\(\int (c x^2)^{3/2} (a+b x) \, dx\) [767]

Optimal result

Integrand size = 15, antiderivative size = 37 \[ \int \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{4} a c x^3 \sqrt {c x^2}+\frac {1}{5} b c x^4 \sqrt {c x^2} \]

[Out]

1/4*a*c*x^3*(c*x^2)^(1/2)+1/5*b*c*x^4*(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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 45} \[ \int \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{4} a c x^3 \sqrt {c x^2}+\frac {1}{5} b c x^4 \sqrt {c x^2} \]

[In]

Int[(c*x^2)^(3/2)*(a + b*x),x]

[Out]

(a*c*x^3*Sqrt[c*x^2])/4 + (b*c*x^4*Sqrt[c*x^2])/5

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^3 (a+b x) \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int \left (a x^3+b x^4\right ) \, dx}{x} \\ & = \frac {1}{4} a c x^3 \sqrt {c x^2}+\frac {1}{5} b c x^4 \sqrt {c x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.59 \[ \int \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{20} x \left (c x^2\right )^{3/2} (5 a+4 b x) \]

[In]

Integrate[(c*x^2)^(3/2)*(a + b*x),x]

[Out]

(x*(c*x^2)^(3/2)*(5*a + 4*b*x))/20

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.51

method	result	size
gosper	\(\frac {x \left (4 b x +5 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{20}\)	\(19\)
default	\(\frac {x \left (4 b x +5 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{20}\)	\(19\)
risch	\(\frac {a c \,x^{3} \sqrt {c \,x^{2}}}{4}+\frac {b c \,x^{4} \sqrt {c \,x^{2}}}{5}\)	\(30\)
trager	\(\frac {c \left (4 b \,x^{4}+5 a \,x^{3}+4 b \,x^{3}+5 a \,x^{2}+4 b \,x^{2}+5 a x +4 b x +5 a +4 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{20 x}\)	\(62\)

[In]

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

[Out]

1/20*x*(4*b*x+5*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 \[ \int \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{20} \, {\left (4 \, b c x^{4} + 5 \, a c x^{3}\right )} \sqrt {c x^{2}} \]

[In]

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

[Out]

1/20*(4*b*c*x^4 + 5*a*c*x^3)*sqrt(c*x^2)

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73 \[ \int \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {a x \left (c x^{2}\right )^{\frac {3}{2}}}{4} + \frac {b x^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{5} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.68 \[ \int \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{4} \, \left (c x^{2}\right )^{\frac {3}{2}} a x + \frac {\left (c x^{2}\right )^{\frac {5}{2}} b}{5 \, c} \]

[In]

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

[Out]

1/4*(c*x^2)^(3/2)*a*x + 1/5*(c*x^2)^(5/2)*b/c

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.59 \[ \int \left (c x^2\right )^{3/2} (a+b x) \, dx=\frac {1}{20} \, {\left (4 \, b x^{5} \mathrm {sgn}\left (x\right ) + 5 \, a x^{4} \mathrm {sgn}\left (x\right )\right )} c^{\frac {3}{2}} \]

[In]

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

[Out]

1/20*(4*b*x^5*sgn(x) + 5*a*x^4*sgn(x))*c^(3/2)

Mupad [F(-1)]

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

[In]

int((c*x^2)^(3/2)*(a + b*x),x)

[Out]

int((c*x^2)^(3/2)*(a + b*x), x)