\(\int \frac {(c x^2)^{5/2} (a+b x)^2}{x^5} \, dx\) [826]

Optimal result

Integrand size = 20, antiderivative size = 29 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^5} \, dx=\frac {c^2 \sqrt {c x^2} (a+b x)^3}{3 b x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 29, 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} \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^5} \, dx=\frac {c^2 \sqrt {c x^2} (a+b x)^3}{3 b x} \]

[In]

Int[((c*x^2)^(5/2)*(a + b*x)^2)/x^5,x]

[Out]

(c^2*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^2 \sqrt {c x^2}\right ) \int (a+b x)^2 \, dx}{x} \\ & = \frac {c^2 \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 = 0.90 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^5} \, dx=\frac {\left (c x^2\right )^{5/2} (a+b x)^3}{3 b x^5} \]

[In]

Integrate[((c*x^2)^(5/2)*(a + b*x)^2)/x^5,x]

[Out]

((c*x^2)^(5/2)*(a + b*x)^3)/(3*b*x^5)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (b x +a \right )^{3}}{3 x^{5} b}\)	\(23\)
risch	\(\frac {c^{2} \left (b x +a \right )^{3} \sqrt {c \,x^{2}}}{3 b x}\)	\(26\)
gosper	\(\frac {\left (b^{2} x^{2}+3 a b x +3 a^{2}\right ) \left (c \,x^{2}\right )^{\frac {5}{2}}}{3 x^{4}}\)	\(31\)
trager	\(\frac {c^{2} \left (b^{2} x^{2}+3 a b x +b^{2} x +3 a^{2}+3 a b +b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{3 x}\)	\(49\)

[In]

int((c*x^2)^(5/2)*(b*x+a)^2/x^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^5} \, dx=\frac {1}{3} \, {\left (b^{2} c^{2} x^{2} + 3 \, a b c^{2} x + 3 \, a^{2} c^{2}\right )} \sqrt {c x^{2}} \]

[In]

integrate((c*x^2)^(5/2)*(b*x+a)^2/x^5,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).

Time = 0.39 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^5} \, dx=\frac {a^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{x^{4}} + \frac {a b \left (c x^{2}\right )^{\frac {5}{2}}}{x^{3}} + \frac {b^{2} \left (c x^{2}\right )^{\frac {5}{2}}}{3 x^{2}} \]

[In]

integrate((c*x**2)**(5/2)*(b*x+a)**2/x**5,x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^5} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((c*x^2)^(5/2)*(b*x+a)^2/x^5,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^5} \, dx=\frac {1}{3} \, {\left (b^{2} c^{2} x^{3} \mathrm {sgn}\left (x\right ) + 3 \, a b c^{2} x^{2} \mathrm {sgn}\left (x\right ) + 3 \, a^{2} c^{2} x \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \]

[In]

integrate((c*x^2)^(5/2)*(b*x+a)^2/x^5,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c x^2\right )^{5/2} (a+b x)^2}{x^5} \, dx=\int \frac {{\left (c\,x^2\right )}^{5/2}\,{\left (a+b\,x\right )}^2}{x^5} \,d x \]

[In]

int(((c*x^2)^(5/2)*(a + b*x)^2)/x^5,x)

[Out]

int(((c*x^2)^(5/2)*(a + b*x)^2)/x^5, x)