\(\int \frac {(b+a x^5)^{3/4}}{x^6} \, dx\) [992]

Optimal result

Integrand size = 15, antiderivative size = 75 \[ \int \frac {\left (b+a x^5\right )^{3/4}}{x^6} \, dx=-\frac {\left (b+a x^5\right )^{3/4}}{5 x^5}+\frac {3 a \arctan \left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{10 \sqrt [4]{b}}-\frac {3 a \text {arctanh}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{10 \sqrt [4]{b}} \]

[Out]

-1/5*(a*x^5+b)^(3/4)/x^5+3/10*a*arctan((a*x^5+b)^(1/4)/b^(1/4))/b^(1/4)-3/10*a*arctanh((a*x^5+b)^(1/4)/b^(1/4)
)/b^(1/4)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 43, 65, 304, 209, 212} \[ \int \frac {\left (b+a x^5\right )^{3/4}}{x^6} \, dx=\frac {3 a \arctan \left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )}{10 \sqrt [4]{b}}-\frac {3 a \text {arctanh}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )}{10 \sqrt [4]{b}}-\frac {\left (a x^5+b\right )^{3/4}}{5 x^5} \]

[In]

Int[(b + a*x^5)^(3/4)/x^6,x]

[Out]

-1/5*(b + a*x^5)^(3/4)/x^5 + (3*a*ArcTan[(b + a*x^5)^(1/4)/b^(1/4)])/(10*b^(1/4)) - (3*a*ArcTanh[(b + a*x^5)^(
1/4)/b^(1/4)])/(10*b^(1/4))

Rule 43

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {(b+a x)^{3/4}}{x^2} \, dx,x,x^5\right ) \\ & = -\frac {\left (b+a x^5\right )^{3/4}}{5 x^5}+\frac {1}{20} (3 a) \text {Subst}\left (\int \frac {1}{x \sqrt [4]{b+a x}} \, dx,x,x^5\right ) \\ & = -\frac {\left (b+a x^5\right )^{3/4}}{5 x^5}+\frac {3}{5} \text {Subst}\left (\int \frac {x^2}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^5}\right ) \\ & = -\frac {\left (b+a x^5\right )^{3/4}}{5 x^5}-\frac {1}{10} (3 a) \text {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^5}\right )+\frac {1}{10} (3 a) \text {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^5}\right ) \\ & = -\frac {\left (b+a x^5\right )^{3/4}}{5 x^5}+\frac {3 a \arctan \left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{10 \sqrt [4]{b}}-\frac {3 a \text {arctanh}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{10 \sqrt [4]{b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {\left (b+a x^5\right )^{3/4}}{x^6} \, dx=-\frac {\left (b+a x^5\right )^{3/4}}{5 x^5}+\frac {3 a \arctan \left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{10 \sqrt [4]{b}}-\frac {3 a \text {arctanh}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{10 \sqrt [4]{b}} \]

[In]

Integrate[(b + a*x^5)^(3/4)/x^6,x]

[Out]

-1/5*(b + a*x^5)^(3/4)/x^5 + (3*a*ArcTan[(b + a*x^5)^(1/4)/b^(1/4)])/(10*b^(1/4)) - (3*a*ArcTanh[(b + a*x^5)^(
1/4)/b^(1/4)])/(10*b^(1/4))

Maple [A] (verified)

Time = 1.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.09

method	result	size
pseudoelliptic	\(\frac {6 \arctan \left (\frac {\left (a \,x^{5}+b \right )^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) a \,x^{5}-3 \ln \left (\frac {\left (a \,x^{5}+b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{\left (a \,x^{5}+b \right )^{\frac {1}{4}}-b^{\frac {1}{4}}}\right ) a \,x^{5}-4 \left (a \,x^{5}+b \right )^{\frac {3}{4}} b^{\frac {1}{4}}}{20 x^{5} b^{\frac {1}{4}}}\)	\(82\)

[In]

int((a*x^5+b)^(3/4)/x^6,x,method=_RETURNVERBOSE)

[Out]

1/20*(6*arctan((a*x^5+b)^(1/4)/b^(1/4))*a*x^5-3*ln(((a*x^5+b)^(1/4)+b^(1/4))/((a*x^5+b)^(1/4)-b^(1/4)))*a*x^5-
4*(a*x^5+b)^(3/4)*b^(1/4))/x^5/b^(1/4)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.47 \[ \int \frac {\left (b+a x^5\right )^{3/4}}{x^6} \, dx=-\frac {3 \, \left (\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{5} \log \left (27 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}} a^{3} + 27 \, \left (\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) - 3 i \, \left (\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{5} \log \left (27 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}} a^{3} + 27 i \, \left (\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) + 3 i \, \left (\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{5} \log \left (27 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}} a^{3} - 27 i \, \left (\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) - 3 \, \left (\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{5} \log \left (27 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}} a^{3} - 27 \, \left (\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) + 4 \, {\left (a x^{5} + b\right )}^{\frac {3}{4}}}{20 \, x^{5}} \]

[In]

integrate((a*x^5+b)^(3/4)/x^6,x, algorithm="fricas")

[Out]

-1/20*(3*(a^4/b)^(1/4)*x^5*log(27*(a*x^5 + b)^(1/4)*a^3 + 27*(a^4/b)^(3/4)*b) - 3*I*(a^4/b)^(1/4)*x^5*log(27*(
a*x^5 + b)^(1/4)*a^3 + 27*I*(a^4/b)^(3/4)*b) + 3*I*(a^4/b)^(1/4)*x^5*log(27*(a*x^5 + b)^(1/4)*a^3 - 27*I*(a^4/
b)^(3/4)*b) - 3*(a^4/b)^(1/4)*x^5*log(27*(a*x^5 + b)^(1/4)*a^3 - 27*(a^4/b)^(3/4)*b) + 4*(a*x^5 + b)^(3/4))/x^
5

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.91 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.56 \[ \int \frac {\left (b+a x^5\right )^{3/4}}{x^6} \, dx=- \frac {a^{\frac {3}{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{5}}} \right )}}{5 x^{\frac {5}{4}} \Gamma \left (\frac {5}{4}\right )} \]

[In]

integrate((a*x**5+b)**(3/4)/x**6,x)

[Out]

-a**(3/4)*gamma(1/4)*hyper((-3/4, 1/4), (5/4,), b*exp_polar(I*pi)/(a*x**5))/(5*x**(5/4)*gamma(5/4))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \frac {\left (b+a x^5\right )^{3/4}}{x^6} \, dx=\frac {3}{20} \, a {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{5} + b\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (\frac {{\left (a x^{5} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}}{{\left (a x^{5} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}}\right )} - \frac {{\left (a x^{5} + b\right )}^{\frac {3}{4}}}{5 \, x^{5}} \]

[In]

integrate((a*x^5+b)^(3/4)/x^6,x, algorithm="maxima")

[Out]

3/20*a*(2*arctan((a*x^5 + b)^(1/4)/b^(1/4))/b^(1/4) + log(((a*x^5 + b)^(1/4) - b^(1/4))/((a*x^5 + b)^(1/4) + b
^(1/4)))/b^(1/4)) - 1/5*(a*x^5 + b)^(3/4)/x^5

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (55) = 110\).

Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.79 \[ \int \frac {\left (b+a x^5\right )^{3/4}}{x^6} \, dx=\frac {\frac {6 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} + 2 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{\left (-b\right )^{\frac {1}{4}}} + \frac {6 \, \sqrt {2} a^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} - 2 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{\left (-b\right )^{\frac {1}{4}}} + \frac {3 \, \sqrt {2} a^{2} \left (-b\right )^{\frac {3}{4}} \log \left (\sqrt {2} {\left (a x^{5} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{5} + b} + \sqrt {-b}\right )}{b} + \frac {3 \, \sqrt {2} a^{2} \log \left (-\sqrt {2} {\left (a x^{5} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{5} + b} + \sqrt {-b}\right )}{\left (-b\right )^{\frac {1}{4}}} - \frac {8 \, {\left (a x^{5} + b\right )}^{\frac {3}{4}} a}{x^{5}}}{40 \, a} \]

[In]

integrate((a*x^5+b)^(3/4)/x^6,x, algorithm="giac")

[Out]

1/40*(6*sqrt(2)*a^2*arctan(1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4) + 2*(a*x^5 + b)^(1/4))/(-b)^(1/4))/(-b)^(1/4) + 6*s
qrt(2)*a^2*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4) - 2*(a*x^5 + b)^(1/4))/(-b)^(1/4))/(-b)^(1/4) + 3*sqrt(2)*a
^2*(-b)^(3/4)*log(sqrt(2)*(a*x^5 + b)^(1/4)*(-b)^(1/4) + sqrt(a*x^5 + b) + sqrt(-b))/b + 3*sqrt(2)*a^2*log(-sq
rt(2)*(a*x^5 + b)^(1/4)*(-b)^(1/4) + sqrt(a*x^5 + b) + sqrt(-b))/(-b)^(1/4) - 8*(a*x^5 + b)^(3/4)*a/x^5)/a

Mupad [B] (verification not implemented)

Time = 6.13 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.73 \[ \int \frac {\left (b+a x^5\right )^{3/4}}{x^6} \, dx=\frac {3\,a\,\mathrm {atan}\left (\frac {{\left (a\,x^5+b\right )}^{1/4}}{b^{1/4}}\right )}{10\,b^{1/4}}-\frac {{\left (a\,x^5+b\right )}^{3/4}}{5\,x^5}-\frac {3\,a\,\mathrm {atanh}\left (\frac {{\left (a\,x^5+b\right )}^{1/4}}{b^{1/4}}\right )}{10\,b^{1/4}} \]

[In]

int((b + a*x^5)^(3/4)/x^6,x)

[Out]

(3*a*atan((b + a*x^5)^(1/4)/b^(1/4)))/(10*b^(1/4)) - (b + a*x^5)^(3/4)/(5*x^5) - (3*a*atanh((b + a*x^5)^(1/4)/
b^(1/4)))/(10*b^(1/4))