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

Optimal. Leaf size=57 \[ \frac {1}{3} a^2 x^2 \sqrt {c x^2}+\frac {1}{2} a b x^3 \sqrt {c x^2}+\frac {1}{5} b^2 x^4 \sqrt {c x^2} \]

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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 43} \begin {gather*} \frac {1}{3} a^2 x^2 \sqrt {c x^2}+\frac {1}{2} a b x^3 \sqrt {c x^2}+\frac {1}{5} b^2 x^4 \sqrt {c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[c*x^2]*(a + b*x)^2,x]

[Out]

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

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 x \sqrt {c x^2} (a+b x)^2 \, dx &=\frac {\sqrt {c x^2} \int x^2 (a+b x)^2 \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (a^2 x^2+2 a b x^3+b^2 x^4\right ) \, dx}{x}\\ &=\frac {1}{3} a^2 x^2 \sqrt {c x^2}+\frac {1}{2} a b x^3 \sqrt {c x^2}+\frac {1}{5} b^2 x^4 \sqrt {c x^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 35, normalized size = 0.61 \begin {gather*} \frac {1}{30} x^2 \sqrt {c x^2} \left (10 a^2+15 a b x+6 b^2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*Sqrt[c*x^2]*(10*a^2 + 15*a*b*x + 6*b^2*x^2))/30

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IntegrateAlgebraic [A]  time = 0.02, size = 35, normalized size = 0.61 \begin {gather*} \frac {1}{30} x^2 \sqrt {c x^2} \left (10 a^2+15 a b x+6 b^2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*Sqrt[c*x^2]*(10*a^2 + 15*a*b*x + 6*b^2*x^2))/30

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fricas [A]  time = 1.11, size = 33, normalized size = 0.58 \begin {gather*} \frac {1}{30} \, {\left (6 \, b^{2} x^{4} + 15 \, a b x^{3} + 10 \, a^{2} x^{2}\right )} \sqrt {c 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, algorithm="fricas")

[Out]

1/30*(6*b^2*x^4 + 15*a*b*x^3 + 10*a^2*x^2)*sqrt(c*x^2)

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giac [A]  time = 1.11, size = 35, normalized size = 0.61 \begin {gather*} \frac {1}{30} \, {\left (6 \, b^{2} x^{5} \mathrm {sgn}\relax (x) + 15 \, a b x^{4} \mathrm {sgn}\relax (x) + 10 \, a^{2} x^{3} \mathrm {sgn}\relax (x)\right )} \sqrt {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="giac")

[Out]

1/30*(6*b^2*x^5*sgn(x) + 15*a*b*x^4*sgn(x) + 10*a^2*x^3*sgn(x))*sqrt(c)

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maple [A]  time = 0.00, size = 32, normalized size = 0.56 \begin {gather*} \frac {\left (6 b^{2} x^{2}+15 a b x +10 a^{2}\right ) \sqrt {c \,x^{2}}\, x^{2}}{30} \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]

1/30*x^2*(6*b^2*x^2+15*a*b*x+10*a^2)*(c*x^2)^(1/2)

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maxima [A]  time = 1.31, size = 49, normalized size = 0.86 \begin {gather*} \frac {\left (c x^{2}\right )^{\frac {3}{2}} b^{2} x^{2}}{5 \, c} + \frac {\left (c x^{2}\right )^{\frac {3}{2}} a b x}{2 \, c} + \frac {\left (c x^{2}\right )^{\frac {3}{2}} a^{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="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c*x^2)^(1/2)*(a + b*x)^2,x)

[Out]

int(x*(c*x^2)^(1/2)*(a + b*x)^2, x)

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sympy [A]  time = 0.37, size = 60, normalized size = 1.05 \begin {gather*} \frac {a^{2} \sqrt {c} x^{2} \sqrt {x^{2}}}{3} + \frac {a b \sqrt {c} x^{3} \sqrt {x^{2}}}{2} + \frac {b^{2} \sqrt {c} x^{4} \sqrt {x^{2}}}{5} \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*sqrt(c)*x**2*sqrt(x**2)/3 + a*b*sqrt(c)*x**3*sqrt(x**2)/2 + b**2*sqrt(c)*x**4*sqrt(x**2)/5

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