\(\int \frac {(c x^2)^{3/2} (a+b x)^2}{x^3} \, dx\) [818]

Optimal result

Integrand size = 20, antiderivative size = 27 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^2}{x^3} \, dx=\frac {c \sqrt {c x^2} (a+b x)^3}{3 b x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32} \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^2}{x^3} \, dx=\frac {c \sqrt {c x^2} (a+b x)^3}{3 b x} \]

[In]

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

[Out]

(c*Sqrt[c*x^2]*(a + b*x)^3)/(3*b*x)

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^2}{x^3} \, dx=\frac {\left (c x^2\right )^{3/2} (a+b x)^3}{3 b x^3} \]

[In]

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

[Out]

((c*x^2)^(3/2)*(a + b*x)^3)/(3*b*x^3)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x +a \right )^{3}}{3 x^{3} b}\)	\(23\)
risch	\(\frac {c \left (b x +a \right )^{3} \sqrt {c \,x^{2}}}{3 b x}\)	\(24\)
gosper	\(\frac {\left (b^{2} x^{2}+3 a b x +3 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{3 x^{2}}\)	\(31\)
trager	\(\frac {c \left (b^{2} x^{2}+3 a b x +b^{2} x +3 a^{2}+3 a b +b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{3 x}\)	\(47\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^2}{x^3} \, dx=\frac {1}{3} \, {\left (b^{2} c x^{2} + 3 \, a b c x + 3 \, a^{2} c\right )} \sqrt {c x^{2}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).

Time = 0.40 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^2}{x^3} \, dx=\frac {a^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{x^{2}} + \frac {a b \left (c x^{2}\right )^{\frac {3}{2}}}{x} + \frac {b^{2} \left (c x^{2}\right )^{\frac {3}{2}}}{3} \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^2}{x^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^2}{x^3} \, dx=\frac {1}{3} \, {\left (\frac {{\left (b x + a\right )}^{3} \mathrm {sgn}\left (x\right )}{b} - \frac {a^{3} \mathrm {sgn}\left (x\right )}{b}\right )} c^{\frac {3}{2}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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