Integrand size = 19, antiderivative size = 75 \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=-\frac {3 \sqrt {a x^{23}} \sqrt {1+x^5}}{20 x^9}+\frac {\sqrt {a x^{23}} \sqrt {1+x^5}}{10 x^4}+\frac {3 \sqrt {a x^{23}} \text {arcsinh}\left (x^{5/2}\right )}{20 x^{23/2}} \] Output:
-3/20*(a*x^23)^(1/2)*(x^5+1)^(1/2)/x^9+1/10*(a*x^23)^(1/2)*(x^5+1)^(1/2)/x ^4+3/20*(a*x^23)^(1/2)*arcsinh(x^(5/2))/x^(23/2)
Time = 1.54 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\frac {\sqrt {a x^{23}} \left (x^{5/2} \sqrt {1+x^5} \left (-3+2 x^5\right )+3 \log \left (x^{5/2}+\sqrt {1+x^5}\right )\right )}{20 x^{23/2}} \] Input:
Integrate[Sqrt[a*x^23]/Sqrt[1 + x^5],x]
Output:
(Sqrt[a*x^23]*(x^(5/2)*Sqrt[1 + x^5]*(-3 + 2*x^5) + 3*Log[x^(5/2) + Sqrt[1 + x^5]]))/(20*x^(23/2))
Time = 0.33 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {34, 843, 843, 851, 807, 222}
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.
\(\displaystyle \int \frac {\sqrt {a x^{23}}}{\sqrt {x^5+1}} \, dx\) |
\(\Big \downarrow \) 34 |
\(\displaystyle \frac {\sqrt {a x^{23}} \int \frac {x^{23/2}}{\sqrt {x^5+1}}dx}{x^{23/2}}\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {\sqrt {a x^{23}} \left (\frac {1}{10} x^{15/2} \sqrt {x^5+1}-\frac {3}{4} \int \frac {x^{13/2}}{\sqrt {x^5+1}}dx\right )}{x^{23/2}}\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {\sqrt {a x^{23}} \left (\frac {1}{10} x^{15/2} \sqrt {x^5+1}-\frac {3}{4} \left (\frac {1}{5} x^{5/2} \sqrt {x^5+1}-\frac {1}{2} \int \frac {x^{3/2}}{\sqrt {x^5+1}}dx\right )\right )}{x^{23/2}}\) |
\(\Big \downarrow \) 851 |
\(\displaystyle \frac {\sqrt {a x^{23}} \left (\frac {1}{10} x^{15/2} \sqrt {x^5+1}-\frac {3}{4} \left (\frac {1}{5} x^{5/2} \sqrt {x^5+1}-\int \frac {x^2}{\sqrt {x^5+1}}d\sqrt {x}\right )\right )}{x^{23/2}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {\sqrt {a x^{23}} \left (\frac {1}{10} x^{15/2} \sqrt {x^5+1}-\frac {3}{4} \left (\frac {1}{5} x^{5/2} \sqrt {x^5+1}-\frac {1}{5} \int \frac {1}{\sqrt {x+1}}dx^{5/2}\right )\right )}{x^{23/2}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {\sqrt {a x^{23}} \left (\frac {1}{10} x^{15/2} \sqrt {x^5+1}-\frac {3}{4} \left (\frac {1}{5} x^{5/2} \sqrt {x^5+1}-\frac {1}{5} \text {arcsinh}\left (x^{5/2}\right )\right )\right )}{x^{23/2}}\) |
Input:
Int[Sqrt[a*x^23]/Sqrt[1 + x^5],x]
Output:
(Sqrt[a*x^23]*((x^(15/2)*Sqrt[1 + x^5])/10 - (3*((x^(5/2)*Sqrt[1 + x^5])/5 - ArcSinh[x^(5/2)]/5))/4))/x^(23/2)
Int[(u_.)*((a_.)*(x_)^(m_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*x^m)^F racPart[p]/x^(m*FracPart[p])) Int[u*x^(m*p), x], x] /; FreeQ[{a, m, p}, x ] && !IntegerQ[p]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.64
method | result | size |
meijerg | \(\frac {\sqrt {a \,x^{23}}\, \left (-\frac {\sqrt {\pi }\, x^{\frac {5}{2}} \left (-10 x^{5}+15\right ) \sqrt {x^{5}+1}}{20}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (x^{\frac {5}{2}}\right )}{4}\right )}{5 x^{\frac {23}{2}} \sqrt {\pi }}\) | \(48\) |
risch | \(\frac {\left (2 x^{5}-3\right ) \sqrt {x^{5}+1}\, \sqrt {a \,x^{23}}}{20 x^{9}}+\frac {3 \,\operatorname {arcsinh}\left (x^{\frac {5}{2}}\right ) \sqrt {a \,x^{23}}\, \sqrt {x a \left (x^{5}+1\right )}}{20 \sqrt {a}\, x^{12} \sqrt {x^{5}+1}}\) | \(64\) |
Input:
int((a*x^23)^(1/2)/(x^5+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/5*(a*x^23)^(1/2)/x^(23/2)/Pi^(1/2)*(-1/20*Pi^(1/2)*x^(5/2)*(-10*x^5+15)* (x^5+1)^(1/2)+3/4*Pi^(1/2)*arcsinh(x^(5/2)))
Time = 0.26 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.25 \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\left [\frac {3 \, \sqrt {a} x^{9} \log \left (-\frac {8 \, a x^{19} + 8 \, a x^{14} + a x^{9} + 4 \, \sqrt {a x^{23}} {\left (2 \, x^{5} + 1\right )} \sqrt {x^{5} + 1} \sqrt {a}}{x^{9}}\right ) + 4 \, \sqrt {a x^{23}} {\left (2 \, x^{5} - 3\right )} \sqrt {x^{5} + 1}}{80 \, x^{9}}, -\frac {3 \, \sqrt {-a} x^{9} \arctan \left (\frac {\sqrt {a x^{23}} {\left (2 \, x^{5} + 1\right )} \sqrt {x^{5} + 1} \sqrt {-a}}{2 \, {\left (a x^{19} + a x^{14}\right )}}\right ) - 2 \, \sqrt {a x^{23}} {\left (2 \, x^{5} - 3\right )} \sqrt {x^{5} + 1}}{40 \, x^{9}}\right ] \] Input:
integrate((a*x^23)^(1/2)/(x^5+1)^(1/2),x, algorithm="fricas")
Output:
[1/80*(3*sqrt(a)*x^9*log(-(8*a*x^19 + 8*a*x^14 + a*x^9 + 4*sqrt(a*x^23)*(2 *x^5 + 1)*sqrt(x^5 + 1)*sqrt(a))/x^9) + 4*sqrt(a*x^23)*(2*x^5 - 3)*sqrt(x^ 5 + 1))/x^9, -1/40*(3*sqrt(-a)*x^9*arctan(1/2*sqrt(a*x^23)*(2*x^5 + 1)*sqr t(x^5 + 1)*sqrt(-a)/(a*x^19 + a*x^14)) - 2*sqrt(a*x^23)*(2*x^5 - 3)*sqrt(x ^5 + 1))/x^9]
\[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\int \frac {\sqrt {a x^{23}}}{\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \] Input:
integrate((a*x**23)**(1/2)/(x**5+1)**(1/2),x)
Output:
Integral(sqrt(a*x**23)/sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1)), x)
\[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\int { \frac {\sqrt {a x^{23}}}{\sqrt {x^{5} + 1}} \,d x } \] Input:
integrate((a*x^23)^(1/2)/(x^5+1)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(a*x^23)/sqrt(x^5 + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (55) = 110\).
Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\frac {3 \, {\left (\frac {a^{\frac {5}{2}} \log \left (a^{2} {\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{{\left | a \right |}} - \frac {a^{\frac {5}{2}} \log \left ({\left | -\sqrt {a x} a^{\frac {5}{2}} x^{2} + \sqrt {a^{6} x^{5} + a^{6}} \right |}\right ) \mathrm {sgn}\left (x\right )}{{\left | a \right |}}\right )} a^{3}}{20 \, {\left | a \right |}^{4}} + \frac {\sqrt {a^{6} x^{5} + a^{6}} {\left (2 \, a^{4} x^{5} - 3 \, a^{4}\right )} \sqrt {a x} x^{2} \mathrm {sgn}\left (x\right )}{20 \, a^{6} {\left | a \right |}} \] Input:
integrate((a*x^23)^(1/2)/(x^5+1)^(1/2),x, algorithm="giac")
Output:
3/20*(a^(5/2)*log(a^2*abs(a))*sgn(x)/abs(a) - a^(5/2)*log(abs(-sqrt(a*x)*a ^(5/2)*x^2 + sqrt(a^6*x^5 + a^6)))*sgn(x)/abs(a))*a^3/abs(a)^4 + 1/20*sqrt (a^6*x^5 + a^6)*(2*a^4*x^5 - 3*a^4)*sqrt(a*x)*x^2*sgn(x)/(a^6*abs(a))
Timed out. \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\int \frac {\sqrt {a\,x^{23}}}{\sqrt {x^5+1}} \,d x \] Input:
int((a*x^23)^(1/2)/(x^5 + 1)^(1/2),x)
Output:
int((a*x^23)^(1/2)/(x^5 + 1)^(1/2), x)
Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {a x^{23}}}{\sqrt {1+x^5}} \, dx=\frac {\sqrt {a}\, \left (4 \sqrt {x}\, \sqrt {x^{5}+1}\, x^{7}-6 \sqrt {x}\, \sqrt {x^{5}+1}\, x^{2}-3 \,\mathrm {log}\left (\sqrt {x^{5}+1}-\sqrt {x}\, x^{2}\right )+3 \,\mathrm {log}\left (\sqrt {x^{5}+1}+\sqrt {x}\, x^{2}\right )\right )}{40} \] Input:
int((a*x^23)^(1/2)/(x^5+1)^(1/2),x)
Output:
(sqrt(a)*(4*sqrt(x)*sqrt(x**5 + 1)*x**7 - 6*sqrt(x)*sqrt(x**5 + 1)*x**2 - 3*log(sqrt(x**5 + 1) - sqrt(x)*x**2) + 3*log(sqrt(x**5 + 1) + sqrt(x)*x**2 )))/40