Integrand size = 19, antiderivative size = 379 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^6} \, dx=-\frac {3 a \sqrt {b x^{2/3}+a x}}{88 x^4}-\frac {3 a^2 \sqrt {b x^{2/3}+a x}}{1760 b x^{11/3}}+\frac {19 a^3 \sqrt {b x^{2/3}+a x}}{10560 b^2 x^{10/3}}-\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{168960 b^3 x^3}+\frac {323 a^5 \sqrt {b x^{2/3}+a x}}{157696 b^4 x^{8/3}}-\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{1892352 b^5 x^{7/3}}+\frac {4199 a^7 \sqrt {b x^{2/3}+a x}}{1720320 b^6 x^2}-\frac {12597 a^8 \sqrt {b x^{2/3}+a x}}{4587520 b^7 x^{5/3}}+\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{1310720 b^8 x^{4/3}}-\frac {4199 a^{10} \sqrt {b x^{2/3}+a x}}{1048576 b^9 x}+\frac {12597 a^{11} \sqrt {b x^{2/3}+a x}}{2097152 b^{10} x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{4 x^5}-\frac {12597 a^{12} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{2097152 b^{21/2}} \] Output:
-3/88*a*(b*x^(2/3)+a*x)^(1/2)/x^4-3/1760*a^2*(b*x^(2/3)+a*x)^(1/2)/b/x^(11 /3)+19/10560*a^3*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(10/3)-323/168960*a^4*(b*x^(2 /3)+a*x)^(1/2)/b^3/x^3+323/157696*a^5*(b*x^(2/3)+a*x)^(1/2)/b^4/x^(8/3)-41 99/1892352*a^6*(b*x^(2/3)+a*x)^(1/2)/b^5/x^(7/3)+4199/1720320*a^7*(b*x^(2/ 3)+a*x)^(1/2)/b^6/x^2-12597/4587520*a^8*(b*x^(2/3)+a*x)^(1/2)/b^7/x^(5/3)+ 4199/1310720*a^9*(b*x^(2/3)+a*x)^(1/2)/b^8/x^(4/3)-4199/1048576*a^10*(b*x^ (2/3)+a*x)^(1/2)/b^9/x+12597/2097152*a^11*(b*x^(2/3)+a*x)^(1/2)/b^10/x^(2/ 3)-1/4*(b*x^(2/3)+a*x)^(3/2)/x^5-12597/2097152*a^12*arctanh(b^(1/2)*x^(1/3 )/(b*x^(2/3)+a*x)^(1/2))/b^(21/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.16 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^6} \, dx=-\frac {6 a^{12} \left (b+a \sqrt [3]{x}\right )^2 \sqrt {b x^{2/3}+a x} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},13,\frac {7}{2},1+\frac {a \sqrt [3]{x}}{b}\right )}{5 b^{13} \sqrt [3]{x}} \] Input:
Integrate[(b*x^(2/3) + a*x)^(3/2)/x^6,x]
Output:
(-6*a^12*(b + a*x^(1/3))^2*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[5/2, 13 , 7/2, 1 + (a*x^(1/3))/b])/(5*b^13*x^(1/3))
Time = 1.42 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {1926, 1926, 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.
| \(\displaystyle \int \frac {\left (a x+b x^{2/3}\right )^{3/2}}{x^6} \, dx\) |
| \(\Big \downarrow \) 1926 |
| \(\displaystyle \frac {1}{8} a \int \frac {\sqrt {x^{2/3} b+a x}}{x^5}dx-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
| \(\Big \downarrow \) 1926 |
| \(\displaystyle \frac {1}{8} a \left (\frac {1}{22} a \int \frac {1}{x^4 \sqrt {x^{2/3} b+a x}}dx-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{8} a \left (\frac {1}{22} a \left (-\frac {19 a \int \frac {1}{x^{11/3} \sqrt {x^{2/3} b+a x}}dx}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{8} a \left (\frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \int \frac {1}{x^{10/3} \sqrt {x^{2/3} b+a x}}dx}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{8} a \left (\frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \int \frac {1}{x^3 \sqrt {x^{2/3} b+a x}}dx}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{8} a \left (\frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \int \frac {1}{x^{8/3} \sqrt {x^{2/3} b+a x}}dx}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{8} a \left (\frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \int \frac {1}{x^{7/3} \sqrt {x^{2/3} b+a x}}dx}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{8} a \left (\frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \int \frac {1}{x^2 \sqrt {x^{2/3} b+a x}}dx}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{8} a \left (\frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \int \frac {1}{x^{5/3} \sqrt {x^{2/3} b+a x}}dx}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{8} a \left (\frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \int \frac {1}{x^{4/3} \sqrt {x^{2/3} b+a x}}dx}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{8} a \left (\frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \int \frac {1}{x \sqrt {x^{2/3} b+a x}}dx}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{8} a \left (\frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (-\frac {a \int \frac {1}{x^{2/3} \sqrt {x^{2/3} b+a x}}dx}{2 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
| \(\Big \downarrow \) 1935 |
| \(\displaystyle \frac {1}{8} a \left (\frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \int \frac {1}{1-\frac {b x^{2/3}}{x^{2/3} b+a x}}d\frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}}{b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{8} a \left (\frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}\) |
Input:
Int[(b*x^(2/3) + a*x)^(3/2)/x^6,x]
Output:
-1/4*(b*x^(2/3) + a*x)^(3/2)/x^5 + (a*((-3*Sqrt[b*x^(2/3) + a*x])/(11*x^4) + (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))/8
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*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + j*p + 1))), x] - Simp[b*p *((n - j)/(c^n*(m + j*p + 1))) Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Integer sQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && LtQ[m + j*p + 1, 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, 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 = 0.36 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.59
| method | result | size |
| derivativedivides | \(\frac {\left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (14549535 b^{\frac {21}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {23}{2}}-169744575 b^{\frac {23}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {21}{2}}+904981077 b^{\frac {25}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {19}{2}}-2913648309 b^{\frac {27}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {17}{2}}+6303782342 b^{\frac {29}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {15}{2}}-9643633350 b^{\frac {31}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {13}{2}}+10677769530 b^{\frac {33}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {11}{2}}-8598579770 b^{\frac {35}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {9}{2}}+4975837515 b^{\frac {37}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {7}{2}}-2001671595 b^{\frac {39}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {5}{2}}-14549535 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) b^{10} a^{12} x^{4}-169744575 b^{\frac {41}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}}+14549535 b^{\frac {43}{2}} \sqrt {x^{\frac {1}{3}} a +b}\right )}{2422210560 x^{5} \left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}} b^{\frac {41}{2}}}\) | \(223\) |
| default | \(\frac {\left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (14549535 b^{\frac {21}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {23}{2}}-169744575 b^{\frac {23}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {21}{2}}+904981077 b^{\frac {25}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {19}{2}}-2913648309 b^{\frac {27}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {17}{2}}+6303782342 b^{\frac {29}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {15}{2}}-9643633350 b^{\frac {31}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {13}{2}}+10677769530 b^{\frac {33}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {11}{2}}-8598579770 b^{\frac {35}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {9}{2}}+4975837515 b^{\frac {37}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {7}{2}}-2001671595 b^{\frac {39}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {5}{2}}-14549535 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) b^{10} a^{12} x^{4}-169744575 b^{\frac {41}{2}} \left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}}+14549535 b^{\frac {43}{2}} \sqrt {x^{\frac {1}{3}} a +b}\right )}{2422210560 x^{5} \left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}} b^{\frac {41}{2}}}\) | \(223\) |
Input:
int((b*x^(2/3)+a*x)^(3/2)/x^6,x,method=_RETURNVERBOSE)
Output:
1/2422210560*(b*x^(2/3)+a*x)^(3/2)*(14549535*b^(21/2)*(x^(1/3)*a+b)^(23/2) -169744575*b^(23/2)*(x^(1/3)*a+b)^(21/2)+904981077*b^(25/2)*(x^(1/3)*a+b)^ (19/2)-2913648309*b^(27/2)*(x^(1/3)*a+b)^(17/2)+6303782342*b^(29/2)*(x^(1/ 3)*a+b)^(15/2)-9643633350*b^(31/2)*(x^(1/3)*a+b)^(13/2)+10677769530*b^(33/ 2)*(x^(1/3)*a+b)^(11/2)-8598579770*b^(35/2)*(x^(1/3)*a+b)^(9/2)+4975837515 *b^(37/2)*(x^(1/3)*a+b)^(7/2)-2001671595*b^(39/2)*(x^(1/3)*a+b)^(5/2)-1454 9535*arctanh((x^(1/3)*a+b)^(1/2)/b^(1/2))*b^10*a^12*x^4-169744575*b^(41/2) *(x^(1/3)*a+b)^(3/2)+14549535*b^(43/2)*(x^(1/3)*a+b)^(1/2))/x^5/(x^(1/3)*a +b)^(3/2)/b^(41/2)
Timed out. \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^6} \, dx=\text {Timed out} \] Input:
integrate((b*x^(2/3)+a*x)^(3/2)/x^6,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^6} \, dx=\text {Timed out} \] Input:
integrate((b*x**(2/3)+a*x)**(3/2)/x**6,x)
Output:
Timed out
\[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \] Input:
integrate((b*x^(2/3)+a*x)^(3/2)/x^6,x, algorithm="maxima")
Output:
integrate((a*x + b*x^(2/3))^(3/2)/x^6, x)
Time = 0.37 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.65 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^6} \, dx=\frac {\frac {14549535 \, a^{13} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{10}} + \frac {14549535 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} a^{13} - 169744575 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} a^{13} b + 904981077 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} a^{13} b^{2} - 2913648309 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{13} b^{3} + 6303782342 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{13} b^{4} - 9643633350 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{13} b^{5} + 10677769530 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{13} b^{6} - 8598579770 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{13} b^{7} + 4975837515 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{13} b^{8} - 2001671595 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{13} b^{9} - 169744575 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{13} b^{10} + 14549535 \, \sqrt {a x^{\frac {1}{3}} + b} a^{13} b^{11}}{a^{12} b^{10} x^{4}}}{2422210560 \, a} \] Input:
integrate((b*x^(2/3)+a*x)^(3/2)/x^6,x, algorithm="giac")
Output:
1/2422210560*(14549535*a^13*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b) *b^10) + (14549535*(a*x^(1/3) + b)^(23/2)*a^13 - 169744575*(a*x^(1/3) + b) ^(21/2)*a^13*b + 904981077*(a*x^(1/3) + b)^(19/2)*a^13*b^2 - 2913648309*(a *x^(1/3) + b)^(17/2)*a^13*b^3 + 6303782342*(a*x^(1/3) + b)^(15/2)*a^13*b^4 - 9643633350*(a*x^(1/3) + b)^(13/2)*a^13*b^5 + 10677769530*(a*x^(1/3) + b )^(11/2)*a^13*b^6 - 8598579770*(a*x^(1/3) + b)^(9/2)*a^13*b^7 + 4975837515 *(a*x^(1/3) + b)^(7/2)*a^13*b^8 - 2001671595*(a*x^(1/3) + b)^(5/2)*a^13*b^ 9 - 169744575*(a*x^(1/3) + b)^(3/2)*a^13*b^10 + 14549535*sqrt(a*x^(1/3) + b)*a^13*b^11)/(a^12*b^10*x^4))/a
Timed out. \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^6} \, dx=\int \frac {{\left (a\,x+b\,x^{2/3}\right )}^{3/2}}{x^6} \,d x \] Input:
int((a*x + b*x^(2/3))^(3/2)/x^6,x)
Output:
int((a*x + b*x^(2/3))^(3/2)/x^6, x)
Time = 0.26 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.72 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^6} \, dx=\frac {15519504 x^{\frac {11}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{9} b^{3}-10749440 x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{6} b^{6}+8716288 x^{\frac {5}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{3} b^{9}-1211105280 x^{\frac {2}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, b^{12}+29099070 x^{\frac {13}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{11} b -13302432 x^{\frac {10}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{8} b^{4}+9922560 x^{\frac {7}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{5} b^{7}-8257536 x^{\frac {4}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{2} b^{10}-19399380 \sqrt {x^{\frac {1}{3}} a +b}\, a^{10} b^{2} x^{4}+11824384 \sqrt {x^{\frac {1}{3}} a +b}\, a^{7} b^{5} x^{3}-9261056 \sqrt {x^{\frac {1}{3}} a +b}\, a^{4} b^{8} x^{2}-1376256000 \sqrt {x^{\frac {1}{3}} a +b}\, a \,b^{11} x +14549535 x^{\frac {14}{3}} \sqrt {b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}-\sqrt {b}\right ) a^{12}-14549535 x^{\frac {14}{3}} \sqrt {b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}+\sqrt {b}\right ) a^{12}}{4844421120 x^{\frac {14}{3}} b^{11}} \] Input:
int((b*x^(2/3)+a*x)^(3/2)/x^6,x)
Output:
(15519504*x**(2/3)*sqrt(x**(1/3)*a + b)*a**9*b**3*x**3 - 10749440*x**(2/3) *sqrt(x**(1/3)*a + b)*a**6*b**6*x**2 + 8716288*x**(2/3)*sqrt(x**(1/3)*a + b)*a**3*b**9*x - 1211105280*x**(2/3)*sqrt(x**(1/3)*a + b)*b**12 + 29099070 *x**(1/3)*sqrt(x**(1/3)*a + b)*a**11*b*x**4 - 13302432*x**(1/3)*sqrt(x**(1 /3)*a + b)*a**8*b**4*x**3 + 9922560*x**(1/3)*sqrt(x**(1/3)*a + b)*a**5*b** 7*x**2 - 8257536*x**(1/3)*sqrt(x**(1/3)*a + b)*a**2*b**10*x - 19399380*sqr t(x**(1/3)*a + b)*a**10*b**2*x**4 + 11824384*sqrt(x**(1/3)*a + b)*a**7*b** 5*x**3 - 9261056*sqrt(x**(1/3)*a + b)*a**4*b**8*x**2 - 1376256000*sqrt(x** (1/3)*a + b)*a*b**11*x + 14549535*x**(2/3)*sqrt(b)*log(sqrt(x**(1/3)*a + b ) - sqrt(b))*a**12*x**4 - 14549535*x**(2/3)*sqrt(b)*log(sqrt(x**(1/3)*a + b) + sqrt(b))*a**12*x**4)/(4844421120*x**(2/3)*b**11*x**4)