Integrand size = 32, antiderivative size = 581 \[ \int \frac {a c+a d x+b c x^3+b d x^4}{\left (a+b x^3\right )^{9/2}} \, dx=\frac {2 x (c+d x)}{15 a \left (a+b x^3\right )^{5/2}}+\frac {2 x (13 c+11 d x)}{135 a^2 \left (a+b x^3\right )^{3/2}}+\frac {2 x (91 c+55 d x)}{405 a^3 \sqrt {a+b x^3}}-\frac {22 d \sqrt {a+b x^3}}{81 a^3 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {11 \sqrt {2-\sqrt {3}} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{27\ 3^{3/4} a^{8/3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (91 \sqrt [3]{b} c+55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{405 \sqrt [4]{3} a^3 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
2/15*x*(d*x+c)/a/(b*x^3+a)^(5/2)+2/135*x*(11*d*x+13*c)/a^2/(b*x^3+a)^(3/2) +2/405*x*(55*d*x+91*c)/a^3/(b*x^3+a)^(1/2)-22/81*d*(b*x^3+a)^(1/2)/a^3/b^( 2/3)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))+11/81*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)*3^(1/4)/a^(8/3)/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 )+2/1215*(a^(1/3)+b^(1/3)*x)*EllipticF((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)*(91*b^(1/3)*c+55*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)*3^(3/4)/a^3/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 = 10.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.24 \[ \int \frac {a c+a d x+b c x^3+b d x^4}{\left (a+b x^3\right )^{9/2}} \, dx=\frac {4 c x \left (157 a^2+221 a b x^3+91 b^2 x^6\right )+182 c x \left (a+b x^3\right )^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a}\right )+405 d x^2 \left (a+b x^3\right )^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{2},\frac {5}{3},-\frac {b x^3}{a}\right )}{810 a^3 \left (a+b x^3\right )^{5/2}} \]
(4*c*x*(157*a^2 + 221*a*b*x^3 + 91*b^2*x^6) + 182*c*x*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 405*d*x^2*( a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 7/2, 5/3, -((b*x^3 )/a)])/(810*a^3*(a + b*x^3)^(5/2))
Time = 0.75 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2019, 2394, 27, 2394, 27, 2394, 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.
| \(\displaystyle \int \frac {a c+a d x+b c x^3+b d x^4}{\left (a+b x^3\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \int \frac {c+d x}{\left (a+b x^3\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {2 x (c+d x)}{15 a \left (a+b x^3\right )^{5/2}}-\frac {2 \int -\frac {13 c+11 d x}{2 \left (b x^3+a\right )^{5/2}}dx}{15 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {13 c+11 d x}{\left (b x^3+a\right )^{5/2}}dx}{15 a}+\frac {2 x (c+d x)}{15 a \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {2 x (13 c+11 d x)}{9 a \left (a+b x^3\right )^{3/2}}-\frac {2 \int -\frac {91 c+55 d x}{2 \left (b x^3+a\right )^{3/2}}dx}{9 a}}{15 a}+\frac {2 x (c+d x)}{15 a \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {91 c+55 d x}{\left (b x^3+a\right )^{3/2}}dx}{9 a}+\frac {2 x (13 c+11 d x)}{9 a \left (a+b x^3\right )^{3/2}}}{15 a}+\frac {2 x (c+d x)}{15 a \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {\frac {2 x (91 c+55 d x)}{3 a \sqrt {a+b x^3}}-\frac {2 \int -\frac {91 c-55 d x}{2 \sqrt {b x^3+a}}dx}{3 a}}{9 a}+\frac {2 x (13 c+11 d x)}{9 a \left (a+b x^3\right )^{3/2}}}{15 a}+\frac {2 x (c+d x)}{15 a \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {91 c-55 d x}{\sqrt {b x^3+a}}dx}{3 a}+\frac {2 x (91 c+55 d x)}{3 a \sqrt {a+b x^3}}}{9 a}+\frac {2 x (13 c+11 d x)}{9 a \left (a+b x^3\right )^{3/2}}}{15 a}+\frac {2 x (c+d x)}{15 a \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {\frac {\frac {\left (\frac {55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}+91 c\right ) \int \frac {1}{\sqrt {b x^3+a}}dx-\frac {55 d \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}}{3 a}+\frac {2 x (91 c+55 d x)}{3 a \sqrt {a+b x^3}}}{9 a}+\frac {2 x (13 c+11 d x)}{9 a \left (a+b x^3\right )^{3/2}}}{15 a}+\frac {2 x (c+d x)}{15 a \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (\frac {55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}+91 c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {55 d \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}}{3 a}+\frac {2 x (91 c+55 d x)}{3 a \sqrt {a+b x^3}}}{9 a}+\frac {2 x (13 c+11 d x)}{9 a \left (a+b x^3\right )^{3/2}}}{15 a}+\frac {2 x (c+d x)}{15 a \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (\frac {55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}+91 c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {55 d \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{\sqrt [3]{b}}}{3 a}+\frac {2 x (91 c+55 d x)}{3 a \sqrt {a+b x^3}}}{9 a}+\frac {2 x (13 c+11 d x)}{9 a \left (a+b x^3\right )^{3/2}}}{15 a}+\frac {2 x (c+d x)}{15 a \left (a+b x^3\right )^{5/2}}\) |
(2*x*(c + d*x))/(15*a*(a + b*x^3)^(5/2)) + ((2*x*(13*c + 11*d*x))/(9*a*(a + b*x^3)^(3/2)) + ((2*x*(91*c + 55*d*x))/(3*a*Sqrt[a + b*x^3]) + ((-55*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]]*(91*c + (55*(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 - 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^(1/3)*Sqr t[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*S qrt[a + b*x^3]))/(3*a))/(9*a))/(15*a)
3.1.65.3.1 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[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b *x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) Int[ExpandToSum[n *(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x ] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 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]
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 = 1.51 (sec) , antiderivative size = 853, normalized size of antiderivative = 1.47
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(853\) |
| default | \(\text {Expression too large to display}\) | \(1902\) |
(2/15*d/a/b^3*x^2+2/15*c/a/b^3*x)*(b*x^3+a)^(1/2)/(x^3+a/b)^3+(22/135/b^2/ a^2*d*x^2+26/135/b^2/a^2*c*x)*(b*x^3+a)^(1/2)/(x^3+a/b)^2-2*b*(-11/81/a^3/ b*d*x^2-91/405/a^3/b*c*x)/((x^3+a/b)*b)^(1/2)-182/1215*I*c/a^3*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))+22/243*I*d/a^3* 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))*Elli pticE(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)*Ell ipticF(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...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.37 \[ \int \frac {a c+a d x+b c x^3+b d x^4}{\left (a+b x^3\right )^{9/2}} \, dx=\frac {2 \, {\left (91 \, {\left (b^{3} c x^{9} + 3 \, a b^{2} c x^{6} + 3 \, a^{2} b c x^{3} + a^{3} c\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 55 \, {\left (b^{3} d x^{9} + 3 \, a b^{2} d x^{6} + 3 \, a^{2} b d x^{3} + a^{3} d\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (55 \, b^{3} d x^{8} + 91 \, b^{3} c x^{7} + 143 \, a b^{2} d x^{5} + 221 \, a b^{2} c x^{4} + 115 \, a^{2} b d x^{2} + 157 \, a^{2} b c x\right )} \sqrt {b x^{3} + a}\right )}}{405 \, {\left (a^{3} b^{4} x^{9} + 3 \, a^{4} b^{3} x^{6} + 3 \, a^{5} b^{2} x^{3} + a^{6} b\right )}} \]
2/405*(91*(b^3*c*x^9 + 3*a*b^2*c*x^6 + 3*a^2*b*c*x^3 + a^3*c)*sqrt(b)*weie rstrassPInverse(0, -4*a/b, x) + 55*(b^3*d*x^9 + 3*a*b^2*d*x^6 + 3*a^2*b*d* x^3 + a^3*d)*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4* a/b, x)) + (55*b^3*d*x^8 + 91*b^3*c*x^7 + 143*a*b^2*d*x^5 + 221*a*b^2*c*x^ 4 + 115*a^2*b*d*x^2 + 157*a^2*b*c*x)*sqrt(b*x^3 + a))/(a^3*b^4*x^9 + 3*a^4 *b^3*x^6 + 3*a^5*b^2*x^3 + a^6*b)
Time = 36.62 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.28 \[ \int \frac {a c+a d x+b c x^3+b d x^4}{\left (a+b x^3\right )^{9/2}} \, dx=\frac {c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {9}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {7}{2}} \Gamma \left (\frac {4}{3}\right )} + \frac {d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {9}{2} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {7}{2}} \Gamma \left (\frac {5}{3}\right )} + \frac {b c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {9}{2} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {9}{2}} \Gamma \left (\frac {7}{3}\right )} + \frac {b d x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{3}, \frac {9}{2} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {9}{2}} \Gamma \left (\frac {8}{3}\right )} \]
c*x*gamma(1/3)*hyper((1/3, 9/2), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**( 7/2)*gamma(4/3)) + d*x**2*gamma(2/3)*hyper((2/3, 9/2), (5/3,), b*x**3*exp_ polar(I*pi)/a)/(3*a**(7/2)*gamma(5/3)) + b*c*x**4*gamma(4/3)*hyper((4/3, 9 /2), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(9/2)*gamma(7/3)) + b*d*x**5* gamma(5/3)*hyper((5/3, 9/2), (8/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(9/2) *gamma(8/3))
\[ \int \frac {a c+a d x+b c x^3+b d x^4}{\left (a+b x^3\right )^{9/2}} \, dx=\int { \frac {b d x^{4} + b c x^{3} + a d x + a c}{{\left (b x^{3} + a\right )}^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {a c+a d x+b c x^3+b d x^4}{\left (a+b x^3\right )^{9/2}} \, dx=\int { \frac {b d x^{4} + b c x^{3} + a d x + a c}{{\left (b x^{3} + a\right )}^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {a c+a d x+b c x^3+b d x^4}{\left (a+b x^3\right )^{9/2}} \, dx=\int \frac {b\,d\,x^4+b\,c\,x^3+a\,d\,x+a\,c}{{\left (b\,x^3+a\right )}^{9/2}} \,d x \]