3.285 \(\int \frac{x^6}{\sqrt{a x^2+b x^5}} \, dx\)

Optimal. Leaf size=52 \[ \frac{2 x^2 \sqrt{a x^2+b x^5}}{9 b}-\frac{4 a \sqrt{a x^2+b x^5}}{9 b^2 x} \]

[Out]

(-4*a*Sqrt[a*x^2 + b*x^5])/(9*b^2*x) + (2*x^2*Sqrt[a*x^2 + b*x^5])/(9*b)

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Rubi [A]  time = 0.0649265, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2016, 1588} \[ \frac{2 x^2 \sqrt{a x^2+b x^5}}{9 b}-\frac{4 a \sqrt{a x^2+b x^5}}{9 b^2 x} \]

Antiderivative was successfully verified.

[In]

Int[x^6/Sqrt[a*x^2 + b*x^5],x]

[Out]

(-4*a*Sqrt[a*x^2 + b*x^5])/(9*b^2*x) + (2*x^2*Sqrt[a*x^2 + b*x^5])/(9*b)

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^6}{\sqrt{a x^2+b x^5}} \, dx &=\frac{2 x^2 \sqrt{a x^2+b x^5}}{9 b}-\frac{(2 a) \int \frac{x^3}{\sqrt{a x^2+b x^5}} \, dx}{3 b}\\ &=-\frac{4 a \sqrt{a x^2+b x^5}}{9 b^2 x}+\frac{2 x^2 \sqrt{a x^2+b x^5}}{9 b}\\ \end{align*}

Mathematica [A]  time = 0.0171806, size = 34, normalized size = 0.65 \[ \frac{2 \left (b x^3-2 a\right ) \sqrt{x^2 \left (a+b x^3\right )}}{9 b^2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/Sqrt[a*x^2 + b*x^5],x]

[Out]

(2*(-2*a + b*x^3)*Sqrt[x^2*(a + b*x^3)])/(9*b^2*x)

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Maple [A]  time = 0.005, size = 37, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,b{x}^{3}+2\,a \right ) \left ( -b{x}^{3}+2\,a \right ) x}{9\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{5}+a{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x^5+a*x^2)^(1/2),x)

[Out]

-2/9*(b*x^3+a)*(-b*x^3+2*a)*x/b^2/(b*x^5+a*x^2)^(1/2)

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Maxima [A]  time = 1.15932, size = 46, normalized size = 0.88 \begin{align*} \frac{2 \,{\left (b^{2} x^{6} - a b x^{3} - 2 \, a^{2}\right )}}{9 \, \sqrt{b x^{3} + a} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^5+a*x^2)^(1/2),x, algorithm="maxima")

[Out]

2/9*(b^2*x^6 - a*b*x^3 - 2*a^2)/(sqrt(b*x^3 + a)*b^2)

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Fricas [A]  time = 0.809841, size = 63, normalized size = 1.21 \begin{align*} \frac{2 \, \sqrt{b x^{5} + a x^{2}}{\left (b x^{3} - 2 \, a\right )}}{9 \, b^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^5+a*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/9*sqrt(b*x^5 + a*x^2)*(b*x^3 - 2*a)/(b^2*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\sqrt{x^{2} \left (a + b x^{3}\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x**5+a*x**2)**(1/2),x)

[Out]

Integral(x**6/sqrt(x**2*(a + b*x**3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\sqrt{b x^{5} + a x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x^5+a*x^2)^(1/2),x, algorithm="giac")

[Out]

integrate(x^6/sqrt(b*x^5 + a*x^2), x)