Integrand size = 15, antiderivative size = 53 \[ \int \frac {1}{x^8 \sqrt [4]{x^3+x^4}} \, dx=\frac {4 \left (x^3+x^4\right )^{3/4} \left (-129789+134596 x-140448 x^2+147840 x^3-157696 x^4+172032 x^5-196608 x^6+262144 x^7\right )}{4023459 x^{10}} \] Output:
4/4023459*(x^4+x^3)^(3/4)*(262144*x^7-196608*x^6+172032*x^5-157696*x^4+147 840*x^3-140448*x^2+134596*x-129789)/x^10
Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^8 \sqrt [4]{x^3+x^4}} \, dx=\frac {4 \left (x^3 (1+x)\right )^{3/4} \left (-129789+134596 x-140448 x^2+147840 x^3-157696 x^4+172032 x^5-196608 x^6+262144 x^7\right )}{4023459 x^{10}} \] Input:
Integrate[1/(x^8*(x^3 + x^4)^(1/4)),x]
Output:
(4*(x^3*(1 + x))^(3/4)*(-129789 + 134596*x - 140448*x^2 + 147840*x^3 - 157 696*x^4 + 172032*x^5 - 196608*x^6 + 262144*x^7))/(4023459*x^10)
Leaf count is larger than twice the leaf count of optimal. \(175\) vs. \(2(53)=106\).
Time = 0.40 (sec) , antiderivative size = 175, normalized size of antiderivative = 3.30, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {1922, 1922, 1922, 1922, 1922, 1922, 1922, 1920}
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.
| \(\displaystyle \int \frac {1}{x^8 \sqrt [4]{x^4+x^3}} \, dx\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {28}{31} \int \frac {1}{x^7 \sqrt [4]{x^4+x^3}}dx-\frac {4 \left (x^4+x^3\right )^{3/4}}{31 x^{10}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {28}{31} \left (-\frac {8}{9} \int \frac {1}{x^6 \sqrt [4]{x^4+x^3}}dx-\frac {4 \left (x^4+x^3\right )^{3/4}}{27 x^9}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{31 x^{10}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {28}{31} \left (-\frac {8}{9} \left (-\frac {20}{23} \int \frac {1}{x^5 \sqrt [4]{x^4+x^3}}dx-\frac {4 \left (x^4+x^3\right )^{3/4}}{23 x^8}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{27 x^9}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{31 x^{10}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {28}{31} \left (-\frac {8}{9} \left (-\frac {20}{23} \left (-\frac {16}{19} \int \frac {1}{x^4 \sqrt [4]{x^4+x^3}}dx-\frac {4 \left (x^4+x^3\right )^{3/4}}{19 x^7}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{23 x^8}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{27 x^9}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{31 x^{10}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {28}{31} \left (-\frac {8}{9} \left (-\frac {20}{23} \left (-\frac {16}{19} \left (-\frac {4}{5} \int \frac {1}{x^3 \sqrt [4]{x^4+x^3}}dx-\frac {4 \left (x^4+x^3\right )^{3/4}}{15 x^6}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{19 x^7}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{23 x^8}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{27 x^9}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{31 x^{10}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {28}{31} \left (-\frac {8}{9} \left (-\frac {20}{23} \left (-\frac {16}{19} \left (-\frac {4}{5} \left (-\frac {8}{11} \int \frac {1}{x^2 \sqrt [4]{x^4+x^3}}dx-\frac {4 \left (x^4+x^3\right )^{3/4}}{11 x^5}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{15 x^6}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{19 x^7}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{23 x^8}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{27 x^9}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{31 x^{10}}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle -\frac {28}{31} \left (-\frac {8}{9} \left (-\frac {20}{23} \left (-\frac {16}{19} \left (-\frac {4}{5} \left (-\frac {8}{11} \left (-\frac {4}{7} \int \frac {1}{x \sqrt [4]{x^4+x^3}}dx-\frac {4 \left (x^4+x^3\right )^{3/4}}{7 x^4}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{11 x^5}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{15 x^6}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{19 x^7}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{23 x^8}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{27 x^9}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{31 x^{10}}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle -\frac {4 \left (x^4+x^3\right )^{3/4}}{31 x^{10}}-\frac {28}{31} \left (-\frac {4 \left (x^4+x^3\right )^{3/4}}{27 x^9}-\frac {8}{9} \left (-\frac {4 \left (x^4+x^3\right )^{3/4}}{23 x^8}-\frac {20}{23} \left (-\frac {4 \left (x^4+x^3\right )^{3/4}}{19 x^7}-\frac {16}{19} \left (-\frac {4 \left (x^4+x^3\right )^{3/4}}{15 x^6}-\frac {4}{5} \left (-\frac {8}{11} \left (\frac {16 \left (x^4+x^3\right )^{3/4}}{21 x^3}-\frac {4 \left (x^4+x^3\right )^{3/4}}{7 x^4}\right )-\frac {4 \left (x^4+x^3\right )^{3/4}}{11 x^5}\right )\right )\right )\right )\right )\) |
Input:
Int[1/(x^8*(x^3 + x^4)^(1/4)),x]
Output:
(-4*(x^3 + x^4)^(3/4))/(31*x^10) - (28*((-4*(x^3 + x^4)^(3/4))/(27*x^9) - (8*((-4*(x^3 + x^4)^(3/4))/(23*x^8) - (20*((-4*(x^3 + x^4)^(3/4))/(19*x^7) - (16*((-4*(x^3 + x^4)^(3/4))/(15*x^6) - (4*((-4*(x^3 + x^4)^(3/4))/(11*x ^5) - (8*((-4*(x^3 + x^4)^(3/4))/(7*x^4) + (16*(x^3 + x^4)^(3/4))/(21*x^3) ))/11))/5))/19))/23))/9))/31
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])
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] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))) I nt[(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])
Result contains higher order function than in optimal. Order 5 vs. order 2.
Time = 0.16 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.28
| method | result | size |
| meijerg | \(-\frac {4 \operatorname {hypergeom}\left (\left [-\frac {31}{4}, \frac {1}{4}\right ], \left [-\frac {27}{4}\right ], -x \right )}{31 x^{\frac {31}{4}}}\) | \(15\) |
| trager | \(\frac {4 \left (x^{4}+x^{3}\right )^{\frac {3}{4}} \left (262144 x^{7}-196608 x^{6}+172032 x^{5}-157696 x^{4}+147840 x^{3}-140448 x^{2}+134596 x -129789\right )}{4023459 x^{10}}\) | \(50\) |
| pseudoelliptic | \(\frac {4 \left (x^{3} \left (1+x \right )\right )^{\frac {3}{4}} \left (262144 x^{7}-196608 x^{6}+172032 x^{5}-157696 x^{4}+147840 x^{3}-140448 x^{2}+134596 x -129789\right )}{4023459 x^{10}}\) | \(50\) |
| gosper | \(\frac {4 \left (1+x \right ) \left (262144 x^{7}-196608 x^{6}+172032 x^{5}-157696 x^{4}+147840 x^{3}-140448 x^{2}+134596 x -129789\right )}{4023459 x^{7} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}\) | \(53\) |
| orering | \(\frac {4 \left (1+x \right ) \left (262144 x^{7}-196608 x^{6}+172032 x^{5}-157696 x^{4}+147840 x^{3}-140448 x^{2}+134596 x -129789\right )}{4023459 x^{7} \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}\) | \(53\) |
| risch | \(\frac {-\frac {4}{31}+\frac {4}{837} x -\frac {112}{19251} x^{2}+\frac {896}{121923} x^{3}-\frac {3584}{365769} x^{4}+\frac {57344}{4023459} x^{5}-\frac {32768}{1341153} x^{6}+\frac {262144}{4023459} x^{7}+\frac {1048576}{4023459} x^{8}}{x^{7} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}\) | \(55\) |
Input:
int(1/x^8/(x^4+x^3)^(1/4),x,method=_RETURNVERBOSE)
Output:
-4/31/x^(31/4)*hypergeom([-31/4,1/4],[-27/4],-x)
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^8 \sqrt [4]{x^3+x^4}} \, dx=\frac {4 \, {\left (262144 \, x^{7} - 196608 \, x^{6} + 172032 \, x^{5} - 157696 \, x^{4} + 147840 \, x^{3} - 140448 \, x^{2} + 134596 \, x - 129789\right )} {\left (x^{4} + x^{3}\right )}^{\frac {3}{4}}}{4023459 \, x^{10}} \] Input:
integrate(1/x^8/(x^4+x^3)^(1/4),x, algorithm="fricas")
Output:
4/4023459*(262144*x^7 - 196608*x^6 + 172032*x^5 - 157696*x^4 + 147840*x^3 - 140448*x^2 + 134596*x - 129789)*(x^4 + x^3)^(3/4)/x^10
\[ \int \frac {1}{x^8 \sqrt [4]{x^3+x^4}} \, dx=\int \frac {1}{x^{8} \sqrt [4]{x^{3} \left (x + 1\right )}}\, dx \] Input:
integrate(1/x**8/(x**4+x**3)**(1/4),x)
Output:
Integral(1/(x**8*(x**3*(x + 1))**(1/4)), x)
\[ \int \frac {1}{x^8 \sqrt [4]{x^3+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} x^{8}} \,d x } \] Input:
integrate(1/x^8/(x^4+x^3)^(1/4),x, algorithm="maxima")
Output:
integrate(1/((x^4 + x^3)^(1/4)*x^8), x)
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x^8 \sqrt [4]{x^3+x^4}} \, dx=-\frac {4}{31} \, {\left (\frac {1}{x} + 1\right )}^{\frac {31}{4}} + \frac {28}{27} \, {\left (\frac {1}{x} + 1\right )}^{\frac {27}{4}} - \frac {84}{23} \, {\left (\frac {1}{x} + 1\right )}^{\frac {23}{4}} + \frac {140}{19} \, {\left (\frac {1}{x} + 1\right )}^{\frac {19}{4}} - \frac {28}{3} \, {\left (\frac {1}{x} + 1\right )}^{\frac {15}{4}} + \frac {84}{11} \, {\left (\frac {1}{x} + 1\right )}^{\frac {11}{4}} - 4 \, {\left (\frac {1}{x} + 1\right )}^{\frac {7}{4}} + \frac {4}{3} \, {\left (\frac {1}{x} + 1\right )}^{\frac {3}{4}} \] Input:
integrate(1/x^8/(x^4+x^3)^(1/4),x, algorithm="giac")
Output:
-4/31*(1/x + 1)^(31/4) + 28/27*(1/x + 1)^(27/4) - 84/23*(1/x + 1)^(23/4) + 140/19*(1/x + 1)^(19/4) - 28/3*(1/x + 1)^(15/4) + 84/11*(1/x + 1)^(11/4) - 4*(1/x + 1)^(7/4) + 4/3*(1/x + 1)^(3/4)
Time = 7.75 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.13 \[ \int \frac {1}{x^8 \sqrt [4]{x^3+x^4}} \, dx=\frac {1048576\,{\left (x^4+x^3\right )}^{3/4}}{4023459\,x^3}-\frac {262144\,{\left (x^4+x^3\right )}^{3/4}}{1341153\,x^4}+\frac {229376\,{\left (x^4+x^3\right )}^{3/4}}{1341153\,x^5}-\frac {57344\,{\left (x^4+x^3\right )}^{3/4}}{365769\,x^6}+\frac {17920\,{\left (x^4+x^3\right )}^{3/4}}{121923\,x^7}-\frac {896\,{\left (x^4+x^3\right )}^{3/4}}{6417\,x^8}+\frac {112\,{\left (x^4+x^3\right )}^{3/4}}{837\,x^9}-\frac {4\,{\left (x^4+x^3\right )}^{3/4}}{31\,x^{10}} \] Input:
int(1/(x^8*(x^3 + x^4)^(1/4)),x)
Output:
(1048576*(x^3 + x^4)^(3/4))/(4023459*x^3) - (262144*(x^3 + x^4)^(3/4))/(13 41153*x^4) + (229376*(x^3 + x^4)^(3/4))/(1341153*x^5) - (57344*(x^3 + x^4) ^(3/4))/(365769*x^6) + (17920*(x^3 + x^4)^(3/4))/(121923*x^7) - (896*(x^3 + x^4)^(3/4))/(6417*x^8) + (112*(x^3 + x^4)^(3/4))/(837*x^9) - (4*(x^3 + x ^4)^(3/4))/(31*x^10)
\[ \int \frac {1}{x^8 \sqrt [4]{x^3+x^4}} \, dx=\int \frac {1}{x^{\frac {35}{4}} \left (x +1\right )^{\frac {1}{4}}}d x \] Input:
int(1/x^8/(x^4+x^3)^(1/4),x)
Output:
int(1/(x**(3/4)*(x + 1)**(1/4)*x**8),x)