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

Optimal. Leaf size=37 \[ \frac {1}{4} a c x^3 \sqrt {c x^2}+\frac {1}{5} b c x^4 \sqrt {c x^2} \]

[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]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 45} \begin {gather*} \frac {1}{4} a c x^3 \sqrt {c x^2}+\frac {1}{5} b c x^4 \sqrt {c x^2} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \left (c x^2\right )^{3/2} (a+b x) \, dx &=\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]
time = 0.00, size = 22, normalized size = 0.59 \begin {gather*} \frac {1}{20} x \left (c x^2\right )^{3/2} (5 a+4 b x) \end {gather*}

Antiderivative was successfully verified.

[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]
time = 0.02, size = 19, normalized size = 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	\(\frac {c \left (4 b \,x^{4}+5 a \,x^{3}+4 b \,x^{3}+5 a \,x^{2}+4 x^{2} b +5 a x +4 b x +5 a +4 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{20 x}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 25, normalized size = 0.68 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]
time = 0.73, size = 24, normalized size = 0.65 \begin {gather*} \frac {1}{20} \, {\left (4 \, b c x^{4} + 5 \, a c x^{3}\right )} \sqrt {c x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.14, size = 27, normalized size = 0.73 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Giac [A]
time = 1.09, size = 22, normalized size = 0.59 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(3/2)*(a + b*x),x)

[Out]

int((c*x^2)^(3/2)*(a + b*x), x)

________________________________________________________________________________________