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

Optimal. Leaf size=57 \[ \frac {a^2 x^3}{2 \sqrt {c x^2}}+\frac {2 a b x^4}{3 \sqrt {c x^2}}+\frac {b^2 x^5}{4 \sqrt {c x^2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \begin {gather*} \frac {a^2 x^3}{2 \sqrt {c x^2}}+\frac {2 a b x^4}{3 \sqrt {c x^2}}+\frac {b^2 x^5}{4 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(a + b*x)^2)/Sqrt[c*x^2],x]

[Out]

(a^2*x^3)/(2*Sqrt[c*x^2]) + (2*a*b*x^4)/(3*Sqrt[c*x^2]) + (b^2*x^5)/(4*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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {x^2 (a+b x)^2}{\sqrt {c x^2}} \, dx &=\frac {x \int x (a+b x)^2 \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {a^2 x^3}{2 \sqrt {c x^2}}+\frac {2 a b x^4}{3 \sqrt {c x^2}}+\frac {b^2 x^5}{4 \sqrt {c x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 35, normalized size = 0.61 \begin {gather*} \frac {x^3 \left (6 a^2+8 a b x+3 b^2 x^2\right )}{12 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(6*a^2 + 8*a*b*x + 3*b^2*x^2))/(12*Sqrt[c*x^2])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.03, size = 36, normalized size = 0.63 \begin {gather*} \frac {x \sqrt {c x^2} \left (6 a^2+8 a b x+3 b^2 x^2\right )}{12 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[c*x^2]*(6*a^2 + 8*a*b*x + 3*b^2*x^2))/(12*c)

________________________________________________________________________________________

fricas [A]  time = 1.07, size = 34, normalized size = 0.60 \begin {gather*} \frac {{\left (3 \, b^{2} x^{3} + 8 \, a b x^{2} + 6 \, a^{2} x\right )} \sqrt {c x^{2}}}{12 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^2*x^3 + 8*a*b*x^2 + 6*a^2*x)*sqrt(c*x^2)/c

________________________________________________________________________________________

giac [A]  time = 1.25, size = 38, normalized size = 0.67 \begin {gather*} \frac {1}{12} \, \sqrt {c x^{2}} {\left ({\left (\frac {3 \, b^{2} x}{c} + \frac {8 \, a b}{c}\right )} x + \frac {6 \, a^{2}}{c}\right )} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(c*x^2)*((3*b^2*x/c + 8*a*b/c)*x + 6*a^2/c)*x

________________________________________________________________________________________

maple [A]  time = 0.00, size = 32, normalized size = 0.56 \begin {gather*} \frac {\left (3 b^{2} x^{2}+8 a b x +6 a^{2}\right ) x^{3}}{12 \sqrt {c \,x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x^3*(3*b^2*x^2+8*a*b*x+6*a^2)/(c*x^2)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.32, size = 47, normalized size = 0.82 \begin {gather*} \frac {\sqrt {c x^{2}} b^{2} x^{3}}{4 \, c} + \frac {2 \, \sqrt {c x^{2}} a b x^{2}}{3 \, c} + \frac {a^{2} x^{2}}{2 \, \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2\,{\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^2*(a + b*x)^2)/(c*x^2)^(1/2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.65, size = 61, normalized size = 1.07 \begin {gather*} \frac {a^{2} x^{3}}{2 \sqrt {c} \sqrt {x^{2}}} + \frac {2 a b x^{4}}{3 \sqrt {c} \sqrt {x^{2}}} + \frac {b^{2} x^{5}}{4 \sqrt {c} \sqrt {x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________