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

Optimal. Leaf size=35 \[ \frac {a x^3}{2 \sqrt {c x^2}}+\frac {b x^4}{3 \sqrt {c x^2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 35, 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 {a x^3}{2 \sqrt {c x^2}}+\frac {b x^4}{3 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(a + b*x))/Sqrt[c*x^2],x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 24, normalized size = 0.69 \begin {gather*} \frac {x^3 (3 a+2 b x)}{6 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(3*a + 2*b*x))/(6*Sqrt[c*x^2])

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IntegrateAlgebraic [A]  time = 0.02, size = 25, normalized size = 0.71 \begin {gather*} \frac {x \sqrt {c x^2} (3 a+2 b x)}{6 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[c*x^2]*(3*a + 2*b*x))/(6*c)

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fricas [A]  time = 1.02, size = 23, normalized size = 0.66 \begin {gather*} \frac {{\left (2 \, b x^{2} + 3 \, a x\right )} \sqrt {c x^{2}}}{6 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b*x^2 + 3*a*x)*sqrt(c*x^2)/c

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giac [A]  time = 1.16, size = 24, normalized size = 0.69 \begin {gather*} \frac {1}{6} \, \sqrt {c x^{2}} {\left (\frac {2 \, b x}{c} + \frac {3 \, a}{c}\right )} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(c*x^2)*(2*b*x/c + 3*a/c)*x

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maple [A]  time = 0.00, size = 21, normalized size = 0.60 \begin {gather*} \frac {\left (2 b x +3 a \right ) x^{3}}{6 \sqrt {c \,x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.30, size = 26, normalized size = 0.74 \begin {gather*} \frac {\sqrt {c x^{2}} b x^{2}}{3 \, c} + \frac {a x^{2}}{2 \, \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.25, size = 23, normalized size = 0.66 \begin {gather*} \frac {2\,b\,\sqrt {x^6}+3\,a\,x\,\sqrt {x^2}}{6\,\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(a + b*x))/(c*x^2)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.52, size = 36, normalized size = 1.03 \begin {gather*} \frac {a x^{3}}{2 \sqrt {c} \sqrt {x^{2}}} + \frac {b x^{4}}{3 \sqrt {c} \sqrt {x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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