\(\int \frac {x (a+b x)^2}{\sqrt {c x^2}} \, dx\) [830]

Optimal result

Integrand size = 18, antiderivative size = 24 \[ \int \frac {x (a+b x)^2}{\sqrt {c x^2}} \, dx=\frac {x (a+b x)^3}{3 b \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 32} \[ \int \frac {x (a+b x)^2}{\sqrt {c x^2}} \, dx=\frac {x (a+b x)^3}{3 b \sqrt {c x^2}} \]

[In]

Int[(x*(a + b*x)^2)/Sqrt[c*x^2],x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x (a+b x)^2}{\sqrt {c x^2}} \, dx=\frac {x (a+b x)^3}{3 b \sqrt {c x^2}} \]

[In]

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

[Out]

(x*(a + b*x)^3)/(3*b*Sqrt[c*x^2])

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {x \left (b x +a \right )^{3}}{3 b \sqrt {c \,x^{2}}}\)	\(21\)
risch	\(\frac {x \left (b x +a \right )^{3}}{3 b \sqrt {c \,x^{2}}}\)	\(21\)
gosper	\(\frac {x^{2} \left (b^{2} x^{2}+3 a b x +3 a^{2}\right )}{3 \sqrt {c \,x^{2}}}\)	\(31\)
trager	\(\frac {\left (b^{2} x^{2}+3 a b x +b^{2} x +3 a^{2}+3 a b +b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{3 c x}\)	\(49\)

[In]

int(x*(b*x+a)^2/(c*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {x (a+b x)^2}{\sqrt {c x^2}} \, dx=\frac {{\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt {c x^{2}}}{3 \, c} \]

[In]

integrate(x*(b*x+a)^2/(c*x^2)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).

Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {x (a+b x)^2}{\sqrt {c x^2}} \, dx=\frac {a^{2} x^{2}}{\sqrt {c x^{2}}} + \frac {a b x^{3}}{\sqrt {c x^{2}}} + \frac {b^{2} x^{4}}{3 \sqrt {c x^{2}}} \]

[In]

integrate(x*(b*x+a)**2/(c*x**2)**(1/2),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).

Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {x (a+b x)^2}{\sqrt {c x^2}} \, dx=\frac {\sqrt {c x^{2}} b^{2} x^{2}}{3 \, c} + \frac {a b x^{2}}{\sqrt {c}} + \frac {\sqrt {c x^{2}} a^{2}}{c} \]

[In]

integrate(x*(b*x+a)^2/(c*x^2)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {x (a+b x)^2}{\sqrt {c x^2}} \, dx=\frac {b^{2} \sqrt {c} x^{3} + 3 \, a b \sqrt {c} x^{2} + 3 \, a^{2} \sqrt {c} x}{3 \, c \mathrm {sgn}\left (x\right )} \]

[In]

integrate(x*(b*x+a)^2/(c*x^2)^(1/2),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*sgn(x))

Mupad [F(-1)]

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

[In]

int((x*(a + b*x)^2)/(c*x^2)^(1/2),x)

[Out]

int((x*(a + b*x)^2)/(c*x^2)^(1/2), x)