Integrand size = 19, antiderivative size = 858 \[ \int \frac {1}{(a+b x)^{5/2} \sqrt [6]{c+d x}} \, dx=-\frac {2 (c+d x)^{5/6}}{3 (b c-a d) (a+b x)^{3/2}}+\frac {8 d (c+d x)^{5/6}}{9 (b c-a d)^2 \sqrt {a+b x}}+\frac {8 \left (1+\sqrt {3}\right ) d^2 \sqrt {a+b x} \sqrt [6]{c+d x}}{9 b^{2/3} (b c-a d)^2 \left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )}+\frac {8 d \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3\ 3^{3/4} b^{2/3} (b c-a d)^{5/3} \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}}+\frac {4 \left (1-\sqrt {3}\right ) d \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{9 \sqrt [4]{3} b^{2/3} (b c-a d)^{5/3} \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \] Output:
-2/3*(d*x+c)^(5/6)/(-a*d+b*c)/(b*x+a)^(3/2)+8/9*d*(d*x+c)^(5/6)/(-a*d+b*c) ^2/(b*x+a)^(1/2)+8/9*(1+3^(1/2))*d^2*(b*x+a)^(1/2)*(d*x+c)^(1/6)/b^(2/3)/( -a*d+b*c)^2/((-a*d+b*c)^(1/3)-(1+3^(1/2))*b^(1/3)*(d*x+c)^(1/3))+8/9*d*(d* x+c)^(1/6)*((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3))*(((-a*d+b*c)^(2/3)+b^( 1/3)*(-a*d+b*c)^(1/3)*(d*x+c)^(1/3)+b^(2/3)*(d*x+c)^(2/3))/((-a*d+b*c)^(1/ 3)-(1+3^(1/2))*b^(1/3)*(d*x+c)^(1/3))^2)^(1/2)*EllipticE((1-((-a*d+b*c)^(1 /3)-(1-3^(1/2))*b^(1/3)*(d*x+c)^(1/3))^2/((-a*d+b*c)^(1/3)-(1+3^(1/2))*b^( 1/3)*(d*x+c)^(1/3))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*3^(1/4)/b^(2/3)/(-a* d+b*c)^(5/3)/(b*x+a)^(1/2)/(-b^(1/3)*(d*x+c)^(1/3)*((-a*d+b*c)^(1/3)-b^(1/ 3)*(d*x+c)^(1/3))/((-a*d+b*c)^(1/3)-(1+3^(1/2))*b^(1/3)*(d*x+c)^(1/3))^2)^ (1/2)+4/27*(1-3^(1/2))*d*(d*x+c)^(1/6)*((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^( 1/3))*(((-a*d+b*c)^(2/3)+b^(1/3)*(-a*d+b*c)^(1/3)*(d*x+c)^(1/3)+b^(2/3)*(d *x+c)^(2/3))/((-a*d+b*c)^(1/3)-(1+3^(1/2))*b^(1/3)*(d*x+c)^(1/3))^2)^(1/2) *InverseJacobiAM(arccos(((-a*d+b*c)^(1/3)-(1-3^(1/2))*b^(1/3)*(d*x+c)^(1/3 ))/((-a*d+b*c)^(1/3)-(1+3^(1/2))*b^(1/3)*(d*x+c)^(1/3))),1/4*6^(1/2)+1/4*2 ^(1/2))*3^(3/4)/b^(2/3)/(-a*d+b*c)^(5/3)/(b*x+a)^(1/2)/(-b^(1/3)*(d*x+c)^( 1/3)*((-a*d+b*c)^(1/3)-b^(1/3)*(d*x+c)^(1/3))/((-a*d+b*c)^(1/3)-(1+3^(1/2) )*b^(1/3)*(d*x+c)^(1/3))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.09 \[ \int \frac {1}{(a+b x)^{5/2} \sqrt [6]{c+d x}} \, dx=-\frac {2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{6},-\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{3 b (a+b x)^{3/2} \sqrt [6]{c+d x}} \] Input:
Integrate[1/((a + b*x)^(5/2)*(c + d*x)^(1/6)),x]
Output:
(-2*((b*(c + d*x))/(b*c - a*d))^(1/6)*Hypergeometric2F1[-3/2, 1/6, -1/2, ( d*(a + b*x))/(-(b*c) + a*d)])/(3*b*(a + b*x)^(3/2)*(c + d*x)^(1/6))
Time = 0.74 (sec) , antiderivative size = 908, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {61, 61, 73, 837, 25, 766, 2420}
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.
| \(\displaystyle \int \frac {1}{(a+b x)^{5/2} \sqrt [6]{c+d x}} \, dx\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {4 d \int \frac {1}{(a+b x)^{3/2} \sqrt [6]{c+d x}}dx}{9 (b c-a d)}-\frac {2 (c+d x)^{5/6}}{3 (a+b x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {4 d \left (\frac {2 d \int \frac {1}{\sqrt {a+b x} \sqrt [6]{c+d x}}dx}{3 (b c-a d)}-\frac {2 (c+d x)^{5/6}}{\sqrt {a+b x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 (c+d x)^{5/6}}{3 (a+b x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {4 d \left (\frac {4 \int \frac {(c+d x)^{2/3}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [6]{c+d x}}{b c-a d}-\frac {2 (c+d x)^{5/6}}{\sqrt {a+b x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 (c+d x)^{5/6}}{3 (a+b x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 837 |
| \(\displaystyle -\frac {4 d \left (\frac {4 \left (-\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3} \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [6]{c+d x}}{2 b^{2/3}}-\frac {\int -\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 b^{2/3} (c+d x)^{2/3}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [6]{c+d x}}{2 b^{2/3}}\right )}{b c-a d}-\frac {2 (c+d x)^{5/6}}{\sqrt {a+b x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 (c+d x)^{5/6}}{3 (a+b x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {4 d \left (\frac {4 \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 b^{2/3} (c+d x)^{2/3}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [6]{c+d x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3} \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [6]{c+d x}}{2 b^{2/3}}\right )}{b c-a d}-\frac {2 (c+d x)^{5/6}}{\sqrt {a+b x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 (c+d x)^{5/6}}{3 (a+b x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle -\frac {4 d \left (\frac {4 \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 b^{2/3} (c+d x)^{2/3}}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [6]{c+d x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [6]{c+d x} \sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b c-a d}-\frac {2 (c+d x)^{5/6}}{\sqrt {a+b x} (b c-a d)}\right )}{9 (b c-a d)}-\frac {2 (c+d x)^{5/6}}{3 (a+b x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 2420 |
| \(\displaystyle -\frac {4 d \left (\frac {4 \left (\frac {-\frac {\left (1+\sqrt {3}\right ) \sqrt [6]{c+d x} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}} d}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}-\frac {\sqrt [4]{3} \sqrt [3]{b c-a d} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d} \sqrt [6]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+b^{2/3} (c+d x)^{2/3}}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{b c-a d}-\left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {-\frac {\sqrt [3]{b} \sqrt [3]{c+d x} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\sqrt [3]{b c-a d}-\left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\right )}{b c-a d}-\frac {2 (c+d x)^{5/6}}{(b c-a d) \sqrt {a+b x}}\right )}{9 (b c-a d)}-\frac {2 (c+d x)^{5/6}}{3 (b c-a d) (a+b x)^{3/2}}\) |
Input:
Int[1/((a + b*x)^(5/2)*(c + d*x)^(1/6)),x]
Output:
(-2*(c + d*x)^(5/6))/(3*(b*c - a*d)*(a + b*x)^(3/2)) - (4*d*((-2*(c + d*x) ^(5/6))/((b*c - a*d)*Sqrt[a + b*x]) + (4*((-(((1 + Sqrt[3])*d*(c + d*x)^(1 /6)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])/((b*c - a*d)^(1/3) - (1 + Sqrt[3] )*b^(1/3)*(c + d*x)^(1/3))) - (3^(1/4)*(b*c - a*d)^(1/3)*(c + d*x)^(1/6)*( (b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a*d)^(2/3) + b^( 1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^(2/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2]*EllipticE[ArcCos[(( b*c - a*d)^(1/3) - (1 - Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))/((b*c - a*d)^(1/ 3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqrt[3])/4])/(Sqrt[-((b ^(1/3)*(c + d*x)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((b* c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2)]*Sqrt[a - (b*c) /d + (b*(c + d*x))/d]))/(2*b^(2/3)) - ((1 - Sqrt[3])*(b*c - a*d)^(1/3)*(c + d*x)^(1/6)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3))*Sqrt[((b*c - a* d)^(2/3) + b^(1/3)*(b*c - a*d)^(1/3)*(c + d*x)^(1/3) + b^(2/3)*(c + d*x)^( 2/3))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2]*Ellip ticF[ArcCos[((b*c - a*d)^(1/3) - (1 - Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))/(( b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))], (2 + Sqrt[3])/ 4])/(4*3^(1/4)*b^(2/3)*Sqrt[-((b^(1/3)*(c + d*x)^(1/3)*((b*c - a*d)^(1/3) - b^(1/3)*(c + d*x)^(1/3)))/((b*c - a*d)^(1/3) - (1 + Sqrt[3])*b^(1/3)*(c + d*x)^(1/3))^2)]*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])))/(b*c - a*d)))/...
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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
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[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ (s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x ]
Int[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2)) Int[1/Sqrt[ a + b*x^6], x], x] - Simp[1/(2*r^2) Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S qrt[a + b*x^6], x], x]] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*x* (s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/(s + (1 + Sqrt[3])*r*x^2)^2]/(2 *r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) )*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]
\[\int \frac {1}{\left (b x +a \right )^{\frac {5}{2}} \left (x d +c \right )^{\frac {1}{6}}}d x\]
Input:
int(1/(b*x+a)^(5/2)/(d*x+c)^(1/6),x)
Output:
int(1/(b*x+a)^(5/2)/(d*x+c)^(1/6),x)
\[ \int \frac {1}{(a+b x)^{5/2} \sqrt [6]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {1}{6}}} \,d x } \] Input:
integrate(1/(b*x+a)^(5/2)/(d*x+c)^(1/6),x, algorithm="fricas")
Output:
integral(sqrt(b*x + a)*(d*x + c)^(5/6)/(b^3*d*x^4 + a^3*c + (b^3*c + 3*a*b ^2*d)*x^3 + 3*(a*b^2*c + a^2*b*d)*x^2 + (3*a^2*b*c + a^3*d)*x), x)
\[ \int \frac {1}{(a+b x)^{5/2} \sqrt [6]{c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \sqrt [6]{c + d x}}\, dx \] Input:
integrate(1/(b*x+a)**(5/2)/(d*x+c)**(1/6),x)
Output:
Integral(1/((a + b*x)**(5/2)*(c + d*x)**(1/6)), x)
\[ \int \frac {1}{(a+b x)^{5/2} \sqrt [6]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {1}{6}}} \,d x } \] Input:
integrate(1/(b*x+a)^(5/2)/(d*x+c)^(1/6),x, algorithm="maxima")
Output:
integrate(1/((b*x + a)^(5/2)*(d*x + c)^(1/6)), x)
\[ \int \frac {1}{(a+b x)^{5/2} \sqrt [6]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} {\left (d x + c\right )}^{\frac {1}{6}}} \,d x } \] Input:
integrate(1/(b*x+a)^(5/2)/(d*x+c)^(1/6),x, algorithm="giac")
Output:
integrate(1/((b*x + a)^(5/2)*(d*x + c)^(1/6)), x)
Timed out. \[ \int \frac {1}{(a+b x)^{5/2} \sqrt [6]{c+d x}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{1/6}} \,d x \] Input:
int(1/((a + b*x)^(5/2)*(c + d*x)^(1/6)),x)
Output:
int(1/((a + b*x)^(5/2)*(c + d*x)^(1/6)), x)
\[ \int \frac {1}{(a+b x)^{5/2} \sqrt [6]{c+d x}} \, dx=\int \frac {\left (d x +c \right )^{\frac {1}{6}} \sqrt {b x +a}}{\left (d x +c \right )^{\frac {1}{3}} a^{3}+3 \left (d x +c \right )^{\frac {1}{3}} a^{2} b x +3 \left (d x +c \right )^{\frac {1}{3}} a \,b^{2} x^{2}+\left (d x +c \right )^{\frac {1}{3}} b^{3} x^{3}}d x \] Input:
int(1/(b*x+a)^(5/2)/(d*x+c)^(1/6),x)
Output:
int(((c + d*x)**(1/6)*sqrt(a + b*x))/((c + d*x)**(1/3)*a**3 + 3*(c + d*x)* *(1/3)*a**2*b*x + 3*(c + d*x)**(1/3)*a*b**2*x**2 + (c + d*x)**(1/3)*b**3*x **3),x)