Integrand size = 22, antiderivative size = 614 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{11}} \, dx=\frac {(11 A b-20 a B) \sqrt {a+b x^3}}{140 a x^7}+\frac {3 b (11 A b-20 a B) \sqrt {a+b x^3}}{1120 a^2 x^4}-\frac {3 b^2 (11 A b-20 a B) \sqrt {a+b x^3}}{448 a^3 x}+\frac {3 b^{7/3} (11 A b-20 a B) \sqrt {a+b x^3}}{448 a^3 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {A \left (a+b x^3\right )^{3/2}}{10 a x^{10}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b^{7/3} (11 A b-20 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{896 a^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {3^{3/4} b^{7/3} (11 A b-20 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{224 \sqrt {2} a^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
1/140*(11*A*b-20*B*a)*(b*x^3+a)^(1/2)/a/x^7+3/1120*b*(11*A*b-20*B*a)*(b*x^ 3+a)^(1/2)/a^2/x^4-3/448*b^2*(11*A*b-20*B*a)*(b*x^3+a)^(1/2)/a^3/x+3/448*b ^(7/3)*(11*A*b-20*B*a)*(b*x^3+a)^(1/2)/a^3/((1+3^(1/2))*a^(1/3)+b^(1/3)*x) -1/10*A*(b*x^3+a)^(3/2)/a/x^10-3/896*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*b^( 7/3)*(11*A*b-20*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)*EllipticE(((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^(8/3)/(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)+1/448*3^(3/4)*b^(7/3)*(11*A*b-20*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)*2^(1/2)/a^(8/3)/(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 = 10.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.13 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{11}} \, dx=\frac {\sqrt {a+b x^3} \left (-7 A \left (a+b x^3\right )+\frac {\left (\frac {11 A b}{2}-10 a B\right ) x^3 \operatorname {Hypergeometric2F1}\left (-\frac {7}{3},-\frac {1}{2},-\frac {4}{3},-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{70 a x^{10}} \] Input:
Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^11,x]
Output:
(Sqrt[a + b*x^3]*(-7*A*(a + b*x^3) + (((11*A*b)/2 - 10*a*B)*x^3*Hypergeome tric2F1[-7/3, -1/2, -4/3, -((b*x^3)/a)])/Sqrt[1 + (b*x^3)/a]))/(70*a*x^10)
Time = 1.04 (sec) , antiderivative size = 605, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {955, 809, 847, 847, 832, 759, 2416}
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.
| \(\displaystyle \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{11}} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle -\frac {(11 A b-20 a B) \int \frac {\sqrt {b x^3+a}}{x^8}dx}{20 a}-\frac {A \left (a+b x^3\right )^{3/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 809 |
| \(\displaystyle -\frac {(11 A b-20 a B) \left (\frac {3}{14} b \int \frac {1}{x^5 \sqrt {b x^3+a}}dx-\frac {\sqrt {a+b x^3}}{7 x^7}\right )}{20 a}-\frac {A \left (a+b x^3\right )^{3/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {(11 A b-20 a B) \left (\frac {3}{14} b \left (-\frac {5 b \int \frac {1}{x^2 \sqrt {b x^3+a}}dx}{8 a}-\frac {\sqrt {a+b x^3}}{4 a x^4}\right )-\frac {\sqrt {a+b x^3}}{7 x^7}\right )}{20 a}-\frac {A \left (a+b x^3\right )^{3/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle -\frac {(11 A b-20 a B) \left (\frac {3}{14} b \left (-\frac {5 b \left (\frac {b \int \frac {x}{\sqrt {b x^3+a}}dx}{2 a}-\frac {\sqrt {a+b x^3}}{a x}\right )}{8 a}-\frac {\sqrt {a+b x^3}}{4 a x^4}\right )-\frac {\sqrt {a+b x^3}}{7 x^7}\right )}{20 a}-\frac {A \left (a+b x^3\right )^{3/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle -\frac {(11 A b-20 a B) \left (\frac {3}{14} b \left (-\frac {5 b \left (\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}\right )}{2 a}-\frac {\sqrt {a+b x^3}}{a x}\right )}{8 a}-\frac {\sqrt {a+b x^3}}{4 a x^4}\right )-\frac {\sqrt {a+b x^3}}{7 x^7}\right )}{20 a}-\frac {A \left (a+b x^3\right )^{3/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle -\frac {(11 A b-20 a B) \left (\frac {3}{14} b \left (-\frac {5 b \left (\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{2 a}-\frac {\sqrt {a+b x^3}}{a x}\right )}{8 a}-\frac {\sqrt {a+b x^3}}{4 a x^4}\right )-\frac {\sqrt {a+b x^3}}{7 x^7}\right )}{20 a}-\frac {A \left (a+b x^3\right )^{3/2}}{10 a x^{10}}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle -\frac {(11 A b-20 a B) \left (\frac {3}{14} b \left (-\frac {5 b \left (\frac {b \left (\frac {\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{2 a}-\frac {\sqrt {a+b x^3}}{a x}\right )}{8 a}-\frac {\sqrt {a+b x^3}}{4 a x^4}\right )-\frac {\sqrt {a+b x^3}}{7 x^7}\right )}{20 a}-\frac {A \left (a+b x^3\right )^{3/2}}{10 a x^{10}}\) |
Input:
Int[(Sqrt[a + b*x^3]*(A + B*x^3))/x^11,x]
Output:
-1/10*(A*(a + b*x^3)^(3/2))/(a*x^10) - ((11*A*b - 20*a*B)*(-1/7*Sqrt[a + b *x^3]/x^7 + (3*b*(-1/4*Sqrt[a + b*x^3]/(a*x^4) - (5*b*(-(Sqrt[a + b*x^3]/( a*x)) + (b*(((2*Sqrt[a + b*x^3])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3) *x)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(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]*EllipticE[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]])/(b^(1/3)*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]))/b^(1/3) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*a^(1/3)*(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]])/(3^(1/4)*b^(2/3)*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])))/(2*a)))/(8*a)))/14))/(20*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[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 - Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], 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 + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[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]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt [3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) *s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Time = 2.81 (sec) , antiderivative size = 530, normalized size of antiderivative = 0.86
| method | result | size |
| risch | \(-\frac {\sqrt {b \,x^{3}+a}\, \left (165 A \,x^{9} b^{3}-300 B \,x^{9} a \,b^{2}-66 A \,x^{6} a \,b^{2}+120 B \,x^{6} a^{2} b +48 a^{2} A b \,x^{3}+320 B \,x^{3} a^{3}+224 a^{3} A \right )}{2240 x^{10} a^{3}}-\frac {i b^{2} \left (11 A b -20 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}}}}\, \left (\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 ) \operatorname {EllipticE}\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 )+\frac {\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 )}{b}\right )}{448 a^{3} \sqrt {b \,x^{3}+a}}\) | \(530\) |
| elliptic | \(-\frac {A \sqrt {b \,x^{3}+a}}{10 x^{10}}-\frac {\left (3 A b +20 B a \right ) \sqrt {b \,x^{3}+a}}{140 a \,x^{7}}+\frac {3 b \left (11 A b -20 B a \right ) \sqrt {b \,x^{3}+a}}{1120 a^{2} x^{4}}-\frac {3 b^{2} \left (11 A b -20 B a \right ) \sqrt {b \,x^{3}+a}}{448 a^{3} x}-\frac {i b^{2} \left (11 A b -20 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}}}}\, \left (\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 ) \operatorname {EllipticE}\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 )+\frac {\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 )}{b}\right )}{448 a^{3} \sqrt {b \,x^{3}+a}}\) | \(545\) |
| default | \(\text {Expression too large to display}\) | \(1006\) |
Input:
int((b*x^3+a)^(1/2)*(B*x^3+A)/x^11,x,method=_RETURNVERBOSE)
Output:
-1/2240*(b*x^3+a)^(1/2)*(165*A*b^3*x^9-300*B*a*b^2*x^9-66*A*a*b^2*x^6+120* B*a^2*b*x^6+48*A*a^2*b*x^3+320*B*a^3*x^3+224*A*a^3)/x^10/a^3-1/448*I*b^2*( 11*A*b-20*B*a)/a^3*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)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a *b^2)^(1/3))*EllipticE(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))+1/b*( -a*b^2)^(1/3)*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 = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.20 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{11}} \, dx=\frac {15 \, {\left (20 \, B a b^{2} - 11 \, A b^{3}\right )} \sqrt {b} x^{10} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (15 \, {\left (20 \, B a b^{2} - 11 \, A b^{3}\right )} x^{9} - 6 \, {\left (20 \, B a^{2} b - 11 \, A a b^{2}\right )} x^{6} - 224 \, A a^{3} - 16 \, {\left (20 \, B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{2240 \, a^{3} x^{10}} \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^11,x, algorithm="fricas")
Output:
1/2240*(15*(20*B*a*b^2 - 11*A*b^3)*sqrt(b)*x^10*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (15*(20*B*a*b^2 - 11*A*b^3)*x^9 - 6* (20*B*a^2*b - 11*A*a*b^2)*x^6 - 224*A*a^3 - 16*(20*B*a^3 + 3*A*a^2*b)*x^3) *sqrt(b*x^3 + a))/(a^3*x^10)
Time = 1.71 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{11}} \, dx=\frac {A \sqrt {a} \Gamma \left (- \frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {10}{3}, - \frac {1}{2} \\ - \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{10} \Gamma \left (- \frac {7}{3}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, - \frac {1}{2} \\ - \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{7} \Gamma \left (- \frac {4}{3}\right )} \] Input:
integrate((b*x**3+a)**(1/2)*(B*x**3+A)/x**11,x)
Output:
A*sqrt(a)*gamma(-10/3)*hyper((-10/3, -1/2), (-7/3,), b*x**3*exp_polar(I*pi )/a)/(3*x**10*gamma(-7/3)) + B*sqrt(a)*gamma(-7/3)*hyper((-7/3, -1/2), (-4 /3,), b*x**3*exp_polar(I*pi)/a)/(3*x**7*gamma(-4/3))
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{11}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{11}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^11,x, algorithm="maxima")
Output:
integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^11, x)
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{11}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{11}} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^11,x, algorithm="giac")
Output:
integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^11, x)
Timed out. \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{11}} \, dx=\int \frac {\left (B\,x^3+A\right )\,\sqrt {b\,x^3+a}}{x^{11}} \,d x \] Input:
int(((A + B*x^3)*(a + b*x^3)^(1/2))/x^11,x)
Output:
int(((A + B*x^3)*(a + b*x^3)^(1/2))/x^11, x)
\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{11}} \, dx=\frac {-16 \sqrt {b \,x^{3}+a}\, a -34 \sqrt {b \,x^{3}+a}\, b \,x^{3}+27 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{14}+a \,x^{11}}d x \right ) a^{2} x^{10}}{187 x^{10}} \] Input:
int((b*x^3+a)^(1/2)*(B*x^3+A)/x^11,x)
Output:
( - 16*sqrt(a + b*x**3)*a - 34*sqrt(a + b*x**3)*b*x**3 + 27*int(sqrt(a + b *x**3)/(a*x**11 + b*x**14),x)*a**2*x**10)/(187*x**10)