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

Optimal. Leaf size=24 \[ \frac {x (a+b x)^3}{3 b \sqrt {c x^2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 32} \begin {gather*} \frac {x (a+b x)^3}{3 b \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {x (a+b x)^2}{\sqrt {c x^2}} \, dx &=\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]  time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} \frac {x (a+b x)^3}{3 b \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*x)^3)/(3*b*Sqrt[c*x^2])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.03, size = 34, normalized size = 1.42 \begin {gather*} \frac {\sqrt {c x^2} \left (3 a^2+3 a b x+b^2 x^2\right )}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.16, size = 30, normalized size = 1.25 \begin {gather*} \frac {{\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt {c x^{2}}}{3 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

giac [A]  time = 1.15, size = 36, normalized size = 1.50 \begin {gather*} \frac {1}{3} \, \sqrt {c x^{2}} {\left ({\left (\frac {b^{2} x}{c} + \frac {3 \, a b}{c}\right )} x + \frac {3 \, a^{2}}{c}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 31, normalized size = 1.29 \begin {gather*} \frac {\left (b^{2} x^{2}+3 a b x +3 a^{2}\right ) x^{2}}{3 \sqrt {c \,x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 1.37, size = 42, normalized size = 1.75 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x\,{\left (a+b\,x\right )}^2}{\sqrt {c\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [B]  time = 0.54, size = 56, normalized size = 2.33 \begin {gather*} \frac {a^{2} x^{2}}{\sqrt {c} \sqrt {x^{2}}} + \frac {a b x^{3}}{\sqrt {c} \sqrt {x^{2}}} + \frac {b^{2} x^{4}}{3 \sqrt {c} \sqrt {x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________