Integrand size = 32, antiderivative size = 676 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{7/2}} \, dx=\frac {2 x \left (b c-a f+(b d-a g) x+b e x^2\right )}{15 a b \left (a+b x^3\right )^{5/2}}+\frac {2 x (7 (13 b c+2 a f)+5 (11 b d+4 a g) x)}{405 a^3 b \sqrt {a+b x^3}}-\frac {2 (11 b d+4 a g) \sqrt {a+b x^3}}{81 a^3 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 (9 a e-x (13 b c+2 a f+(11 b d+4 a g) x))}{135 a^2 b \left (a+b x^3\right )^{3/2}}+\frac {\sqrt {2-\sqrt {3}} (11 b d+4 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{27\ 3^{3/4} a^{8/3} b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (7 \sqrt [3]{b} (13 b c+2 a f)+5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (11 b d+4 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{405 \sqrt [4]{3} a^3 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]
2/15*x*(b*c-a*f+(-a*g+b*d)*x+b*e*x^2)/a/b/(b*x^3+a)^(5/2)-2/135*(9*a*e-x*( 13*b*c+2*a*f+(4*a*g+11*b*d)*x))/a^2/b/(b*x^3+a)^(3/2)+2/405*x*(14*a*f+91*b *c+5*(4*a*g+11*b*d)*x)/a^3/b/(b*x^3+a)^(1/2)-2/81*(4*a*g+11*b*d)*(b*x^3+a) ^(1/2)/a^3/b^(5/3)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))+1/81*(4*a*g+11*b*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^(5/3)/(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)*(7*b ^(1/3)*(2*a*f+13*b*c)+5*a^(1/3)*(4*a*g+11*b*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^(5/3)/(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 = 10.25 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.29 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{7/2}} \, dx=\frac {4004 b^3 c x^7+44 a b^2 x^4 \left (221 c+14 f x^3\right )+44 a^2 b x \left (157 c+34 f x^3\right )-4 a^3 (297 e+x (77 f+405 g x))+154 (13 b c+2 a f) x \left (a+b x^3\right )^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a}\right )+405 (11 b d+4 a g) x^2 \left (a+b x^3\right )^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {7}{2},\frac {5}{3},-\frac {b x^3}{a}\right )}{8910 a^3 b \left (a+b x^3\right )^{5/2}} \]
(4004*b^3*c*x^7 + 44*a*b^2*x^4*(221*c + 14*f*x^3) + 44*a^2*b*x*(157*c + 34 *f*x^3) - 4*a^3*(297*e + x*(77*f + 405*g*x)) + 154*(13*b*c + 2*a*f)*x*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a )] + 405*(11*b*d + 4*a*g)*x^2*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*Hypergeome tric2F1[2/3, 7/2, 5/3, -((b*x^3)/a)])/(8910*a^3*b*(a + b*x^3)^(5/2))
Time = 0.91 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2397, 27, 2393, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{15 a b \left (a+b x^3\right )^{5/2}}-\frac {2 \int -\frac {9 b^2 e x^2+b (11 b d+4 a g) x+b (13 b c+2 a f)}{2 \left (b x^3+a\right )^{5/2}}dx}{15 a b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {9 b^2 e x^2+b (11 b d+4 a g) x+b (13 b c+2 a f)}{\left (b x^3+a\right )^{5/2}}dx}{15 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{15 a b \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {b (7 (13 b c+2 a f)+5 (11 b d+4 a g) x)}{2 \left (b x^3+a\right )^{3/2}}dx}{9 a}-\frac {2 (9 a b e-b x (x (4 a g+11 b d)+2 a f+13 b c))}{9 a \left (a+b x^3\right )^{3/2}}}{15 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{15 a b \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b \int \frac {7 (13 b c+2 a f)+5 (11 b d+4 a g) x}{\left (b x^3+a\right )^{3/2}}dx}{9 a}-\frac {2 (9 a b e-b x (x (4 a g+11 b d)+2 a f+13 b c))}{9 a \left (a+b x^3\right )^{3/2}}}{15 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{15 a b \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {b \left (\frac {2 x (7 (2 a f+13 b c)+5 x (4 a g+11 b d))}{3 a \sqrt {a+b x^3}}-\frac {2 \int -\frac {7 (13 b c+2 a f)-5 (11 b d+4 a g) x}{2 \sqrt {b x^3+a}}dx}{3 a}\right )}{9 a}-\frac {2 (9 a b e-b x (x (4 a g+11 b d)+2 a f+13 b c))}{9 a \left (a+b x^3\right )^{3/2}}}{15 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{15 a b \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b \left (\frac {\int \frac {7 (13 b c+2 a f)-5 (11 b d+4 a g) x}{\sqrt {b x^3+a}}dx}{3 a}+\frac {2 x (7 (2 a f+13 b c)+5 x (4 a g+11 b d))}{3 a \sqrt {a+b x^3}}\right )}{9 a}-\frac {2 (9 a b e-b x (x (4 a g+11 b d)+2 a f+13 b c))}{9 a \left (a+b x^3\right )^{3/2}}}{15 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{15 a b \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {\frac {b \left (\frac {\left (\frac {5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (4 a g+11 b d)}{\sqrt [3]{b}}+14 a f+91 b c\right ) \int \frac {1}{\sqrt {b x^3+a}}dx-\frac {5 (4 a g+11 b d) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}}{3 a}+\frac {2 x (7 (2 a f+13 b c)+5 x (4 a g+11 b d))}{3 a \sqrt {a+b x^3}}\right )}{9 a}-\frac {2 (9 a b e-b x (x (4 a g+11 b d)+2 a f+13 b c))}{9 a \left (a+b x^3\right )^{3/2}}}{15 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{15 a b \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\frac {b \left (\frac {\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (\frac {5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (4 a g+11 b d)}{\sqrt [3]{b}}+14 a f+91 b c\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {5 (4 a g+11 b d) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}}{3 a}+\frac {2 x (7 (2 a f+13 b c)+5 x (4 a g+11 b d))}{3 a \sqrt {a+b x^3}}\right )}{9 a}-\frac {2 (9 a b e-b x (x (4 a g+11 b d)+2 a f+13 b c))}{9 a \left (a+b x^3\right )^{3/2}}}{15 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{15 a b \left (a+b x^3\right )^{5/2}}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {\frac {b \left (\frac {\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (\frac {5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (4 a g+11 b d)}{\sqrt [3]{b}}+14 a f+91 b c\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {5 (4 a g+11 b d) \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{\sqrt [3]{b}}}{3 a}+\frac {2 x (7 (2 a f+13 b c)+5 x (4 a g+11 b d))}{3 a \sqrt {a+b x^3}}\right )}{9 a}-\frac {2 (9 a b e-b x (x (4 a g+11 b d)+2 a f+13 b c))}{9 a \left (a+b x^3\right )^{3/2}}}{15 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{15 a b \left (a+b x^3\right )^{5/2}}\) |
(2*x*(b*c - a*f + (b*d - a*g)*x + b*e*x^2))/(15*a*b*(a + b*x^3)^(5/2)) + ( (-2*(9*a*b*e - b*x*(13*b*c + 2*a*f + (11*b*d + 4*a*g)*x)))/(9*a*(a + b*x^3 )^(3/2)) + (b*((2*x*(7*(13*b*c + 2*a*f) + 5*(11*b*d + 4*a*g)*x))/(3*a*Sqrt [a + b*x^3]) + ((-5*(11*b*d + 4*a*g)*((2*Sqrt[a + b*x^3])/(b^(1/3)*((1 + S qrt[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)*Sq rt[(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*b*c + 14*a*f + (5*(1 - Sqrt[3])*a^(1/3)*(11*b*d + 4*a*g))/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)*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*a)))/(9*a))/(15*a*b^2)
3.1.69.3.1 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] :> Module[{q = Expon[Pq, x], i}, Simp[(a*Coeff[Pq, x, q] - b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q , x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) In t[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^( p + 1), x], x] /; q == n - 1] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n , 0] && LtQ[p, -1]
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[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[(a + b*x^n)^(p + 1)* ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -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 = 1.55 (sec) , antiderivative size = 921, normalized size of antiderivative = 1.36
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(921\) |
| default | \(\text {Expression too large to display}\) | \(1793\) |
(-2/15/a/b^4*(a*g-b*d)*x^2-2/15/a/b^4*(a*f-b*c)*x-2/15/b^4*e)*(b*x^3+a)^(1 /2)/(x^3+a/b)^3+(2/135/a^2/b^3*(4*a*g+11*b*d)*x^2+2/135/a^2/b^3*(2*a*f+13* b*c)*x)*(b*x^3+a)^(1/2)/(x^3+a/b)^2-2*b*(-1/81*(4*a*g+11*b*d)/a^3/b^2*x^2- 7/405*(2*a*f+13*b*c)/a^3/b^2*x)/((x^3+a/b)*b)^(1/2)-14/1215*I*(2*a*f+13*b* c)/a^3/b^2*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))+2/243*I*(4*a*g+11*b*d)/a^3/b^2*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^...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.55 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{7/2}} \, dx=\frac {2 \, {\left (7 \, {\left ({\left (13 \, b^{4} c + 2 \, a b^{3} f\right )} x^{9} + 3 \, {\left (13 \, a b^{3} c + 2 \, a^{2} b^{2} f\right )} x^{6} + 13 \, a^{3} b c + 2 \, a^{4} f + 3 \, {\left (13 \, a^{2} b^{2} c + 2 \, a^{3} b f\right )} x^{3}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 5 \, {\left ({\left (11 \, b^{4} d + 4 \, a b^{3} g\right )} x^{9} + 3 \, {\left (11 \, a b^{3} d + 4 \, a^{2} b^{2} g\right )} x^{6} + 11 \, a^{3} b d + 4 \, a^{4} g + 3 \, {\left (11 \, a^{2} b^{2} d + 4 \, a^{3} b g\right )} x^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (5 \, {\left (11 \, b^{4} d + 4 \, a b^{3} g\right )} x^{8} + 7 \, {\left (13 \, b^{4} c + 2 \, a b^{3} f\right )} x^{7} + 13 \, {\left (11 \, a b^{3} d + 4 \, a^{2} b^{2} g\right )} x^{5} - 27 \, a^{3} b e + 17 \, {\left (13 \, a b^{3} c + 2 \, a^{2} b^{2} f\right )} x^{4} + 5 \, {\left (23 \, a^{2} b^{2} d + a^{3} b g\right )} x^{2} + {\left (157 \, a^{2} b^{2} c - 7 \, a^{3} b f\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{405 \, {\left (a^{3} b^{5} x^{9} + 3 \, a^{4} b^{4} x^{6} + 3 \, a^{5} b^{3} x^{3} + a^{6} b^{2}\right )}} \]
2/405*(7*((13*b^4*c + 2*a*b^3*f)*x^9 + 3*(13*a*b^3*c + 2*a^2*b^2*f)*x^6 + 13*a^3*b*c + 2*a^4*f + 3*(13*a^2*b^2*c + 2*a^3*b*f)*x^3)*sqrt(b)*weierstra ssPInverse(0, -4*a/b, x) + 5*((11*b^4*d + 4*a*b^3*g)*x^9 + 3*(11*a*b^3*d + 4*a^2*b^2*g)*x^6 + 11*a^3*b*d + 4*a^4*g + 3*(11*a^2*b^2*d + 4*a^3*b*g)*x^ 3)*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (5*(11*b^4*d + 4*a*b^3*g)*x^8 + 7*(13*b^4*c + 2*a*b^3*f)*x^7 + 13*(11*a*b ^3*d + 4*a^2*b^2*g)*x^5 - 27*a^3*b*e + 17*(13*a*b^3*c + 2*a^2*b^2*f)*x^4 + 5*(23*a^2*b^2*d + a^3*b*g)*x^2 + (157*a^2*b^2*c - 7*a^3*b*f)*x)*sqrt(b*x^ 3 + a))/(a^3*b^5*x^9 + 3*a^4*b^4*x^6 + 3*a^5*b^3*x^3 + a^6*b^2)
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{7/2}} \, dx=\int { \frac {g x^{4} + f x^{3} + e x^{2} + d x + c}{{\left (b x^{3} + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{7/2}} \, dx=\int { \frac {g x^{4} + f x^{3} + e x^{2} + d x + c}{{\left (b x^{3} + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{7/2}} \, dx=\int \frac {g\,x^4+f\,x^3+e\,x^2+d\,x+c}{{\left (b\,x^3+a\right )}^{7/2}} \,d x \]