3.3.85 \(\int \frac {x^6}{\sqrt {a x^2+b x^5}} \, dx\) [285]

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

[Out]

-4/9*a*(b*x^5+a*x^2)^(1/2)/b^2/x+2/9*x^2*(b*x^5+a*x^2)^(1/2)/b

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Rubi [A]
time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, 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 = {2041, 1602} \begin {gather*} \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} \end {gather*}

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 1602

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]

Rule 2041

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])

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*}

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Mathematica [A]
time = 0.02, size = 34, normalized size = 0.65 \begin {gather*} \frac {2 \left (-2 a+b x^3\right ) \sqrt {x^2 \left (a+b x^3\right )}}{9 b^2 x} \end {gather*}

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 = 0.37, size = 37, normalized size = 0.71

method	result	size
trager	\(-\frac {2 \left (-b \,x^{3}+2 a \right ) \sqrt {b \,x^{5}+a \,x^{2}}}{9 b^{2} x}\)	\(32\)
gosper	\(-\frac {2 \left (b \,x^{3}+a \right ) \left (-b \,x^{3}+2 a \right ) x}{9 b^{2} \sqrt {b \,x^{5}+a \,x^{2}}}\)	\(37\)
default	\(-\frac {2 \left (b \,x^{3}+a \right ) \left (-b \,x^{3}+2 a \right ) x}{9 b^{2} \sqrt {b \,x^{5}+a \,x^{2}}}\)	\(37\)
risch	\(-\frac {2 x \left (b \,x^{3}+a \right ) \left (-b \,x^{3}+2 a \right )}{9 \sqrt {x^{2} \left (b \,x^{3}+a \right )}\, b^{2}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.30, size = 34, normalized size = 0.65 \begin {gather*} \frac {2 \, {\left (b^{2} x^{6} - a b x^{3} - 2 \, a^{2}\right )}}{9 \, \sqrt {b x^{3} + a} b^{2}} \end {gather*}

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 = 3.07, size = 30, normalized size = 0.58 \begin {gather*} \frac {2 \, \sqrt {b x^{5} + a x^{2}} {\left (b x^{3} - 2 \, a\right )}}{9 \, b^{2} x} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6}}{\sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \end {gather*}

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 [A]
time = 1.95, size = 48, normalized size = 0.92 \begin {gather*} \frac {4 \, a^{\frac {3}{2}} \mathrm {sgn}\left (x\right )}{9 \, b^{2}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{9 \, b^{2} \mathrm {sgn}\left (x\right )} - \frac {2 \, \sqrt {b x^{3} + a} a}{3 \, b^{2} \mathrm {sgn}\left (x\right )} \end {gather*}

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]

4/9*a^(3/2)*sgn(x)/b^2 + 2/9*(b*x^3 + a)^(3/2)/(b^2*sgn(x)) - 2/3*sqrt(b*x^3 + a)*a/(b^2*sgn(x))

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Mupad [B]
time = 5.33, size = 33, normalized size = 0.63 \begin {gather*} -\frac {\sqrt {b\,x^5+a\,x^2}\,\left (\frac {4\,a}{9\,b^2}-\frac {2\,x^3}{9\,b}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(a*x^2 + b*x^5)^(1/2),x)

[Out]

-((a*x^2 + b*x^5)^(1/2)*((4*a)/(9*b^2) - (2*x^3)/(9*b)))/x

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