\(\int \frac {x^3 (a+b x)^2}{(c x^2)^{3/2}} \, dx\) [836]

Optimal result

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

[Out]

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

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

[In]

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

[Out]

(x*(a + b*x)^3)/(3*b*c*Sqrt[c*x^2])

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

Mathematica [A] (verified)

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

[In]

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

[Out]

(x^3*(a + b*x)^3)/(3*b*(c*x^2)^(3/2))

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {\left (b x +a \right )^{3} x^{3}}{3 \left (c \,x^{2}\right )^{\frac {3}{2}} b}\)	\(23\)
risch	\(\frac {x \left (b x +a \right )^{3}}{3 b c \sqrt {c \,x^{2}}}\)	\(24\)
gosper	\(\frac {x^{4} \left (b^{2} x^{2}+3 a b x +3 a^{2}\right )}{3 \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(31\)
trager	\(\frac {\left (b^{2} x^{2}+3 a b x +b^{2} x +3 a^{2}+3 a b +b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{3 c^{2} x}\)	\(49\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

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

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (20) = 40\).

Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {x^3 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {a^{2} x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}}} + \frac {a b x^{5}}{\left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} x^{6}}{3 \left (c x^{2}\right )^{\frac {3}{2}}} \]

[In]

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

[Out]

a**2*x**4/(c*x**2)**(3/2) + a*b*x**5/(c*x**2)**(3/2) + b**2*x**6/(3*(c*x**2)**(3/2))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (23) = 46\).

Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.93 \[ \int \frac {x^3 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {b^{2} x^{4}}{3 \, \sqrt {c x^{2}} c} + \frac {a b x^{3}}{\sqrt {c x^{2}} c} + \frac {a^{2} x^{2}}{\sqrt {c x^{2}} c} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {x^3 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx=\frac {b^{2} \sqrt {c} x^{3} + 3 \, a b \sqrt {c} x^{2} + 3 \, a^{2} \sqrt {c} x}{3 \, c^{2} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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