Integrand size = 32, antiderivative size = 585 \[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {810 a^3 d \sqrt {a+b x^3}}{1729 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {54 a^2 \left (1729 c x+935 d x^2\right ) \sqrt {a+b x^3}}{323323}+\frac {30 a \left (247 c x+187 d x^2\right ) \left (a+b x^3\right )^{3/2}}{46189}+\frac {2}{323} \left (19 c x+17 d x^2\right ) \left (a+b x^3\right )^{5/2}-\frac {405 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{1729 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {54\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (1729 \sqrt [3]{b} c-935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{323323 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
30/46189*a*(187*d*x^2+247*c*x)*(b*x^3+a)^(3/2)+2/323*(17*d*x^2+19*c*x)*(b* x^3+a)^(5/2)+54/323323*a^2*(935*d*x^2+1729*c*x)*(b*x^3+a)^(1/2)+810/1729*a ^3*d*(b*x^3+a)^(1/2)/b^(2/3)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))-405/1729*3^(1 /4)*a^(10/3)*d*(a^(1/3)+b^(1/3)*x)*EllipticE((b^(1/3)*x+a^(1/3)*(1-3^(1/2) ))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2) )*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2))) ^2)^(1/2)/b^(2/3)/(b*x^3+a)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+ a^(1/3)*(1+3^(1/2)))^2)^(1/2)+54/323323*3^(3/4)*a^3*(a^(1/3)+b^(1/3)*x)*El lipticF((b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2))),I* 3^(1/2)+2*I)*(1729*b^(1/3)*c-935*a^(1/3)*d*(1-3^(1/2)))*(1/2*6^(1/2)+1/2*2 ^(1/2))*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+3^( 1/2)))^2)^(1/2)/b^(2/3)/(b*x^3+a)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1 /3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.59 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.13 \[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {a^2 x \sqrt {a+b x^3} \left (2 c \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )+d x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{2 \sqrt {1+\frac {b x^3}{a}}} \]
(a^2*x*Sqrt[a + b*x^3]*(2*c*Hypergeometric2F1[-5/2, 1/3, 4/3, -((b*x^3)/a) ] + d*x*Hypergeometric2F1[-5/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[1 + (b*x ^3)/a])
Time = 0.76 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2391, 2392, 27, 2392, 27, 2392, 27, 2417, 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.
| \(\displaystyle \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2391 |
| \(\displaystyle \int \left (a+b x^3\right )^{5/2} (c+d x)dx\) |
| \(\Big \downarrow \) 2392 |
| \(\displaystyle \frac {15}{2} a \int \frac {2}{323} (19 c+17 d x) \left (b x^3+a\right )^{3/2}dx+\frac {2}{323} \left (a+b x^3\right )^{5/2} \left (19 c x+17 d x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {15}{323} a \int (19 c+17 d x) \left (b x^3+a\right )^{3/2}dx+\frac {2}{323} \left (a+b x^3\right )^{5/2} \left (19 c x+17 d x^2\right )\) |
| \(\Big \downarrow \) 2392 |
| \(\displaystyle \frac {15}{323} a \left (\frac {9}{2} a \int \frac {2}{143} (247 c+187 d x) \sqrt {b x^3+a}dx+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (247 c x+187 d x^2\right )\right )+\frac {2}{323} \left (a+b x^3\right )^{5/2} \left (19 c x+17 d x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {15}{323} a \left (\frac {9}{143} a \int (247 c+187 d x) \sqrt {b x^3+a}dx+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (247 c x+187 d x^2\right )\right )+\frac {2}{323} \left (a+b x^3\right )^{5/2} \left (19 c x+17 d x^2\right )\) |
| \(\Big \downarrow \) 2392 |
| \(\displaystyle \frac {15}{323} a \left (\frac {9}{143} a \left (\frac {3}{2} a \int \frac {2 (1729 c+935 d x)}{35 \sqrt {b x^3+a}}dx+\frac {2}{35} \sqrt {a+b x^3} \left (1729 c x+935 d x^2\right )\right )+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (247 c x+187 d x^2\right )\right )+\frac {2}{323} \left (a+b x^3\right )^{5/2} \left (19 c x+17 d x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {15}{323} a \left (\frac {9}{143} a \left (\frac {3}{35} a \int \frac {1729 c+935 d x}{\sqrt {b x^3+a}}dx+\frac {2}{35} \sqrt {a+b x^3} \left (1729 c x+935 d x^2\right )\right )+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (247 c x+187 d x^2\right )\right )+\frac {2}{323} \left (a+b x^3\right )^{5/2} \left (19 c x+17 d x^2\right )\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {15}{323} a \left (\frac {9}{143} a \left (\frac {3}{35} a \left (\left (1729 c-\frac {935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt {b x^3+a}}dx+\frac {935 d \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}\right )+\frac {2}{35} \sqrt {a+b x^3} \left (1729 c x+935 d x^2\right )\right )+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (247 c x+187 d x^2\right )\right )+\frac {2}{323} \left (a+b x^3\right )^{5/2} \left (19 c x+17 d x^2\right )\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {15}{323} a \left (\frac {9}{143} a \left (\frac {3}{35} a \left (\frac {935 d \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}+\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (1729 c-\frac {935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )+\frac {2}{35} \sqrt {a+b x^3} \left (1729 c x+935 d x^2\right )\right )+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (247 c x+187 d x^2\right )\right )+\frac {2}{323} \left (a+b x^3\right )^{5/2} \left (19 c x+17 d x^2\right )\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {15}{323} a \left (\frac {9}{143} a \left (\frac {3}{35} a \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (1729 c-\frac {935 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {935 d \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{\sqrt [3]{b}}\right )+\frac {2}{35} \sqrt {a+b x^3} \left (1729 c x+935 d x^2\right )\right )+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (247 c x+187 d x^2\right )\right )+\frac {2}{323} \left (a+b x^3\right )^{5/2} \left (19 c x+17 d x^2\right )\) |
(2*(19*c*x + 17*d*x^2)*(a + b*x^3)^(5/2))/323 + (15*a*((2*(247*c*x + 187*d *x^2)*(a + b*x^3)^(3/2))/143 + (9*a*((2*(1729*c*x + 935*d*x^2)*Sqrt[a + b* x^3])/35 + (3*a*((935*d*((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*Sqrt[2 + Sqrt[3]]*(1729*c - (935*(1 - Sqrt[3])*a^(1/3)* d)/b^(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 - Sqr t[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sq rt[3]])/(3^(1/4)*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])))/35))/143))/323
3.1.59.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[PolynomialQuoti ent[Pq, a + b*x^n, x]*(a + b*x^n)^(p + 1), x] /; FreeQ[{a, b, p}, x] && Pol yQ[Pq, x] && IGtQ[n, 0] && GeQ[Expon[Pq, x], n] && EqQ[PolynomialRemainder[ Pq, a + b*x^n, x], 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{q = Expon[Pq , x], i}, Simp[(a + b*x^n)^p*Sum[Coeff[Pq, x, i]*(x^(i + 1)/(n*p + i + 1)), {i, 0, q}], x] + Simp[a*n*p Int[(a + b*x^n)^(p - 1)*Sum[Coeff[Pq, x, i]* (x^i/(n*p + i + 1)), {i, 0, q}], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x ] && IGtQ[(n - 1)/2, 0] && GtQ[p, 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]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(c*r - (1 - Sqrt[3])*d*s)/r Int[1/Sqrt[a + b*x^3], x], x] + Simp[d/r Int[((1 - Sqrt[3])*s + r*x)/Sq rt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Time = 1.72 (sec) , antiderivative size = 786, normalized size of antiderivative = 1.34
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(786\) |
| elliptic | \(\text {Expression too large to display}\) | \(830\) |
| default | \(\text {Expression too large to display}\) | \(1618\) |
2/323323*x*(17017*b^2*d*x^7+19019*b^2*c*x^6+53669*a*b*d*x^4+63973*a*b*c*x^ 3+61897*a^2*d*x+91637*a^2*c)*(b*x^3+a)^(1/2)+81/323323*a^3*(-3458/3*I*c*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)*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))-1870/3 *I*d*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))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.20 \[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {2 \, {\left (140049 \, a^{3} \sqrt {b} c {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 75735 \, a^{3} \sqrt {b} d {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (17017 \, b^{3} d x^{8} + 19019 \, b^{3} c x^{7} + 53669 \, a b^{2} d x^{5} + 63973 \, a b^{2} c x^{4} + 61897 \, a^{2} b d x^{2} + 91637 \, a^{2} b c x\right )} \sqrt {b x^{3} + a}\right )}}{323323 \, b} \]
2/323323*(140049*a^3*sqrt(b)*c*weierstrassPInverse(0, -4*a/b, x) - 75735*a ^3*sqrt(b)*d*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (17017*b^3*d*x^8 + 19019*b^3*c*x^7 + 53669*a*b^2*d*x^5 + 63973*a*b^2*c* x^4 + 61897*a^2*b*d*x^2 + 91637*a^2*b*c*x)*sqrt(b*x^3 + a))/b
Time = 2.64 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.45 \[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {a^{\frac {5}{2}} c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a^{\frac {5}{2}} d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {2 a^{\frac {3}{2}} b c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {2 a^{\frac {3}{2}} b d x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {\sqrt {a} b^{2} c x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {\sqrt {a} b^{2} d x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} \]
a**(5/2)*c*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/ a)/(3*gamma(4/3)) + a**(5/2)*d*x**2*gamma(2/3)*hyper((-1/2, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(5/3)) + 2*a**(3/2)*b*c*x**4*gamma(4/3)* hyper((-1/2, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3)) + 2*a* *(3/2)*b*d*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)) + sqrt(a)*b**2*c*x**7*gamma(7/3)*hyper((-1/2, 7/3), ( 10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3)) + sqrt(a)*b**2*d*x**8*ga mma(8/3)*hyper((-1/2, 8/3), (11/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(11 /3))
\[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\int { {\left (b d x^{4} + b c x^{3} + a d x + a c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} \,d x } \]
\[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\int { {\left (b d x^{4} + b c x^{3} + a d x + a c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \left (a+b x^3\right )^{3/2} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\int {\left (b\,x^3+a\right )}^{3/2}\,\left (b\,d\,x^4+b\,c\,x^3+a\,d\,x+a\,c\right ) \,d x \]