3.1.95 \(\int \frac {1}{x^5 \sqrt {a x+b x^4}} \, dx\) [95]

Optimal. Leaf size=48 \[ -\frac {2 \sqrt {a x+b x^4}}{9 a x^5}+\frac {4 b \sqrt {a x+b x^4}}{9 a^2 x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2041, 2039} \begin {gather*} \frac {4 b \sqrt {a x+b x^4}}{9 a^2 x^2}-\frac {2 \sqrt {a x+b x^4}}{9 a x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*Sqrt[a*x + b*x^4]),x]

[Out]

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

Rule 2039

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] &&
NeQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

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

________________________________________________________________________________________

Mathematica [A]
time = 0.18, size = 31, normalized size = 0.65 \begin {gather*} -\frac {2 \left (a-2 b x^3\right ) \sqrt {x \left (a+b x^3\right )}}{9 a^2 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*Sqrt[a*x + b*x^4]),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.36, size = 41, normalized size = 0.85

method	result	size
trager	\(-\frac {2 \left (-2 b \,x^{3}+a \right ) \sqrt {b \,x^{4}+a x}}{9 a^{2} x^{5}}\)	\(28\)
gosper	\(-\frac {2 \left (b \,x^{3}+a \right ) \left (-2 b \,x^{3}+a \right )}{9 x^{4} a^{2} \sqrt {b \,x^{4}+a x}}\)	\(35\)
risch	\(-\frac {2 \left (b \,x^{3}+a \right ) \left (-2 b \,x^{3}+a \right )}{9 a^{2} x^{4} \sqrt {x \left (b \,x^{3}+a \right )}}\)	\(35\)
default	\(-\frac {2 \sqrt {b \,x^{4}+a x}}{9 a \,x^{5}}+\frac {4 b \sqrt {b \,x^{4}+a x}}{9 a^{2} x^{2}}\)	\(41\)
elliptic	\(-\frac {2 \sqrt {b \,x^{4}+a x}}{9 a \,x^{5}}+\frac {4 b \sqrt {b \,x^{4}+a x}}{9 a^{2} x^{2}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(b*x^4+a*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 38, normalized size = 0.79 \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^5/(b*x^4+a*x)^(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))

________________________________________________________________________________________

Fricas [A]
time = 1.42, size = 29, normalized size = 0.60 \begin {gather*} \frac {2 \, \sqrt {b x^{4} + a x} {\left (2 \, b x^{3} - a\right )}}{9 \, a^{2} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^4+a*x)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{5} \sqrt {x \left (a + b x^{3}\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(b*x**4+a*x)**(1/2),x)

[Out]

Integral(1/(x**5*sqrt(x*(a + b*x**3))), x)

________________________________________________________________________________________

Giac [A]
time = 0.89, size = 30, normalized size = 0.62 \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(b*x^4+a*x)^(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

________________________________________________________________________________________

Mupad [B]
time = 5.13, size = 27, normalized size = 0.56 \begin {gather*} -\frac {2\,\sqrt {b\,x^4+a\,x}\,\left (a-2\,b\,x^3\right )}{9\,a^2\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(a*x + b*x^4)^(1/2)),x)

[Out]

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

________________________________________________________________________________________