(3/2)*(a + b*x), x)