Integrand size = 22, antiderivative size = 272 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^6} \, dx=\frac {(A b-10 a B) \sqrt {a+b x^3}}{20 a x^2}-\frac {A \left (a+b x^3\right )^{3/2}}{5 a x^5}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} (A b-10 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{20 a \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
1/20*(A*b-10*B*a)*(b*x^3+a)^(1/2)/a/x^2-1/5*A*(b*x^3+a)^(3/2)/a/x^5-1/20*3 ^(3/4)*(1/2*6^(1/2)+1/2*2^(1/2))*b^(2/3)*(A*b-10*B*a)*(a^(1/3)+b^(1/3)*x)* ((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2 )^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3)+b^( 1/3)*x),I*3^(1/2)+2*I)/a/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^(1/3) +b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^6} \, dx=\frac {\sqrt {a+b x^3} \left (-2 A \left (a+b x^3\right )+\frac {\left (\frac {A b}{2}-5 a B\right ) x^3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{2},\frac {1}{3},-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{10 a x^5} \] Input:
Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^6,x]
Output:
(Sqrt[a + b*x^3]*(-2*A*(a + b*x^3) + (((A*b)/2 - 5*a*B)*x^3*Hypergeometric 2F1[-2/3, -1/2, 1/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(10*a*x^5)
Time = 0.51 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {955, 809, 759}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle -\frac {(A b-10 a B) \int \frac {\sqrt {b x^3+a}}{x^3}dx}{10 a}-\frac {A \left (a+b x^3\right )^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 809 |
| \(\displaystyle -\frac {(A b-10 a B) \left (\frac {3}{4} b \int \frac {1}{\sqrt {b x^3+a}}dx-\frac {\sqrt {a+b x^3}}{2 x^2}\right )}{10 a}-\frac {A \left (a+b x^3\right )^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle -\frac {(A b-10 a B) \left (\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\sqrt {a+b x^3}}{2 x^2}\right )}{10 a}-\frac {A \left (a+b x^3\right )^{3/2}}{5 a x^5}\) |
Input:
Int[(Sqrt[a + b*x^3]*(A + B*x^3))/x^6,x]
Output:
-1/5*(A*(a + b*x^3)^(3/2))/(a*x^5) - ((A*b - 10*a*B)*(-1/2*Sqrt[a + b*x^3] /x^2 + (3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(2/3)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2 /3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x) ^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^ (1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(2*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)* x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])))/(10*a)
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1))), x] - Simp[b*n*(p/(c^n*(m + 1))) I nt[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && IGtQ [n, 0] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + n*p + n + 1)/n, 0] && IntB inomialQ[a, b, c, n, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Time = 1.39 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.21
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{3}+a}\, \left (3 A b \,x^{3}+10 B a \,x^{3}+4 A a \right )}{20 x^{5} a}+\frac {i \left (A b -10 B a \right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{20 a \sqrt {b \,x^{3}+a}}\) | \(328\) |
| elliptic | \(-\frac {A \sqrt {b \,x^{3}+a}}{5 x^{5}}-\frac {\left (3 A b +10 B a \right ) \sqrt {b \,x^{3}+a}}{20 a \,x^{2}}-\frac {2 i \left (B b -\frac {b \left (3 A b +10 B a \right )}{40 a}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{3 b \sqrt {b \,x^{3}+a}}\) | \(344\) |
| default | \(A \left (-\frac {\sqrt {b \,x^{3}+a}}{5 x^{5}}-\frac {3 b \sqrt {b \,x^{3}+a}}{20 a \,x^{2}}+\frac {i b \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{20 a \sqrt {b \,x^{3}+a}}\right )+B \left (-\frac {\sqrt {b \,x^{3}+a}}{2 x^{2}}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{2 \sqrt {b \,x^{3}+a}}\right )\) | \(616\) |
Input:
int((b*x^3+a)^(1/2)*(B*x^3+A)/x^6,x,method=_RETURNVERBOSE)
Output:
-1/20*(b*x^3+a)^(1/2)*(3*A*b*x^3+10*B*a*x^3+4*A*a)/x^5/a+1/20*I*(A*b-10*B* a)/a*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a *b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2 /b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a* b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2) /(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^ (1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b ^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))
Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.24 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^6} \, dx=\frac {3 \, {\left (10 \, B a - A b\right )} \sqrt {b} x^{5} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - {\left ({\left (10 \, B a + 3 \, A b\right )} x^{3} + 4 \, A a\right )} \sqrt {b x^{3} + a}}{20 \, a x^{5}} \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^6,x, algorithm="fricas")
Output:
1/20*(3*(10*B*a - A*b)*sqrt(b)*x^5*weierstrassPInverse(0, -4*a/b, x) - ((1 0*B*a + 3*A*b)*x^3 + 4*A*a)*sqrt(b*x^3 + a))/(a*x^5)
Time = 1.43 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.35 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^6} \, dx=\frac {A \sqrt {a} \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} \] Input:
integrate((b*x**3+a)**(1/2)*(B*x**3+A)/x**6,x)
Output:
A*sqrt(a)*gamma(-5/3)*hyper((-5/3, -1/2), (-2/3,), b*x**3*exp_polar(I*pi)/ a)/(3*x**5*gamma(-2/3)) + B*sqrt(a)*gamma(-2/3)*hyper((-2/3, -1/2), (1/3,) , b*x**3*exp_polar(I*pi)/a)/(3*x**2*gamma(1/3))
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^6} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{6}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^6,x, algorithm="maxima")
Output:
integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^6, x)
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^6} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{6}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^6,x, algorithm="giac")
Output:
integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^6, x)
Timed out. \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^6} \, dx=\int \frac {\left (B\,x^3+A\right )\,\sqrt {b\,x^3+a}}{x^6} \,d x \] Input:
int(((A + B*x^3)*(a + b*x^3)^(1/2))/x^6,x)
Output:
int(((A + B*x^3)*(a + b*x^3)^(1/2))/x^6, x)
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^6} \, dx=\frac {4 \sqrt {b \,x^{3}+a}\, a -14 \sqrt {b \,x^{3}+a}\, b \,x^{3}+27 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{9}+a \,x^{6}}d x \right ) a^{2} x^{5}}{7 x^{5}} \] Input:
int((b*x^3+a)^(1/2)*(B*x^3+A)/x^6,x)
Output:
(4*sqrt(a + b*x**3)*a - 14*sqrt(a + b*x**3)*b*x**3 + 27*int(sqrt(a + b*x** 3)/(a*x**6 + b*x**9),x)*a**2*x**5)/(7*x**5)