Integrand size = 22, antiderivative size = 614 \[ \int x^4 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {54 a^2 (5 A b-2 a B) x^2 \sqrt {a+b x^3}}{8645 b^2}+\frac {18 a (5 A b-2 a B) x^5 \sqrt {a+b x^3}}{1235 b}-\frac {216 a^3 (5 A b-2 a B) \sqrt {a+b x^3}}{8645 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 (5 A b-2 a B) x^5 \left (a+b x^3\right )^{3/2}}{95 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}+\frac {108 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} (5 A b-2 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{8645 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {72 \sqrt {2} 3^{3/4} a^{10/3} (5 A b-2 a B) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{8645 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
54/8645*a^2*(5*A*b-2*B*a)*x^2*(b*x^3+a)^(1/2)/b^2+18/1235*a*(5*A*b-2*B*a)* x^5*(b*x^3+a)^(1/2)/b-216/8645*a^3*(5*A*b-2*B*a)*(b*x^3+a)^(1/2)/b^(8/3)/( (1+3^(1/2))*a^(1/3)+b^(1/3)*x)+2/95*(5*A*b-2*B*a)*x^5*(b*x^3+a)^(3/2)/b+2/ 25*B*x^5*(b*x^3+a)^(5/2)/b+108/8645*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*a^(1 0/3)*(5*A*b-2*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)/b^(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)-72/8645*2^(1/2)*3^(3/4)*a^(10/3)*(5*A*b-2*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^(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 = 8.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.16 \[ \int x^4 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {2 x^2 \sqrt {a+b x^3} \left (-\left (a+b x^3\right )^2 \left (-25 A b+10 a B-19 b B x^3\right )+\frac {5 a^2 (-5 A b+2 a B) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{\sqrt {1+\frac {b x^3}{a}}}\right )}{475 b^2} \] Input:
Integrate[x^4*(a + b*x^3)^(3/2)*(A + B*x^3),x]
Output:
(2*x^2*Sqrt[a + b*x^3]*(-((a + b*x^3)^2*(-25*A*b + 10*a*B - 19*b*B*x^3)) + (5*a^2*(-5*A*b + 2*a*B)*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)])/ Sqrt[1 + (b*x^3)/a]))/(475*b^2)
Time = 1.03 (sec) , antiderivative size = 601, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {959, 811, 811, 843, 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.
| \(\displaystyle \int x^4 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {(5 A b-2 a B) \int x^4 \left (b x^3+a\right )^{3/2}dx}{5 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {9}{19} a \int x^4 \sqrt {b x^3+a}dx+\frac {2}{19} x^5 \left (a+b x^3\right )^{3/2}\right )}{5 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}\) |
| \(\Big \downarrow \) 811 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {9}{19} a \left (\frac {3}{13} a \int \frac {x^4}{\sqrt {b x^3+a}}dx+\frac {2}{13} x^5 \sqrt {a+b x^3}\right )+\frac {2}{19} x^5 \left (a+b x^3\right )^{3/2}\right )}{5 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {9}{19} a \left (\frac {3}{13} a \left (\frac {2 x^2 \sqrt {a+b x^3}}{7 b}-\frac {4 a \int \frac {x}{\sqrt {b x^3+a}}dx}{7 b}\right )+\frac {2}{13} x^5 \sqrt {a+b x^3}\right )+\frac {2}{19} x^5 \left (a+b x^3\right )^{3/2}\right )}{5 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}\) |
| \(\Big \downarrow \) 832 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {9}{19} a \left (\frac {3}{13} a \left (\frac {2 x^2 \sqrt {a+b x^3}}{7 b}-\frac {4 a \left (\frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}\right )}{7 b}\right )+\frac {2}{13} x^5 \sqrt {a+b x^3}\right )+\frac {2}{19} x^5 \left (a+b x^3\right )^{3/2}\right )}{5 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {9}{19} a \left (\frac {3}{13} a \left (\frac {2 x^2 \sqrt {a+b x^3}}{7 b}-\frac {4 a \left (\frac {\int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{7 b}\right )+\frac {2}{13} x^5 \sqrt {a+b x^3}\right )+\frac {2}{19} x^5 \left (a+b x^3\right )^{3/2}\right )}{5 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {(5 A b-2 a B) \left (\frac {9}{19} a \left (\frac {3}{13} a \left (\frac {2 x^2 \sqrt {a+b x^3}}{7 b}-\frac {4 a \left (\frac {\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{7 b}\right )+\frac {2}{13} x^5 \sqrt {a+b x^3}\right )+\frac {2}{19} x^5 \left (a+b x^3\right )^{3/2}\right )}{5 b}+\frac {2 B x^5 \left (a+b x^3\right )^{5/2}}{25 b}\) |
Input:
Int[x^4*(a + b*x^3)^(3/2)*(A + B*x^3),x]
Output:
(2*B*x^5*(a + b*x^3)^(5/2))/(25*b) + ((5*A*b - 2*a*B)*((2*x^5*(a + b*x^3)^ (3/2))/19 + (9*a*((2*x^5*Sqrt[a + b*x^3])/13 + (3*a*((2*x^2*Sqrt[a + b*x^3 ])/(7*b) - (4*a*(((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)*S qrt[(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 + Sq rt[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])))/(7*b)))/13))/19))/(5*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[(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^(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]
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 = 1.46 (sec) , antiderivative size = 527, normalized size of antiderivative = 0.86
| method | result | size |
| risch | \(\frac {2 x^{2} \left (1729 b^{3} B \,x^{9}+2275 A \,b^{3} x^{6}+2548 B a \,b^{2} x^{6}+3850 a A \,b^{2} x^{3}+189 B \,a^{2} b \,x^{3}+675 a^{2} b A -270 a^{3} B \right ) \sqrt {b \,x^{3}+a}}{43225 b^{2}}+\frac {72 i a^{3} \left (5 A b -2 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 )}{8645 b^{3} \sqrt {b \,x^{3}+a}}\) | \(527\) |
| elliptic | \(\frac {2 B b \,x^{11} \sqrt {b \,x^{3}+a}}{25}+\frac {2 \left (b^{2} A +\frac {28}{25} a b B \right ) x^{8} \sqrt {b \,x^{3}+a}}{19 b}+\frac {2 \left (2 a b A +a^{2} B -\frac {16 a \left (b^{2} A +\frac {28}{25} a b B \right )}{19 b}\right ) x^{5} \sqrt {b \,x^{3}+a}}{13 b}+\frac {2 \left (a^{2} A -\frac {10 a \left (2 a b A +a^{2} B -\frac {16 a \left (b^{2} A +\frac {28}{25} a b B \right )}{19 b}\right )}{13 b}\right ) x^{2} \sqrt {b \,x^{3}+a}}{7 b}+\frac {8 i a \left (a^{2} A -\frac {10 a \left (2 a b A +a^{2} B -\frac {16 a \left (b^{2} A +\frac {28}{25} a b B \right )}{19 b}\right )}{13 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}}}}\, \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 )}{21 b^{2} \sqrt {b \,x^{3}+a}}\) | \(623\) |
| default | \(\text {Expression too large to display}\) | \(1002\) |
Input:
int(x^4*(b*x^3+a)^(3/2)*(B*x^3+A),x,method=_RETURNVERBOSE)
Output:
2/43225/b^2*x^2*(1729*B*b^3*x^9+2275*A*b^3*x^6+2548*B*a*b^2*x^6+3850*A*a*b ^2*x^3+189*B*a^2*b*x^3+675*A*a^2*b-270*B*a^3)*(b*x^3+a)^(1/2)+72/8645*I*a^ 3*(5*A*b-2*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)*((-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 = 0.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.21 \[ \int x^4 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=-\frac {2 \, {\left (540 \, {\left (2 \, B a^{4} - 5 \, A a^{3} b\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (1729 \, B b^{4} x^{11} + 91 \, {\left (28 \, B a b^{3} + 25 \, A b^{4}\right )} x^{8} + 7 \, {\left (27 \, B a^{2} b^{2} + 550 \, A a b^{3}\right )} x^{5} - 135 \, {\left (2 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{3} + a}\right )}}{43225 \, b^{3}} \] Input:
integrate(x^4*(b*x^3+a)^(3/2)*(B*x^3+A),x, algorithm="fricas")
Output:
-2/43225*(540*(2*B*a^4 - 5*A*a^3*b)*sqrt(b)*weierstrassZeta(0, -4*a/b, wei erstrassPInverse(0, -4*a/b, x)) - (1729*B*b^4*x^11 + 91*(28*B*a*b^3 + 25*A *b^4)*x^8 + 7*(27*B*a^2*b^2 + 550*A*a*b^3)*x^5 - 135*(2*B*a^3*b - 5*A*a^2* b^2)*x^2)*sqrt(b*x^3 + a))/b^3
Time = 2.11 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.28 \[ \int x^4 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {A a^{\frac {3}{2}} x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {A \sqrt {a} b x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} + \frac {B a^{\frac {3}{2}} x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} + \frac {B \sqrt {a} b x^{11} \Gamma \left (\frac {11}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{3} \\ \frac {14}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {14}{3}\right )} \] Input:
integrate(x**4*(b*x**3+a)**(3/2)*(B*x**3+A),x)
Output:
A*a**(3/2)*x**5*gamma(5/3)*hyper((-1/2, 5/3), (8/3,), b*x**3*exp_polar(I*p i)/a)/(3*gamma(8/3)) + A*sqrt(a)*b*x**8*gamma(8/3)*hyper((-1/2, 8/3), (11/ 3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(11/3)) + B*a**(3/2)*x**8*gamma(8/3 )*hyper((-1/2, 8/3), (11/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(11/3)) + B*sqrt(a)*b*x**11*gamma(11/3)*hyper((-1/2, 11/3), (14/3,), b*x**3*exp_pola r(I*pi)/a)/(3*gamma(14/3))
\[ \int x^4 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{4} \,d x } \] Input:
integrate(x^4*(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^4, x)
\[ \int x^4 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{4} \,d x } \] Input:
integrate(x^4*(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^4, x)
Timed out. \[ \int x^4 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\int x^4\,\left (B\,x^3+A\right )\,{\left (b\,x^3+a\right )}^{3/2} \,d x \] Input:
int(x^4*(A + B*x^3)*(a + b*x^3)^(3/2),x)
Output:
int(x^4*(A + B*x^3)*(a + b*x^3)^(3/2), x)
\[ \int x^4 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx=\frac {\frac {162 \sqrt {b \,x^{3}+a}\, a^{3} x^{2}}{8645}+\frac {1154 \sqrt {b \,x^{3}+a}\, a^{2} b \,x^{5}}{6175}+\frac {106 \sqrt {b \,x^{3}+a}\, a \,b^{2} x^{8}}{475}+\frac {2 \sqrt {b \,x^{3}+a}\, b^{3} x^{11}}{25}-\frac {324 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b \,x^{3}+a}d x \right ) a^{4}}{8645}}{b} \] Input:
int(x^4*(b*x^3+a)^(3/2)*(B*x^3+A),x)
Output:
(2*(405*sqrt(a + b*x**3)*a**3*x**2 + 4039*sqrt(a + b*x**3)*a**2*b*x**5 + 4 823*sqrt(a + b*x**3)*a*b**2*x**8 + 1729*sqrt(a + b*x**3)*b**3*x**11 - 810* int((sqrt(a + b*x**3)*x)/(a + b*x**3),x)*a**4))/(43225*b)