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

Optimal result

Integrand size = 17, antiderivative size = 48 \[ \int \frac {1}{x^5 \sqrt {a x+b x^4}} \, dx=-\frac {2 \sqrt {a x+b x^4}}{9 a x^5}+\frac {4 b \sqrt {a x+b x^4}}{9 a^2 x^2} \]

[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] (verified)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2041, 2039} \[ \int \frac {1}{x^5 \sqrt {a x+b x^4}} \, dx=\frac {4 b \sqrt {a x+b x^4}}{9 a^2 x^2}-\frac {2 \sqrt {a x+b x^4}}{9 a x^5} \]

[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*} \text {integral}& = -\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] (verified)

Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x^5 \sqrt {a x+b x^4}} \, dx=-\frac {2 \left (a-2 b x^3\right ) \sqrt {x \left (a+b x^3\right )}}{9 a^2 x^5} \]

[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] (verified)

Time = 2.59 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.58

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.60 \[ \int \frac {1}{x^5 \sqrt {a x+b x^4}} \, dx=\frac {2 \, \sqrt {b x^{4} + a x} {\left (2 \, b x^{3} - a\right )}}{9 \, a^{2} x^{5}} \]

[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]

\[ \int \frac {1}{x^5 \sqrt {a x+b x^4}} \, dx=\int \frac {1}{x^{5} \sqrt {x \left (a + b x^{3}\right )}}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^5 \sqrt {a x+b x^4}} \, dx=\frac {2 \, {\left (2 \, b^{2} x^{7} + a b x^{4} - a^{2} x\right )}}{9 \, \sqrt {b x^{3} + a} a^{2} x^{\frac {11}{2}}} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x^5 \sqrt {a x+b x^4}} \, dx=-\frac {2 \, {\left (b + \frac {a}{x^{3}}\right )}^{\frac {3}{2}}}{9 \, a^{2}} + \frac {2 \, \sqrt {b + \frac {a}{x^{3}}} b}{3 \, a^{2}} \]

[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] (verification not implemented)

Time = 9.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^5 \sqrt {a x+b x^4}} \, dx=-\frac {2\,\sqrt {b\,x^4+a\,x}\,\left (a-2\,b\,x^3\right )}{9\,a^2\,x^5} \]

[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)