Integrand size = 15, antiderivative size = 92 \[ \int \frac {1}{x^{11} \left (b+a x^5\right )^{3/4}} \, dx=\frac {\sqrt [4]{b+a x^5} \left (-4 b+7 a x^5\right )}{40 b^2 x^{10}}-\frac {21 a^2 \arctan \left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{80 b^{11/4}}-\frac {21 a^2 \text {arctanh}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{80 b^{11/4}} \] Output:
1/40*(a*x^5+b)^(1/4)*(7*a*x^5-4*b)/b^2/x^10-21/80*a^2*arctan((a*x^5+b)^(1/ 4)/b^(1/4))/b^(11/4)-21/80*a^2*arctanh((a*x^5+b)^(1/4)/b^(1/4))/b^(11/4)
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{11} \left (b+a x^5\right )^{3/4}} \, dx=\frac {\sqrt [4]{b+a x^5} \left (-4 b+7 a x^5\right )}{40 b^2 x^{10}}-\frac {21 a^2 \arctan \left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{80 b^{11/4}}-\frac {21 a^2 \text {arctanh}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{80 b^{11/4}} \] Input:
Integrate[1/(x^11*(b + a*x^5)^(3/4)),x]
Output:
((b + a*x^5)^(1/4)*(-4*b + 7*a*x^5))/(40*b^2*x^10) - (21*a^2*ArcTan[(b + a *x^5)^(1/4)/b^(1/4)])/(80*b^(11/4)) - (21*a^2*ArcTanh[(b + a*x^5)^(1/4)/b^ (1/4)])/(80*b^(11/4))
Time = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.26, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {798, 52, 52, 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.
\(\displaystyle \int \frac {1}{x^{11} \left (a x^5+b\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{5} \int \frac {1}{x^{15} \left (a x^5+b\right )^{3/4}}dx^5\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{5} \left (-\frac {7 a \int \frac {1}{x^{10} \left (a x^5+b\right )^{3/4}}dx^5}{8 b}-\frac {\sqrt [4]{a x^5+b}}{2 b x^{10}}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{5} \left (-\frac {7 a \left (-\frac {3 a \int \frac {1}{x^5 \left (a x^5+b\right )^{3/4}}dx^5}{4 b}-\frac {\sqrt [4]{a x^5+b}}{b x^5}\right )}{8 b}-\frac {\sqrt [4]{a x^5+b}}{2 b x^{10}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{5} \left (-\frac {7 a \left (-\frac {3 \int \frac {1}{\frac {x^{20}}{a}-\frac {b}{a}}d\sqrt [4]{a x^5+b}}{b}-\frac {\sqrt [4]{a x^5+b}}{b x^5}\right )}{8 b}-\frac {\sqrt [4]{a x^5+b}}{2 b x^{10}}\right )\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {1}{5} \left (-\frac {7 a \left (-\frac {3 \left (-\frac {a \int \frac {1}{\sqrt {b}-x^{10}}d\sqrt [4]{a x^5+b}}{2 \sqrt {b}}-\frac {a \int \frac {1}{x^{10}+\sqrt {b}}d\sqrt [4]{a x^5+b}}{2 \sqrt {b}}\right )}{b}-\frac {\sqrt [4]{a x^5+b}}{b x^5}\right )}{8 b}-\frac {\sqrt [4]{a x^5+b}}{2 b x^{10}}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{5} \left (-\frac {7 a \left (-\frac {3 \left (-\frac {a \int \frac {1}{\sqrt {b}-x^{10}}d\sqrt [4]{a x^5+b}}{2 \sqrt {b}}-\frac {a \arctan \left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )}{2 b^{3/4}}\right )}{b}-\frac {\sqrt [4]{a x^5+b}}{b x^5}\right )}{8 b}-\frac {\sqrt [4]{a x^5+b}}{2 b x^{10}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{5} \left (-\frac {7 a \left (-\frac {3 \left (-\frac {a \arctan \left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )}{2 b^{3/4}}-\frac {a \text {arctanh}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )}{2 b^{3/4}}\right )}{b}-\frac {\sqrt [4]{a x^5+b}}{b x^5}\right )}{8 b}-\frac {\sqrt [4]{a x^5+b}}{2 b x^{10}}\right )\) |
Input:
Int[1/(x^11*(b + a*x^5)^(3/4)),x]
Output:
(-1/2*(b + a*x^5)^(1/4)/(b*x^10) - (7*a*(-((b + a*x^5)^(1/4)/(b*x^5)) - (3 *(-1/2*(a*ArcTan[(b + a*x^5)^(1/4)/b^(1/4)])/b^(3/4) - (a*ArcTanh[(b + a*x ^5)^(1/4)/b^(1/4)])/(2*b^(3/4))))/b))/(8*b))/5
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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 = 0.39 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.17
method | result | size |
pseudoelliptic | \(\frac {-21 \ln \left (\frac {-\left (a \,x^{5}+b \right )^{\frac {1}{4}}-b^{\frac {1}{4}}}{-\left (a \,x^{5}+b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}\right ) a^{2} x^{10}-42 \arctan \left (\frac {\left (a \,x^{5}+b \right )^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) a^{2} x^{10}+28 \left (a \,x^{5}+b \right )^{\frac {1}{4}} a \,x^{5} b^{\frac {3}{4}}-16 \left (a \,x^{5}+b \right )^{\frac {1}{4}} b^{\frac {7}{4}}}{160 b^{\frac {11}{4}} x^{10}}\) | \(108\) |
Input:
int(1/x^11/(a*x^5+b)^(3/4),x,method=_RETURNVERBOSE)
Output:
1/160*(-21*ln((-(a*x^5+b)^(1/4)-b^(1/4))/(-(a*x^5+b)^(1/4)+b^(1/4)))*a^2*x ^10-42*arctan((a*x^5+b)^(1/4)/b^(1/4))*a^2*x^10+28*(a*x^5+b)^(1/4)*a*x^5*b ^(3/4)-16*(a*x^5+b)^(1/4)*b^(7/4))/b^(11/4)/x^10
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.37 \[ \int \frac {1}{x^{11} \left (b+a x^5\right )^{3/4}} \, dx=-\frac {21 \, b^{2} x^{10} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \log \left (21 \, b^{3} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} + 21 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}} a^{2}\right ) + 21 i \, b^{2} x^{10} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \log \left (21 i \, b^{3} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} + 21 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}} a^{2}\right ) - 21 i \, b^{2} x^{10} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-21 i \, b^{3} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} + 21 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}} a^{2}\right ) - 21 \, b^{2} x^{10} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} \log \left (-21 \, b^{3} \left (\frac {a^{8}}{b^{11}}\right )^{\frac {1}{4}} + 21 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}} a^{2}\right ) - 4 \, {\left (7 \, a x^{5} - 4 \, b\right )} {\left (a x^{5} + b\right )}^{\frac {1}{4}}}{160 \, b^{2} x^{10}} \] Input:
integrate(1/x^11/(a*x^5+b)^(3/4),x, algorithm="fricas")
Output:
-1/160*(21*b^2*x^10*(a^8/b^11)^(1/4)*log(21*b^3*(a^8/b^11)^(1/4) + 21*(a*x ^5 + b)^(1/4)*a^2) + 21*I*b^2*x^10*(a^8/b^11)^(1/4)*log(21*I*b^3*(a^8/b^11 )^(1/4) + 21*(a*x^5 + b)^(1/4)*a^2) - 21*I*b^2*x^10*(a^8/b^11)^(1/4)*log(- 21*I*b^3*(a^8/b^11)^(1/4) + 21*(a*x^5 + b)^(1/4)*a^2) - 21*b^2*x^10*(a^8/b ^11)^(1/4)*log(-21*b^3*(a^8/b^11)^(1/4) + 21*(a*x^5 + b)^(1/4)*a^2) - 4*(7 *a*x^5 - 4*b)*(a*x^5 + b)^(1/4))/(b^2*x^10)
Result contains complex when optimal does not.
Time = 1.88 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.45 \[ \int \frac {1}{x^{11} \left (b+a x^5\right )^{3/4}} \, dx=- \frac {\Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{5}}} \right )}}{5 a^{\frac {3}{4}} x^{\frac {55}{4}} \Gamma \left (\frac {15}{4}\right )} \] Input:
integrate(1/x**11/(a*x**5+b)**(3/4),x)
Output:
-gamma(11/4)*hyper((3/4, 11/4), (15/4,), b*exp_polar(I*pi)/(a*x**5))/(5*a* *(3/4)*x**(55/4)*gamma(15/4))
Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^{11} \left (b+a x^5\right )^{3/4}} \, dx=\frac {7 \, {\left (a x^{5} + b\right )}^{\frac {5}{4}} a^{2} - 11 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}} a^{2} b}{40 \, {\left ({\left (a x^{5} + b\right )}^{2} b^{2} - 2 \, {\left (a x^{5} + b\right )} b^{3} + b^{4}\right )}} - \frac {21 \, {\left (\frac {2 \, a^{2} \arctan \left (\frac {{\left (a x^{5} + b\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} - \frac {a^{2} \log \left (\frac {{\left (a x^{5} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}}{{\left (a x^{5} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}}\right )}}{160 \, b^{2}} \] Input:
integrate(1/x^11/(a*x^5+b)^(3/4),x, algorithm="maxima")
Output:
1/40*(7*(a*x^5 + b)^(5/4)*a^2 - 11*(a*x^5 + b)^(1/4)*a^2*b)/((a*x^5 + b)^2 *b^2 - 2*(a*x^5 + b)*b^3 + b^4) - 21/160*(2*a^2*arctan((a*x^5 + b)^(1/4)/b ^(1/4))/b^(3/4) - a^2*log(((a*x^5 + b)^(1/4) - b^(1/4))/((a*x^5 + b)^(1/4) + b^(1/4)))/b^(3/4))/b^2
Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (72) = 144\).
Time = 0.25 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.65 \[ \int \frac {1}{x^{11} \left (b+a x^5\right )^{3/4}} \, dx=\frac {\frac {42 \, \sqrt {2} a^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} + 2 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{\left (-b\right )^{\frac {3}{4}} b^{2}} + \frac {42 \, \sqrt {2} a^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} - 2 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right )}{\left (-b\right )^{\frac {3}{4}} b^{2}} + \frac {21 \, \sqrt {2} a^{3} \log \left (\sqrt {2} {\left (a x^{5} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{5} + b} + \sqrt {-b}\right )}{\left (-b\right )^{\frac {3}{4}} b^{2}} + \frac {21 \, \sqrt {2} a^{3} \left (-b\right )^{\frac {1}{4}} \log \left (-\sqrt {2} {\left (a x^{5} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{5} + b} + \sqrt {-b}\right )}{b^{3}} + \frac {8 \, {\left (7 \, {\left (a x^{5} + b\right )}^{\frac {5}{4}} a^{3} - 11 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}} a^{3} b\right )}}{a^{2} b^{2} x^{10}}}{320 \, a} \] Input:
integrate(1/x^11/(a*x^5+b)^(3/4),x, algorithm="giac")
Output:
1/320*(42*sqrt(2)*a^3*arctan(1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4) + 2*(a*x^5 + b)^(1/4))/(-b)^(1/4))/((-b)^(3/4)*b^2) + 42*sqrt(2)*a^3*arctan(-1/2*sqrt(2 )*(sqrt(2)*(-b)^(1/4) - 2*(a*x^5 + b)^(1/4))/(-b)^(1/4))/((-b)^(3/4)*b^2) + 21*sqrt(2)*a^3*log(sqrt(2)*(a*x^5 + b)^(1/4)*(-b)^(1/4) + sqrt(a*x^5 + b ) + sqrt(-b))/((-b)^(3/4)*b^2) + 21*sqrt(2)*a^3*(-b)^(1/4)*log(-sqrt(2)*(a *x^5 + b)^(1/4)*(-b)^(1/4) + sqrt(a*x^5 + b) + sqrt(-b))/b^3 + 8*(7*(a*x^5 + b)^(5/4)*a^3 - 11*(a*x^5 + b)^(1/4)*a^3*b)/(a^2*b^2*x^10))/a
Time = 8.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^{11} \left (b+a x^5\right )^{3/4}} \, dx=\frac {7\,{\left (a\,x^5+b\right )}^{5/4}}{40\,b^2\,x^{10}}-\frac {11\,{\left (a\,x^5+b\right )}^{1/4}}{40\,b\,x^{10}}-\frac {21\,a^2\,\mathrm {atan}\left (\frac {{\left (a\,x^5+b\right )}^{1/4}}{b^{1/4}}\right )}{80\,b^{11/4}}+\frac {a^2\,\mathrm {atan}\left (\frac {{\left (a\,x^5+b\right )}^{1/4}\,1{}\mathrm {i}}{b^{1/4}}\right )\,21{}\mathrm {i}}{80\,b^{11/4}} \] Input:
int(1/(x^11*(b + a*x^5)^(3/4)),x)
Output:
(a^2*atan(((b + a*x^5)^(1/4)*1i)/b^(1/4))*21i)/(80*b^(11/4)) - (21*a^2*ata n((b + a*x^5)^(1/4)/b^(1/4)))/(80*b^(11/4)) - (11*(b + a*x^5)^(1/4))/(40*b *x^10) + (7*(b + a*x^5)^(5/4))/(40*b^2*x^10)
\[ \int \frac {1}{x^{11} \left (b+a x^5\right )^{3/4}} \, dx=\int \frac {1}{\left (a \,x^{5}+b \right )^{\frac {3}{4}} x^{11}}d x \] Input:
int(1/x^11/(a*x^5+b)^(3/4),x)
Output:
int(1/((a*x**5 + b)**(3/4)*x**11),x)