3.3.10 \(\int \frac {1}{x^{9/2} \sqrt {a x^2+b x^5}} \, dx\)

Optimal. Leaf size=56 \[ \frac {4 b \sqrt {a x^2+b x^5}}{9 a^2 x^{5/2}}-\frac {2 \sqrt {a x^2+b x^5}}{9 a x^{11/2}} \]

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Rubi [A]  time = 0.08, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2016, 2014} \begin {gather*} \frac {4 b \sqrt {a x^2+b x^5}}{9 a^2 x^{5/2}}-\frac {2 \sqrt {a x^2+b x^5}}{9 a x^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(9/2)*Sqrt[a*x^2 + b*x^5]),x]

[Out]

(-2*Sqrt[a*x^2 + b*x^5])/(9*a*x^(11/2)) + (4*b*Sqrt[a*x^2 + b*x^5])/(9*a^2*x^(5/2))

Rule 2014

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*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

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])

Rubi steps

\begin {align*} \int \frac {1}{x^{9/2} \sqrt {a x^2+b x^5}} \, dx &=-\frac {2 \sqrt {a x^2+b x^5}}{9 a x^{11/2}}-\frac {(2 b) \int \frac {1}{x^{3/2} \sqrt {a x^2+b x^5}} \, dx}{3 a}\\ &=-\frac {2 \sqrt {a x^2+b x^5}}{9 a x^{11/2}}+\frac {4 b \sqrt {a x^2+b x^5}}{9 a^2 x^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 35, normalized size = 0.62 \begin {gather*} -\frac {2 \left (a-2 b x^3\right ) \sqrt {x^2 \left (a+b x^3\right )}}{9 a^2 x^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(9/2)*Sqrt[a*x^2 + b*x^5]),x]

[Out]

(-2*(a - 2*b*x^3)*Sqrt[x^2*(a + b*x^3)])/(9*a^2*x^(11/2))

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IntegrateAlgebraic [A]  time = 0.29, size = 37, normalized size = 0.66 \begin {gather*} \frac {2 \left (2 b x^3-a\right ) \sqrt {a x^2+b x^5}}{9 a^2 x^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^(9/2)*Sqrt[a*x^2 + b*x^5]),x]

[Out]

(2*(-a + 2*b*x^3)*Sqrt[a*x^2 + b*x^5])/(9*a^2*x^(11/2))

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fricas [A]  time = 0.40, size = 31, normalized size = 0.55 \begin {gather*} \frac {2 \, \sqrt {b x^{5} + a x^{2}} {\left (2 \, b x^{3} - a\right )}}{9 \, a^{2} x^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(9/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/9*sqrt(b*x^5 + a*x^2)*(2*b*x^3 - a)/(a^2*x^(11/2))

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giac [A]  time = 0.23, size = 38, normalized size = 0.68 \begin {gather*} -\frac {2 \, {\left (b + \frac {a}{x^{3}}\right )}^{\frac {3}{2}}}{9 \, a^{2}} + \frac {2 \, \sqrt {b + \frac {a}{x^{3}}} b}{3 \, a^{2}} - \frac {4 \, b^{\frac {3}{2}}}{9 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(9/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 37, normalized size = 0.66 \begin {gather*} -\frac {2 \left (b \,x^{3}+a \right ) \left (-2 b \,x^{3}+a \right )}{9 \sqrt {b \,x^{5}+a \,x^{2}}\, a^{2} x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(9/2)/(b*x^5+a*x^2)^(1/2),x)

[Out]

-2/9*(b*x^3+a)*(-2*b*x^3+a)/x^(7/2)/a^2/(b*x^5+a*x^2)^(1/2)

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maxima [A]  time = 1.42, size = 38, normalized size = 0.68 \begin {gather*} \frac {2 \, {\left (2 \, b^{2} x^{7} + a b x^{4} - a^{2} x\right )}}{9 \, \sqrt {b x^{3} + a} a^{2} x^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(9/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="maxima")

[Out]

2/9*(2*b^2*x^7 + a*b*x^4 - a^2*x)/(sqrt(b*x^3 + a)*a^2*x^(11/2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x^{9/2}\,\sqrt {b\,x^5+a\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(9/2)*(a*x^2 + b*x^5)^(1/2)),x)

[Out]

int(1/(x^(9/2)*(a*x^2 + b*x^5)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{\frac {9}{2}} \sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(9/2)/(b*x**5+a*x**2)**(1/2),x)

[Out]

Integral(1/(x**(9/2)*sqrt(x**2*(a + b*x**3))), x)

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