Integrand size = 15, antiderivative size = 91 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^5} \, dx=\frac {5}{4} b \sqrt [4]{a+b x^4}-\frac {\left (a+b x^4\right )^{5/4}}{4 x^4}-\frac {5}{8} \sqrt [4]{a} b \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac {5}{8} \sqrt [4]{a} b \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right ) \]
5/4*b*(b*x^4+a)^(1/4)-1/4*(b*x^4+a)^(5/4)/x^4-5/8*a^(1/4)*b*arctan((b*x^4+ a)^(1/4)/a^(1/4))-5/8*a^(1/4)*b*arctanh((b*x^4+a)^(1/4)/a^(1/4))
Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^5} \, dx=\frac {1}{8} \left (-\frac {2 \left (a-4 b x^4\right ) \sqrt [4]{a+b x^4}}{x^4}-5 \sqrt [4]{a} b \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-5 \sqrt [4]{a} b \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )\right ) \]
((-2*(a - 4*b*x^4)*(a + b*x^4)^(1/4))/x^4 - 5*a^(1/4)*b*ArcTan[(a + b*x^4) ^(1/4)/a^(1/4)] - 5*a^(1/4)*b*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4)])/8
Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {798, 51, 60, 73, 756, 216, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+b x^4\right )^{5/4}}{x^5} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int \frac {\left (b x^4+a\right )^{5/4}}{x^8}dx^4\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{4} \left (\frac {5}{4} b \int \frac {\sqrt [4]{b x^4+a}}{x^4}dx^4-\frac {\left (a+b x^4\right )^{5/4}}{x^4}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (\frac {5}{4} b \left (a \int \frac {1}{x^4 \left (b x^4+a\right )^{3/4}}dx^4+4 \sqrt [4]{a+b x^4}\right )-\frac {\left (a+b x^4\right )^{5/4}}{x^4}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (\frac {5}{4} b \left (\frac {4 a \int \frac {1}{\frac {x^{16}}{b}-\frac {a}{b}}d\sqrt [4]{b x^4+a}}{b}+4 \sqrt [4]{a+b x^4}\right )-\frac {\left (a+b x^4\right )^{5/4}}{x^4}\right )\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {1}{4} \left (\frac {5}{4} b \left (\frac {4 a \left (-\frac {b \int \frac {1}{\sqrt {a}-x^8}d\sqrt [4]{b x^4+a}}{2 \sqrt {a}}-\frac {b \int \frac {1}{x^8+\sqrt {a}}d\sqrt [4]{b x^4+a}}{2 \sqrt {a}}\right )}{b}+4 \sqrt [4]{a+b x^4}\right )-\frac {\left (a+b x^4\right )^{5/4}}{x^4}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{4} \left (\frac {5}{4} b \left (\frac {4 a \left (-\frac {b \int \frac {1}{\sqrt {a}-x^8}d\sqrt [4]{b x^4+a}}{2 \sqrt {a}}-\frac {b \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}\right )}{b}+4 \sqrt [4]{a+b x^4}\right )-\frac {\left (a+b x^4\right )^{5/4}}{x^4}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} \left (\frac {5}{4} b \left (\frac {4 a \left (-\frac {b \arctan \left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}-\frac {b \text {arctanh}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{3/4}}\right )}{b}+4 \sqrt [4]{a+b x^4}\right )-\frac {\left (a+b x^4\right )^{5/4}}{x^4}\right )\) |
(-((a + b*x^4)^(5/4)/x^4) + (5*b*(4*(a + b*x^4)^(1/4) + (4*a*(-1/2*(b*ArcT an[(a + b*x^4)^(1/4)/a^(1/4)])/a^(3/4) - (b*ArcTanh[(a + b*x^4)^(1/4)/a^(1 /4)])/(2*a^(3/4))))/b))/4)/4
3.11.53.3.1 Defintions of rubi rules used
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] - Simp[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 ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Time = 4.41 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(\frac {-5 b \,x^{4} \left (\ln \left (\frac {-\left (b \,x^{4}+a \right )^{\frac {1}{4}}-a^{\frac {1}{4}}}{-\left (b \,x^{4}+a \right )^{\frac {1}{4}}+a^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {1}{4}}-4 \left (b \,x^{4}+a \right )^{\frac {1}{4}} \left (-4 b \,x^{4}+a \right )}{16 x^{4}}\) | \(88\) |
1/16*(-5*b*x^4*(ln((-(b*x^4+a)^(1/4)-a^(1/4))/(-(b*x^4+a)^(1/4)+a^(1/4)))+ 2*arctan((b*x^4+a)^(1/4)/a^(1/4)))*a^(1/4)-4*(b*x^4+a)^(1/4)*(-4*b*x^4+a)) /x^4
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.84 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^5} \, dx=-\frac {5 \, \left (a b^{4}\right )^{\frac {1}{4}} x^{4} \log \left (5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b + 5 \, \left (a b^{4}\right )^{\frac {1}{4}}\right ) + 5 i \, \left (a b^{4}\right )^{\frac {1}{4}} x^{4} \log \left (5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b + 5 i \, \left (a b^{4}\right )^{\frac {1}{4}}\right ) - 5 i \, \left (a b^{4}\right )^{\frac {1}{4}} x^{4} \log \left (5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b - 5 i \, \left (a b^{4}\right )^{\frac {1}{4}}\right ) - 5 \, \left (a b^{4}\right )^{\frac {1}{4}} x^{4} \log \left (5 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b - 5 \, \left (a b^{4}\right )^{\frac {1}{4}}\right ) - 4 \, {\left (4 \, b x^{4} - a\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{16 \, x^{4}} \]
-1/16*(5*(a*b^4)^(1/4)*x^4*log(5*(b*x^4 + a)^(1/4)*b + 5*(a*b^4)^(1/4)) + 5*I*(a*b^4)^(1/4)*x^4*log(5*(b*x^4 + a)^(1/4)*b + 5*I*(a*b^4)^(1/4)) - 5*I *(a*b^4)^(1/4)*x^4*log(5*(b*x^4 + a)^(1/4)*b - 5*I*(a*b^4)^(1/4)) - 5*(a*b ^4)^(1/4)*x^4*log(5*(b*x^4 + a)^(1/4)*b - 5*(a*b^4)^(1/4)) - 4*(4*b*x^4 - a)*(b*x^4 + a)^(1/4))/x^4
Result contains complex when optimal does not.
Time = 1.00 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.46 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^5} \, dx=- \frac {b^{\frac {5}{4}} x \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 \Gamma \left (\frac {3}{4}\right )} \]
-b**(5/4)*x*gamma(-1/4)*hyper((-5/4, -1/4), (3/4,), a*exp_polar(I*pi)/(b*x **4))/(4*gamma(3/4))
Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^5} \, dx=-\frac {5}{16} \, {\left (\frac {2 \, b \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}} - \frac {b \log \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}}\right )} a + {\left (b x^{4} + a\right )}^{\frac {1}{4}} b - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a}{4 \, x^{4}} \]
-5/16*(2*b*arctan((b*x^4 + a)^(1/4)/a^(1/4))/a^(3/4) - b*log(((b*x^4 + a)^ (1/4) - a^(1/4))/((b*x^4 + a)^(1/4) + a^(1/4)))/a^(3/4))*a + (b*x^4 + a)^( 1/4)*b - 1/4*(b*x^4 + a)^(1/4)*a/x^4
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (67) = 134\).
Time = 0.31 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.43 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^5} \, dx=-\frac {10 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + 10 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + 5 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right ) - 5 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (-\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right ) - 32 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2} + \frac {8 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} a b}{x^{4}}}{32 \, b} \]
-1/32*(10*sqrt(2)*(-a)^(1/4)*b^2*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(b*x^4 + a)^(1/4))/(-a)^(1/4)) + 10*sqrt(2)*(-a)^(1/4)*b^2*arctan(-1/2*s qrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(b*x^4 + a)^(1/4))/(-a)^(1/4)) + 5*sqrt(2)* (-a)^(1/4)*b^2*log(sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a)) - 5*sqrt(2)*(-a)^(1/4)*b^2*log(-sqrt(2)*(b*x^4 + a)^(1/4)*(-a) ^(1/4) + sqrt(b*x^4 + a) + sqrt(-a)) - 32*(b*x^4 + a)^(1/4)*b^2 + 8*(b*x^4 + a)^(1/4)*a*b/x^4)/b
Time = 6.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{x^5} \, dx=b\,{\left (b\,x^4+a\right )}^{1/4}-\frac {a\,{\left (b\,x^4+a\right )}^{1/4}}{4\,x^4}-\frac {5\,a^{1/4}\,b\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{8}+\frac {a^{1/4}\,b\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}\,1{}\mathrm {i}}{a^{1/4}}\right )\,5{}\mathrm {i}}{8} \]