Integrand size = 22, antiderivative size = 336 \[ \int x^3 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {54 a^2 (23 A b-8 a B) x \sqrt {a+b x^3}}{21505 b^2}+\frac {18 a (23 A b-8 a B) x^4 \sqrt {a+b x^3}}{4301 b}+\frac {2 (23 A b-8 a B) x^4 \left (a+b x^3\right )^{3/2}}{391 b}+\frac {2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b}-\frac {36\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 (23 A b-8 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{21505 b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
54/21505*a^2*(23*A*b-8*B*a)*x*(b*x^3+a)^(1/2)/b^2+18/4301*a*(23*A*b-8*B*a) *x^4*(b*x^3+a)^(1/2)/b+2/391*(23*A*b-8*B*a)*x^4*(b*x^3+a)^(3/2)/b+2/23*B*x ^4*(b*x^3+a)^(5/2)/b-36/21505*3^(3/4)*(1/2*6^(1/2)+1/2*2^(1/2))*a^3*(23*A* b-8*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)/b^(7/3)/(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 = 7.80 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.28 \[ \int x^3 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {2 x \sqrt {a+b x^3} \left (-\left (a+b x^3\right )^2 \left (-23 A b+8 a B-17 b B x^3\right )+\frac {a^2 (-23 A b+8 a B) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{391 b^2} \] Input:
Integrate[x^3*(a + b*x^3)^(3/2)*(A + B*x^3),x]
Output:
(2*x*Sqrt[a + b*x^3]*(-((a + b*x^3)^2*(-23*A*b + 8*a*B - 17*b*B*x^3)) + (a ^2*(-23*A*b + 8*a*B)*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)])/Sqrt [1 + (b*x^3)/a]))/(391*b^2)
Time = 0.59 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {959, 811, 811, 843, 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.
| \(\displaystyle \int x^3 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {(23 A b-8 a B) \int x^3 \left (b x^3+a\right )^{3/2}dx}{23 b}+\frac {2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {(23 A b-8 a B) \left (\frac {9}{17} a \int x^3 \sqrt {b x^3+a}dx+\frac {2}{17} x^4 \left (a+b x^3\right )^{3/2}\right )}{23 b}+\frac {2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {(23 A b-8 a B) \left (\frac {9}{17} a \left (\frac {3}{11} a \int \frac {x^3}{\sqrt {b x^3+a}}dx+\frac {2}{11} x^4 \sqrt {a+b x^3}\right )+\frac {2}{17} x^4 \left (a+b x^3\right )^{3/2}\right )}{23 b}+\frac {2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {(23 A b-8 a B) \left (\frac {9}{17} a \left (\frac {3}{11} a \left (\frac {2 x \sqrt {a+b x^3}}{5 b}-\frac {2 a \int \frac {1}{\sqrt {b x^3+a}}dx}{5 b}\right )+\frac {2}{11} x^4 \sqrt {a+b x^3}\right )+\frac {2}{17} x^4 \left (a+b x^3\right )^{3/2}\right )}{23 b}+\frac {2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {(23 A b-8 a B) \left (\frac {9}{17} a \left (\frac {3}{11} a \left (\frac {2 x \sqrt {a+b x^3}}{5 b}-\frac {4 \sqrt {2+\sqrt {3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} b^{4/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )+\frac {2}{11} x^4 \sqrt {a+b x^3}\right )+\frac {2}{17} x^4 \left (a+b x^3\right )^{3/2}\right )}{23 b}+\frac {2 B x^4 \left (a+b x^3\right )^{5/2}}{23 b}\) |
Input:
Int[x^3*(a + b*x^3)^(3/2)*(A + B*x^3),x]
Output:
(2*B*x^4*(a + b*x^3)^(5/2))/(23*b) + ((23*A*b - 8*a*B)*((2*x^4*(a + b*x^3) ^(3/2))/17 + (9*a*((2*x^4*Sqrt[a + b*x^3])/11 + (3*a*((2*x*Sqrt[a + b*x^3] )/(5*b) - (4*Sqrt[2 + Sqrt[3]]*a*(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]*Ellip ticF[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]])/(5*3^(1/4)*b^(4/3)*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])))/11))/ 17))/(23*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* x)^(m + 1)*((a + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a*n*(p/(m + n*p + 1 )) Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
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*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
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]
Time = 1.27 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.11
| method | result | size |
| risch | \(\frac {2 x \left (935 b^{3} B \,x^{9}+1265 A \,b^{3} x^{6}+1430 B a \,b^{2} x^{6}+2300 a A \,b^{2} x^{3}+135 B \,a^{2} b \,x^{3}+621 a^{2} b A -216 a^{3} B \right ) \sqrt {b \,x^{3}+a}}{21505 b^{2}}+\frac {36 i a^{3} \left (23 A b -8 B a \right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{21505 b^{3} \sqrt {b \,x^{3}+a}}\) | \(373\) |
| elliptic | \(\frac {2 B b \,x^{10} \sqrt {b \,x^{3}+a}}{23}+\frac {2 \left (b^{2} A +\frac {26}{23} a b B \right ) x^{7} \sqrt {b \,x^{3}+a}}{17 b}+\frac {2 \left (2 a b A +a^{2} B -\frac {14 a \left (b^{2} A +\frac {26}{23} a b B \right )}{17 b}\right ) x^{4} \sqrt {b \,x^{3}+a}}{11 b}+\frac {2 \left (a^{2} A -\frac {8 a \left (2 a b A +a^{2} B -\frac {14 a \left (b^{2} A +\frac {26}{23} a b B \right )}{17 b}\right )}{11 b}\right ) x \sqrt {b \,x^{3}+a}}{5 b}+\frac {4 i a \left (a^{2} A -\frac {8 a \left (2 a b A +a^{2} B -\frac {14 a \left (b^{2} A +\frac {26}{23} a b B \right )}{17 b}\right )}{11 b}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{15 b^{2} \sqrt {b \,x^{3}+a}}\) | \(469\) |
| default | \(A \left (\frac {2 x^{7} \sqrt {b \,x^{3}+a}\, b}{17}+\frac {40 a \,x^{4} \sqrt {b \,x^{3}+a}}{187}+\frac {54 a^{2} x \sqrt {b \,x^{3}+a}}{935 b}+\frac {36 i a^{3} \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 )}{935 b^{2} \sqrt {b \,x^{3}+a}}\right )+B \left (\frac {2 x^{10} \sqrt {b \,x^{3}+a}\, b}{23}+\frac {52 a \,x^{7} \sqrt {b \,x^{3}+a}}{391}+\frac {54 a^{2} x^{4} \sqrt {b \,x^{3}+a}}{4301 b}-\frac {432 a^{3} x \sqrt {b \,x^{3}+a}}{21505 b^{2}}-\frac {288 i a^{4} \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 )}{21505 b^{3} \sqrt {b \,x^{3}+a}}\right )\) | \(694\) |
Input:
int(x^3*(b*x^3+a)^(3/2)*(B*x^3+A),x,method=_RETURNVERBOSE)
Output:
2/21505/b^2*x*(935*B*b^3*x^9+1265*A*b^3*x^6+1430*B*a*b^2*x^6+2300*A*a*b^2* x^3+135*B*a^2*b*x^3+621*A*a^2*b-216*B*a^3)*(b*x^3+a)^(1/2)+36/21505*I*a^3* (23*A*b-8*B*a)/b^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)*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 = 115, normalized size of antiderivative = 0.34 \[ \int x^3 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {2 \, {\left (54 \, {\left (8 \, B a^{4} - 23 \, A a^{3} b\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (935 \, B b^{4} x^{10} + 55 \, {\left (26 \, B a b^{3} + 23 \, A b^{4}\right )} x^{7} + 5 \, {\left (27 \, B a^{2} b^{2} + 460 \, A a b^{3}\right )} x^{4} - 27 \, {\left (8 \, B a^{3} b - 23 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{21505 \, b^{3}} \] Input:
integrate(x^3*(b*x^3+a)^(3/2)*(B*x^3+A),x, algorithm="fricas")
Output:
2/21505*(54*(8*B*a^4 - 23*A*a^3*b)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) + (935*B*b^4*x^10 + 55*(26*B*a*b^3 + 23*A*b^4)*x^7 + 5*(27*B*a^2*b^2 + 460*A*a*b^3)*x^4 - 27*(8*B*a^3*b - 23*A*a^2*b^2)*x)*sqrt(b*x^3 + a))/b^3
Time = 2.04 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.51 \[ \int x^3 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {A a^{\frac {3}{2}} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {A \sqrt {a} b x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {B a^{\frac {3}{2}} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {B \sqrt {a} b x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {13}{3}\right )} \] Input:
integrate(x**3*(b*x**3+a)**(3/2)*(B*x**3+A),x)
Output:
A*a**(3/2)*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), b*x**3*exp_polar(I*p i)/a)/(3*gamma(7/3)) + A*sqrt(a)*b*x**7*gamma(7/3)*hyper((-1/2, 7/3), (10/ 3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3)) + B*a**(3/2)*x**7*gamma(7/3 )*hyper((-1/2, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3)) + B*sqrt(a)*b*x**10*gamma(10/3)*hyper((-1/2, 10/3), (13/3,), b*x**3*exp_pola r(I*pi)/a)/(3*gamma(13/3))
\[ \int x^3 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{3} \,d x } \] Input:
integrate(x^3*(b*x^3+a)^(3/2)*(B*x^3+A),x, algorithm="maxima")
Output:
integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*x^3, x)
\[ \int x^3 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{3} \,d x } \] Input:
integrate(x^3*(b*x^3+a)^(3/2)*(B*x^3+A),x, algorithm="giac")
Output:
integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*x^3, x)
Timed out. \[ \int x^3 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\int x^3\,\left (B\,x^3+A\right )\,{\left (b\,x^3+a\right )}^{3/2} \,d x \] Input:
int(x^3*(A + B*x^3)*(a + b*x^3)^(3/2),x)
Output:
int(x^3*(A + B*x^3)*(a + b*x^3)^(3/2), x)
\[ \int x^3 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {\frac {162 \sqrt {b \,x^{3}+a}\, a^{3} x}{4301}+\frac {974 \sqrt {b \,x^{3}+a}\, a^{2} b \,x^{4}}{4301}+\frac {98 \sqrt {b \,x^{3}+a}\, a \,b^{2} x^{7}}{391}+\frac {2 \sqrt {b \,x^{3}+a}\, b^{3} x^{10}}{23}-\frac {162 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) a^{4}}{4301}}{b} \] Input:
int(x^3*(b*x^3+a)^(3/2)*(B*x^3+A),x)
Output:
(2*(81*sqrt(a + b*x**3)*a**3*x + 487*sqrt(a + b*x**3)*a**2*b*x**4 + 539*sq rt(a + b*x**3)*a*b**2*x**7 + 187*sqrt(a + b*x**3)*b**3*x**10 - 81*int(sqrt (a + b*x**3)/(a + b*x**3),x)*a**4))/(4301*b)