Integrand size = 22, antiderivative size = 576 \[ \int \frac {x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=-\frac {2 (A b-a B) x^5}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac {2 (10 A b-19 a B) x^2}{27 b^3 \sqrt {a+b x^3}}+\frac {2 B x^2 \sqrt {a+b x^3}}{7 b^3}+\frac {80 (7 A b-16 a B) \sqrt {a+b x^3}}{189 b^{11/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {40 \sqrt {2-\sqrt {3}} \sqrt [3]{a} (7 A b-16 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{63\ 3^{3/4} b^{11/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {80 \sqrt {2} \sqrt [3]{a} (7 A b-16 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{189 \sqrt [4]{3} b^{11/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
-2/9*(A*b-B*a)*x^5/b^2/(b*x^3+a)^(3/2)-2/27*(10*A*b-19*B*a)*x^2/b^3/(b*x^3 +a)^(1/2)+2/7*B*x^2*(b*x^3+a)^(1/2)/b^3+80/189*(7*A*b-16*B*a)*(b*x^3+a)^(1 /2)/b^(11/3)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)-40/189*(1/2*6^(1/2)-1/2*2^(1/ 2))*a^(1/3)*(7*A*b-16*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)*3^(1 /4)/b^(11/3)/(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)+80/567*2^(1/2)*a^(1/3)*(7*A*b-16*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)*3^(3/4)/b^(11/3)/(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 = 109, normalized size of antiderivative = 0.19 \[ \int \frac {x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {2 x^2 \left (-32 a^2 B+2 a b \left (7 A-8 B x^3\right )+b^2 x^3 \left (7 A+B x^3\right )+2 (-7 A b+16 a B) \left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {5}{2},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{7 b^3 \left (a+b x^3\right )^{3/2}} \] Input:
Integrate[(x^7*(A + B*x^3))/(a + b*x^3)^(5/2),x]
Output:
(2*x^2*(-32*a^2*B + 2*a*b*(7*A - 8*B*x^3) + b^2*x^3*(7*A + B*x^3) + 2*(-7* A*b + 16*a*B)*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 5/2, 5/3, -((b*x^3)/a)]))/(7*b^3*(a + b*x^3)^(3/2))
Time = 1.02 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {959, 817, 817, 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.
| \(\displaystyle \int \frac {x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {(7 A b-16 a B) \int \frac {x^7}{\left (b x^3+a\right )^{5/2}}dx}{7 b}+\frac {2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {(7 A b-16 a B) \left (\frac {10 \int \frac {x^4}{\left (b x^3+a\right )^{3/2}}dx}{9 b}-\frac {2 x^5}{9 b \left (a+b x^3\right )^{3/2}}\right )}{7 b}+\frac {2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {(7 A b-16 a B) \left (\frac {10 \left (\frac {4 \int \frac {x}{\sqrt {b x^3+a}}dx}{3 b}-\frac {2 x^2}{3 b \sqrt {a+b x^3}}\right )}{9 b}-\frac {2 x^5}{9 b \left (a+b x^3\right )^{3/2}}\right )}{7 b}+\frac {2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle \frac {(7 A b-16 a B) \left (\frac {10 \left (\frac {4 \left (\frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}\right )}{3 b}-\frac {2 x^2}{3 b \sqrt {a+b x^3}}\right )}{9 b}-\frac {2 x^5}{9 b \left (a+b x^3\right )^{3/2}}\right )}{7 b}+\frac {2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {(7 A b-16 a B) \left (\frac {10 \left (\frac {4 \left (\frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{3 b}-\frac {2 x^2}{3 b \sqrt {a+b x^3}}\right )}{9 b}-\frac {2 x^5}{9 b \left (a+b x^3\right )^{3/2}}\right )}{7 b}+\frac {2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {(7 A b-16 a B) \left (\frac {10 \left (\frac {4 \left (\frac {\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{3 b}-\frac {2 x^2}{3 b \sqrt {a+b x^3}}\right )}{9 b}-\frac {2 x^5}{9 b \left (a+b x^3\right )^{3/2}}\right )}{7 b}+\frac {2 B x^8}{7 b \left (a+b x^3\right )^{3/2}}\) |
Input:
Int[(x^7*(A + B*x^3))/(a + b*x^3)^(5/2),x]
Output:
(2*B*x^8)/(7*b*(a + b*x^3)^(3/2)) + ((7*A*b - 16*a*B)*((-2*x^5)/(9*b*(a + b*x^3)^(3/2)) + (10*((-2*x^2)/(3*b*Sqrt[a + b*x^3]) + (4*(((2*Sqrt[a + b*x ^3])/(b^(1/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (3^(1/4)*Sqrt[2 - Sqr t[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 - S qrt[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])))/(3*b)))/(9*b)))/(7*b)
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^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
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 = 3.46 (sec) , antiderivative size = 555, normalized size of antiderivative = 0.96
| method | result | size |
| elliptic | \(\frac {2 x^{2} a \left (A b -B a \right ) \sqrt {b \,x^{3}+a}}{9 b^{5} \left (x^{3}+\frac {a}{b}\right )^{2}}-\frac {2 x^{2} \left (13 A b -22 B a \right )}{27 b^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}+\frac {2 B \,x^{2} \sqrt {b \,x^{3}+a}}{7 b^{3}}-\frac {2 i \left (\frac {A b -2 B a}{b^{3}}+\frac {13 A b -22 B a}{27 b^{3}}-\frac {4 B a}{7 b^{3}}\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 )}{3 b \sqrt {b \,x^{3}+a}}\) | \(555\) |
| default | \(\text {Expression too large to display}\) | \(997\) |
| risch | \(\text {Expression too large to display}\) | \(1443\) |
Input:
int(x^7*(B*x^3+A)/(b*x^3+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/9*x^2*a/b^5*(A*b-B*a)*(b*x^3+a)^(1/2)/(x^3+a/b)^2-2/27/b^3*x^2*(13*A*b-2 2*B*a)/((x^3+a/b)*b)^(1/2)+2/7*B*x^2*(b*x^3+a)^(1/2)/b^3-2/3*I*((A*b-2*B*a )/b^3+1/27/b^3*(13*A*b-22*B*a)-4/7*B/b^3*a)*3^(1/2)/b*(-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 = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.28 \[ \int \frac {x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {2 \, {\left (40 \, {\left ({\left (16 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 16 \, B a^{3} - 7 \, A a^{2} b + 2 \, {\left (16 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (27 \, B b^{3} x^{8} + 13 \, {\left (16 \, B a b^{2} - 7 \, A b^{3}\right )} x^{5} + 10 \, {\left (16 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b x^{3} + a}\right )}}{189 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \] Input:
integrate(x^7*(B*x^3+A)/(b*x^3+a)^(5/2),x, algorithm="fricas")
Output:
2/189*(40*((16*B*a*b^2 - 7*A*b^3)*x^6 + 16*B*a^3 - 7*A*a^2*b + 2*(16*B*a^2 *b - 7*A*a*b^2)*x^3)*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstrassPInvers e(0, -4*a/b, x)) + (27*B*b^3*x^8 + 13*(16*B*a*b^2 - 7*A*b^3)*x^5 + 10*(16* B*a^2*b - 7*A*a*b^2)*x^2)*sqrt(b*x^3 + a))/(b^6*x^6 + 2*a*b^5*x^3 + a^2*b^ 4)
Time = 82.42 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.14 \[ \int \frac {x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {A x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{2}} \Gamma \left (\frac {11}{3}\right )} + \frac {B x^{11} \Gamma \left (\frac {11}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, \frac {11}{3} \\ \frac {14}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{2}} \Gamma \left (\frac {14}{3}\right )} \] Input:
integrate(x**7*(B*x**3+A)/(b*x**3+a)**(5/2),x)
Output:
A*x**8*gamma(8/3)*hyper((5/2, 8/3), (11/3,), b*x**3*exp_polar(I*pi)/a)/(3* a**(5/2)*gamma(11/3)) + B*x**11*gamma(11/3)*hyper((5/2, 11/3), (14/3,), b* x**3*exp_polar(I*pi)/a)/(3*a**(5/2)*gamma(14/3))
\[ \int \frac {x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} x^{7}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^7*(B*x^3+A)/(b*x^3+a)^(5/2),x, algorithm="maxima")
Output:
integrate((B*x^3 + A)*x^7/(b*x^3 + a)^(5/2), x)
\[ \int \frac {x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\int { \frac {{\left (B x^{3} + A\right )} x^{7}}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^7*(B*x^3+A)/(b*x^3+a)^(5/2),x, algorithm="giac")
Output:
integrate((B*x^3 + A)*x^7/(b*x^3 + a)^(5/2), x)
Timed out. \[ \int \frac {x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\int \frac {x^7\,\left (B\,x^3+A\right )}{{\left (b\,x^3+a\right )}^{5/2}} \,d x \] Input:
int((x^7*(A + B*x^3))/(a + b*x^3)^(5/2),x)
Output:
int((x^7*(A + B*x^3))/(a + b*x^3)^(5/2), x)
\[ \int \frac {x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {-\frac {20 \sqrt {b \,x^{3}+a}\, a \,x^{2}}{7}+\frac {2 \sqrt {b \,x^{3}+a}\, b \,x^{5}}{7}+\frac {40 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{3}}{7}+\frac {40 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{2} b \,x^{3}}{7}}{b^{2} \left (b \,x^{3}+a \right )} \] Input:
int(x^7*(B*x^3+A)/(b*x^3+a)^(5/2),x)
Output:
(2*( - 10*sqrt(a + b*x**3)*a*x**2 + sqrt(a + b*x**3)*b*x**5 + 20*int((sqrt (a + b*x**3)*x)/(a**2 + 2*a*b*x**3 + b**2*x**6),x)*a**3 + 20*int((sqrt(a + b*x**3)*x)/(a**2 + 2*a*b*x**3 + b**2*x**6),x)*a**2*b*x**3))/(7*b**2*(a + b*x**3))