Integrand size = 19, antiderivative size = 412 \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {6}{b x^{11/3} \sqrt {b x^{2/3}+a x}}-\frac {25 \sqrt {b x^{2/3}+a x}}{4 b^2 x^{13/3}}+\frac {575 a \sqrt {b x^{2/3}+a x}}{88 b^3 x^4}-\frac {2415 a^2 \sqrt {b x^{2/3}+a x}}{352 b^4 x^{11/3}}+\frac {15295 a^3 \sqrt {b x^{2/3}+a x}}{2112 b^5 x^{10/3}}-\frac {260015 a^4 \sqrt {b x^{2/3}+a x}}{33792 b^6 x^3}+\frac {185725 a^5 \sqrt {b x^{2/3}+a x}}{22528 b^7 x^{8/3}}-\frac {2414425 a^6 \sqrt {b x^{2/3}+a x}}{270336 b^8 x^{7/3}}+\frac {482885 a^7 \sqrt {b x^{2/3}+a x}}{49152 b^9 x^2}-\frac {1448655 a^8 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{5/3}}+\frac {3380195 a^9 \sqrt {b x^{2/3}+a x}}{262144 b^{11} x^{4/3}}-\frac {16900975 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^{12} x}+\frac {50702925 a^{11} \sqrt {b x^{2/3}+a x}}{2097152 b^{13} x^{2/3}}-\frac {50702925 a^{12} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{2097152 b^{27/2}} \] Output:
6/b/x^(11/3)/(b*x^(2/3)+a*x)^(1/2)-25/4*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(13/3) +575/88*a*(b*x^(2/3)+a*x)^(1/2)/b^3/x^4-2415/352*a^2*(b*x^(2/3)+a*x)^(1/2) /b^4/x^(11/3)+15295/2112*a^3*(b*x^(2/3)+a*x)^(1/2)/b^5/x^(10/3)-260015/337 92*a^4*(b*x^(2/3)+a*x)^(1/2)/b^6/x^3+185725/22528*a^5*(b*x^(2/3)+a*x)^(1/2 )/b^7/x^(8/3)-2414425/270336*a^6*(b*x^(2/3)+a*x)^(1/2)/b^8/x^(7/3)+482885/ 49152*a^7*(b*x^(2/3)+a*x)^(1/2)/b^9/x^2-1448655/131072*a^8*(b*x^(2/3)+a*x) ^(1/2)/b^10/x^(5/3)+3380195/262144*a^9*(b*x^(2/3)+a*x)^(1/2)/b^11/x^(4/3)- 16900975/1048576*a^10*(b*x^(2/3)+a*x)^(1/2)/b^12/x+50702925/2097152*a^11*( b*x^(2/3)+a*x)^(1/2)/b^13/x^(2/3)-50702925/2097152*a^12*arctanh(b^(1/2)*x^ (1/3)/(b*x^(2/3)+a*x)^(1/2))/b^(27/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {6 a^{12} \sqrt [3]{x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},13,\frac {1}{2},1+\frac {a \sqrt [3]{x}}{b}\right )}{b^{13} \sqrt {b x^{2/3}+a x}} \] Input:
Integrate[1/(x^4*(b*x^(2/3) + a*x)^(3/2)),x]
Output:
(6*a^12*x^(1/3)*Hypergeometric2F1[-1/2, 13, 1/2, 1 + (a*x^(1/3))/b])/(b^13 *Sqrt[b*x^(2/3) + a*x])
Time = 1.56 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.16, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {1929, 1931, 1931, 1931, 1931, 1931, 1931, 1931, 1931, 1931, 1931, 1931, 1931, 1935, 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.
| \(\displaystyle \int \frac {1}{x^4 \left (a x+b x^{2/3}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1929 |
| \(\displaystyle \frac {25 \int \frac {1}{x^{14/3} \sqrt {x^{2/3} b+a x}}dx}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \int \frac {1}{x^{13/3} \sqrt {x^{2/3} b+a x}}dx}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \left (-\frac {21 a \int \frac {1}{x^4 \sqrt {x^{2/3} b+a x}}dx}{22 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 b x^4}\right )}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \left (-\frac {21 a \left (-\frac {19 a \int \frac {1}{x^{11/3} \sqrt {x^{2/3} b+a x}}dx}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )}{22 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 b x^4}\right )}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \left (-\frac {21 a \left (-\frac {19 a \left (-\frac {17 a \int \frac {1}{x^{10/3} \sqrt {x^{2/3} b+a x}}dx}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )}{22 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 b x^4}\right )}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \left (-\frac {21 a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \int \frac {1}{x^3 \sqrt {x^{2/3} b+a x}}dx}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )}{22 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 b x^4}\right )}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \left (-\frac {21 a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \int \frac {1}{x^{8/3} \sqrt {x^{2/3} b+a x}}dx}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )}{22 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 b x^4}\right )}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \left (-\frac {21 a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \int \frac {1}{x^{7/3} \sqrt {x^{2/3} b+a x}}dx}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )}{22 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 b x^4}\right )}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \left (-\frac {21 a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \int \frac {1}{x^2 \sqrt {x^{2/3} b+a x}}dx}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )}{22 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 b x^4}\right )}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \left (-\frac {21 a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \int \frac {1}{x^{5/3} \sqrt {x^{2/3} b+a x}}dx}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )}{22 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 b x^4}\right )}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \left (-\frac {21 a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \int \frac {1}{x^{4/3} \sqrt {x^{2/3} b+a x}}dx}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )}{22 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 b x^4}\right )}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \left (-\frac {21 a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \int \frac {1}{x \sqrt {x^{2/3} b+a x}}dx}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )}{22 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 b x^4}\right )}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \left (-\frac {21 a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (-\frac {a \int \frac {1}{x^{2/3} \sqrt {x^{2/3} b+a x}}dx}{2 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )}{22 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 b x^4}\right )}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1935 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \left (-\frac {21 a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \int \frac {1}{1-\frac {b x^{2/3}}{x^{2/3} b+a x}}d\frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}}{b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )}{22 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 b x^4}\right )}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {25 \left (-\frac {23 a \left (-\frac {21 a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )}{22 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{11 b x^4}\right )}{24 b}-\frac {\sqrt {a x+b x^{2/3}}}{4 b x^{13/3}}\right )}{b}+\frac {6}{b x^{11/3} \sqrt {a x+b x^{2/3}}}\) |
Input:
Int[1/(x^4*(b*x^(2/3) + a*x)^(3/2)),x]
Output:
6/(b*x^(11/3)*Sqrt[b*x^(2/3) + a*x]) + (25*(-1/4*Sqrt[b*x^(2/3) + a*x]/(b* x^(13/3)) - (23*a*((-3*Sqrt[b*x^(2/3) + a*x])/(11*b*x^4) - (21*a*((-3*Sqrt [b*x^(2/3) + a*x])/(10*b*x^(11/3)) - (19*a*(-1/3*Sqrt[b*x^(2/3) + a*x]/(b* x^(10/3)) - (17*a*((-3*Sqrt[b*x^(2/3) + a*x])/(8*b*x^3) - (15*a*((-3*Sqrt[ b*x^(2/3) + a*x])/(7*b*x^(8/3)) - (13*a*(-1/2*Sqrt[b*x^(2/3) + a*x]/(b*x^( 7/3)) - (11*a*((-3*Sqrt[b*x^(2/3) + a*x])/(5*b*x^2) - (9*a*((-3*Sqrt[b*x^( 2/3) + a*x])/(4*b*x^(5/3)) - (7*a*(-(Sqrt[b*x^(2/3) + a*x]/(b*x^(4/3))) - (5*a*((-3*Sqrt[b*x^(2/3) + a*x])/(2*b*x) - (3*a*((-3*Sqrt[b*x^(2/3) + a*x] )/(b*x^(2/3)) + (3*a*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/b^( 3/2)))/(4*b)))/(6*b)))/(8*b)))/(10*b)))/(12*b)))/(14*b)))/(16*b)))/(18*b)) )/(20*b)))/(22*b)))/(24*b)))/b
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[((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] + Simp[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1))) In t[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, m}, x] & & !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, -1]
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, m, p}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[ m + j*p + 1, 0]
Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp [-2/(n - j) Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
Time = 0.34 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.47
| method | result | size |
| derivativedivides | \(-\frac {\left (x^{\frac {1}{3}} a +b \right ) \left (1673196525 \sqrt {x^{\frac {1}{3}} a +b}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) a^{12} x^{4}+17301504 b^{\frac {25}{2}}+31324160 b^{\frac {17}{2}} a^{4} x^{\frac {4}{3}}-38036480 b^{\frac {15}{2}} a^{5} x^{\frac {5}{3}}+47545600 b^{\frac {13}{2}} a^{6} x^{2}-61809280 b^{\frac {11}{2}} a^{7} x^{\frac {7}{3}}+84987760 b^{\frac {9}{2}} a^{8} x^{\frac {8}{3}}-127481640 b^{\frac {7}{2}} a^{9} x^{3}+223092870 b^{\frac {5}{2}} a^{10} x^{\frac {10}{3}}-557732175 b^{\frac {3}{2}} a^{11} x^{\frac {11}{3}}-19660800 b^{\frac {23}{2}} a \,x^{\frac {1}{3}}+22609920 b^{\frac {21}{2}} a^{2} x^{\frac {2}{3}}-26378240 b^{\frac {19}{2}} a^{3} x -1673196525 a^{12} x^{4} \sqrt {b}\right )}{69206016 x^{3} \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} b^{\frac {27}{2}}}\) | \(192\) |
| default | \(-\frac {\left (x^{\frac {1}{3}} a +b \right ) \left (1673196525 \sqrt {x^{\frac {1}{3}} a +b}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) a^{12} x^{4}+17301504 b^{\frac {25}{2}}+31324160 b^{\frac {17}{2}} a^{4} x^{\frac {4}{3}}-38036480 b^{\frac {15}{2}} a^{5} x^{\frac {5}{3}}+47545600 b^{\frac {13}{2}} a^{6} x^{2}-61809280 b^{\frac {11}{2}} a^{7} x^{\frac {7}{3}}+84987760 b^{\frac {9}{2}} a^{8} x^{\frac {8}{3}}-127481640 b^{\frac {7}{2}} a^{9} x^{3}+223092870 b^{\frac {5}{2}} a^{10} x^{\frac {10}{3}}-557732175 b^{\frac {3}{2}} a^{11} x^{\frac {11}{3}}-19660800 b^{\frac {23}{2}} a \,x^{\frac {1}{3}}+22609920 b^{\frac {21}{2}} a^{2} x^{\frac {2}{3}}-26378240 b^{\frac {19}{2}} a^{3} x -1673196525 a^{12} x^{4} \sqrt {b}\right )}{69206016 x^{3} \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} b^{\frac {27}{2}}}\) | \(192\) |
Input:
int(1/x^4/(b*x^(2/3)+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/69206016*(x^(1/3)*a+b)*(1673196525*(x^(1/3)*a+b)^(1/2)*arctanh((x^(1/3) *a+b)^(1/2)/b^(1/2))*a^12*x^4+17301504*b^(25/2)+31324160*b^(17/2)*a^4*x^(4 /3)-38036480*b^(15/2)*a^5*x^(5/3)+47545600*b^(13/2)*a^6*x^2-61809280*b^(11 /2)*a^7*x^(7/3)+84987760*b^(9/2)*a^8*x^(8/3)-127481640*b^(7/2)*a^9*x^3+223 092870*b^(5/2)*a^10*x^(10/3)-557732175*b^(3/2)*a^11*x^(11/3)-19660800*b^(2 3/2)*a*x^(1/3)+22609920*b^(21/2)*a^2*x^(2/3)-26378240*b^(19/2)*a^3*x-16731 96525*a^12*x^4*b^(1/2))/x^3/(b*x^(2/3)+a*x)^(3/2)/b^(27/2)
Timed out. \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x^4/(b*x^(2/3)+a*x)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{4} \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x**4/(b*x**(2/3)+a*x)**(3/2),x)
Output:
Integral(1/(x**4*(a*x + b*x**(2/3))**(3/2)), x)
\[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(b*x^(2/3)+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((a*x + b*x^(2/3))^(3/2)*x^4), x)
Time = 0.38 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {50702925 \, a^{12} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{2097152 \, \sqrt {-b} b^{13}} + \frac {6 \, a^{12}}{\sqrt {a x^{\frac {1}{3}} + b} b^{13}} + \frac {1257960429 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} a^{12} - 14537792973 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} a^{12} b + 76667241519 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} a^{12} b^{2} - 243717614415 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{12} b^{3} + 519393101810 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{12} b^{4} - 780150847218 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{12} b^{5} + 844265343246 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{12} b^{6} - 659969685518 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{12} b^{7} + 366679446705 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{12} b^{8} - 138840292305 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{12} b^{9} + 32660709939 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{12} b^{10} - 3724872723 \, \sqrt {a x^{\frac {1}{3}} + b} a^{12} b^{11}}{69206016 \, a^{12} b^{13} x^{4}} \] Input:
integrate(1/x^4/(b*x^(2/3)+a*x)^(3/2),x, algorithm="giac")
Output:
50702925/2097152*a^12*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^13) + 6*a^12/(sqrt(a*x^(1/3) + b)*b^13) + 1/69206016*(1257960429*(a*x^(1/3) + b)^(23/2)*a^12 - 14537792973*(a*x^(1/3) + b)^(21/2)*a^12*b + 76667241519* (a*x^(1/3) + b)^(19/2)*a^12*b^2 - 243717614415*(a*x^(1/3) + b)^(17/2)*a^12 *b^3 + 519393101810*(a*x^(1/3) + b)^(15/2)*a^12*b^4 - 780150847218*(a*x^(1 /3) + b)^(13/2)*a^12*b^5 + 844265343246*(a*x^(1/3) + b)^(11/2)*a^12*b^6 - 659969685518*(a*x^(1/3) + b)^(9/2)*a^12*b^7 + 366679446705*(a*x^(1/3) + b) ^(7/2)*a^12*b^8 - 138840292305*(a*x^(1/3) + b)^(5/2)*a^12*b^9 + 3266070993 9*(a*x^(1/3) + b)^(3/2)*a^12*b^10 - 3724872723*sqrt(a*x^(1/3) + b)*a^12*b^ 11)/(a^12*b^13*x^4)
Timed out. \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^4\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2}} \,d x \] Input:
int(1/(x^4*(a*x + b*x^(2/3))^(3/2)),x)
Output:
int(1/(x^4*(a*x + b*x^(2/3))^(3/2)), x)
Time = 0.22 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.51 \[ \int \frac {1}{x^4 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {1673196525 \sqrt {b}\, \sqrt {x^{\frac {1}{3}} a +b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}-\sqrt {b}\right ) a^{12} x^{4}-1673196525 \sqrt {b}\, \sqrt {x^{\frac {1}{3}} a +b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}+\sqrt {b}\right ) a^{12} x^{4}+1115464350 x^{\frac {11}{3}} a^{11} b^{2}-169975520 x^{\frac {8}{3}} a^{8} b^{5}+76072960 x^{\frac {5}{3}} a^{5} b^{8}-45219840 x^{\frac {2}{3}} a^{2} b^{11}-446185740 x^{\frac {10}{3}} a^{10} b^{3}+123618560 x^{\frac {7}{3}} a^{7} b^{6}-62648320 x^{\frac {4}{3}} a^{4} b^{9}+39321600 x^{\frac {1}{3}} a \,b^{12}+3346393050 a^{12} b \,x^{4}+254963280 a^{9} b^{4} x^{3}-95091200 a^{6} b^{7} x^{2}+52756480 a^{3} b^{10} x -34603008 b^{13}}{138412032 \sqrt {x^{\frac {1}{3}} a +b}\, b^{14} x^{4}} \] Input:
int(1/x^4/(b*x^(2/3)+a*x)^(3/2),x)
Output:
(1673196525*sqrt(b)*sqrt(x**(1/3)*a + b)*log(sqrt(x**(1/3)*a + b) - sqrt(b ))*a**12*x**4 - 1673196525*sqrt(b)*sqrt(x**(1/3)*a + b)*log(sqrt(x**(1/3)* a + b) + sqrt(b))*a**12*x**4 + 1115464350*x**(2/3)*a**11*b**2*x**3 - 16997 5520*x**(2/3)*a**8*b**5*x**2 + 76072960*x**(2/3)*a**5*b**8*x - 45219840*x* *(2/3)*a**2*b**11 - 446185740*x**(1/3)*a**10*b**3*x**3 + 123618560*x**(1/3 )*a**7*b**6*x**2 - 62648320*x**(1/3)*a**4*b**9*x + 39321600*x**(1/3)*a*b** 12 + 3346393050*a**12*b*x**4 + 254963280*a**9*b**4*x**3 - 95091200*a**6*b* *7*x**2 + 52756480*a**3*b**10*x - 34603008*b**13)/(138412032*sqrt(x**(1/3) *a + b)*b**14*x**4)