Integrand size = 32, antiderivative size = 624 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{5/2}} \, dx=-\frac {2 \left (a e-(b c-a f) x-(b d-a g) x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}}+\frac {2 x (7 b c+2 a f+(5 b d+4 a g) x)}{27 a^2 b \sqrt {a+b x^3}}-\frac {2 (5 b d+4 a g) \sqrt {a+b x^3}}{27 a^2 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\sqrt {2-\sqrt {3}} (5 b d+4 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{9\ 3^{3/4} a^{5/3} b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{b} (7 b c+2 a f)+\left (1-\sqrt {3}\right ) \sqrt [3]{a} (5 b d+4 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{27 \sqrt [4]{3} a^2 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
1/9*(-2*a*e+2*(-a*f+b*c)*x+2*(-a*g+b*d)*x^2)/a/b/(b*x^3+a)^(3/2)+2/27*x*(7 *b*c+2*a*f+(4*a*g+5*b*d)*x)/a^2/b/(b*x^3+a)^(1/2)-2/27*(4*a*g+5*b*d)*(b*x^ 3+a)^(1/2)/a^2/b^(5/3)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)+1/27*(1/2*6^(1/2)-1 /2*2^(1/2))*(4*a*g+5*b*d)*(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)*3^(1/ 4)/a^(5/3)/b^(5/3)/(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)+2/81*(1/2*6^(1/2)+1/2*2^(1/2))*(b^(1/3)*(2* a*f+7*b*c)+(1-3^(1/2))*a^(1/3)*(4*a*g+5*b*d))*(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)*3^(3/4)/a^2/b^(5/3)/(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 = 10.14 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.27 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {140 b^2 c x^4+40 a b x \left (5 c+f x^3\right )-4 a^2 (15 e+x (5 f+27 g x))+10 (7 b c+2 a f) x \left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a}\right )+27 (5 b d+4 a g) x^2 \left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {5}{2},\frac {5}{3},-\frac {b x^3}{a}\right )}{270 a^2 b \left (a+b x^3\right )^{3/2}} \] Input:
Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^3)^(5/2),x]
Output:
(140*b^2*c*x^4 + 40*a*b*x*(5*c + f*x^3) - 4*a^2*(15*e + x*(5*f + 27*g*x)) + 10*(7*b*c + 2*a*f)*x*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1 /3, 1/2, 4/3, -((b*x^3)/a)] + 27*(5*b*d + 4*a*g)*x^2*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 5/2, 5/3, -((b*x^3)/a)])/(270*a^2*b*(a + b*x^3)^(3/2))
Time = 1.39 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2397, 27, 2393, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}}-\frac {2 \int -\frac {3 b^2 e x^2+b (5 b d+4 a g) x+b (7 b c+2 a f)}{2 \left (b x^3+a\right )^{3/2}}dx}{9 a b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 b^2 e x^2+b (5 b d+4 a g) x+b (7 b c+2 a f)}{\left (b x^3+a\right )^{3/2}}dx}{9 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {b (7 b c+2 a f-(5 b d+4 a g) x)}{2 \sqrt {b x^3+a}}dx}{3 a}-\frac {2 (3 a b e-b x (x (4 a g+5 b d)+2 a f+7 b c))}{3 a \sqrt {a+b x^3}}}{9 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {b \int \frac {7 b c+2 a f-(5 b d+4 a g) x}{\sqrt {b x^3+a}}dx}{3 a}-\frac {2 (3 a b e-b x (x (4 a g+5 b d)+2 a f+7 b c))}{3 a \sqrt {a+b x^3}}}{9 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {\frac {b \left (\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} (4 a g+5 b d)}{\sqrt [3]{b}}+2 a f+7 b c\right ) \int \frac {1}{\sqrt {b x^3+a}}dx-\frac {(4 a g+5 b d) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}\right )}{3 a}-\frac {2 (3 a b e-b x (x (4 a g+5 b d)+2 a f+7 b c))}{3 a \sqrt {a+b x^3}}}{9 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\frac {b \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} (4 a g+5 b d)}{\sqrt [3]{b}}+2 a f+7 b c\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {(4 a g+5 b d) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}\right )}{3 a}-\frac {2 (3 a b e-b x (x (4 a g+5 b d)+2 a f+7 b c))}{3 a \sqrt {a+b x^3}}}{9 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {\frac {b \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} (4 a g+5 b d)}{\sqrt [3]{b}}+2 a f+7 b c\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {(4 a g+5 b d) \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{\sqrt [3]{b}}\right )}{3 a}-\frac {2 (3 a b e-b x (x (4 a g+5 b d)+2 a f+7 b c))}{3 a \sqrt {a+b x^3}}}{9 a b^2}+\frac {2 x \left (x (b d-a g)-a f+b c+b e x^2\right )}{9 a b \left (a+b x^3\right )^{3/2}}\) |
Input:
Int[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^3)^(5/2),x]
Output:
(2*x*(b*c - a*f + (b*d - a*g)*x + b*e*x^2))/(9*a*b*(a + b*x^3)^(3/2)) + (( -2*(3*a*b*e - b*x*(7*b*c + 2*a*f + (5*b*d + 4*a*g)*x)))/(3*a*Sqrt[a + b*x^ 3]) + (b*(-(((5*b*d + 4*a*g)*((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]]*(7*b*c + 2*a*f + ((1 - Sqrt[3]) *a^(1/3)*(5*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]*Ell ipticF[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*b^2)
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] :> 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 = 0.61 (sec) , antiderivative size = 861, normalized size of antiderivative = 1.38
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(861\) |
| default | \(\text {Expression too large to display}\) | \(1673\) |
Input:
int((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
(-2/9/a/b^3*(a*g-b*d)*x^2-2/9/a/b^3*(a*f-b*c)*x-2/9/b^3*e)*(b*x^3+a)^(1/2) /(x^3+a/b)^2-2*b*(-1/27/a^2/b^2*(4*a*g+5*b*d)*x^2-1/27/a^2/b^2*(2*a*f+7*b* c)*x)/((x^3+a/b)*b)^(1/2)-2/81*I/b^2/a^2*(2*a*f+7*b*c)*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)*Ellipti cF(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))+2/81*I/b^2/a^2*(4*a*g+5*b *d)*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))*El lipticE(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)*E llipticF(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...
Time = 0.10 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.42 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (7 \, b^{3} c + 2 \, a b^{2} f\right )} x^{6} + 7 \, a^{2} b c + 2 \, a^{3} f + 2 \, {\left (7 \, a b^{2} c + 2 \, a^{2} b f\right )} x^{3}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left ({\left (5 \, b^{3} d + 4 \, a b^{2} g\right )} x^{6} + 5 \, a^{2} b d + 4 \, a^{3} g + 2 \, {\left (5 \, a b^{2} d + 4 \, a^{2} b g\right )} x^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left ({\left (5 \, b^{3} d + 4 \, a b^{2} g\right )} x^{5} + {\left (7 \, b^{3} c + 2 \, a b^{2} f\right )} x^{4} - 3 \, a^{2} b e + {\left (8 \, a b^{2} d + a^{2} b g\right )} x^{2} + {\left (10 \, a b^{2} c - a^{2} b f\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{27 \, {\left (a^{2} b^{4} x^{6} + 2 \, a^{3} b^{3} x^{3} + a^{4} b^{2}\right )}} \] Input:
integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^(5/2),x, algorithm="fricas")
Output:
2/27*(((7*b^3*c + 2*a*b^2*f)*x^6 + 7*a^2*b*c + 2*a^3*f + 2*(7*a*b^2*c + 2* a^2*b*f)*x^3)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) + ((5*b^3*d + 4*a* b^2*g)*x^6 + 5*a^2*b*d + 4*a^3*g + 2*(5*a*b^2*d + 4*a^2*b*g)*x^3)*sqrt(b)* weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + ((5*b^3*d + 4*a*b^2*g)*x^5 + (7*b^3*c + 2*a*b^2*f)*x^4 - 3*a^2*b*e + (8*a*b^2*d + a^ 2*b*g)*x^2 + (10*a*b^2*c - a^2*b*f)*x)*sqrt(b*x^3 + a))/(a^2*b^4*x^6 + 2*a ^3*b^3*x^3 + a^4*b^2)
Time = 60.20 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.33 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{5/2}} \, dx=e \left (\begin {cases} - \frac {2}{9 a b \sqrt {a + b x^{3}} + 9 b^{2} x^{3} \sqrt {a + b x^{3}}} & \text {for}\: b \neq 0 \\\frac {x^{3}}{3 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {5}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{2}} \Gamma \left (\frac {4}{3}\right )} + \frac {d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {5}{2} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{2}} \Gamma \left (\frac {5}{3}\right )} + \frac {f x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {5}{2} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{2}} \Gamma \left (\frac {7}{3}\right )} + \frac {g x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{3}, \frac {5}{2} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{2}} \Gamma \left (\frac {8}{3}\right )} \] Input:
integrate((g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a)**(5/2),x)
Output:
e*Piecewise((-2/(9*a*b*sqrt(a + b*x**3) + 9*b**2*x**3*sqrt(a + b*x**3)), N e(b, 0)), (x**3/(3*a**(5/2)), True)) + c*x*gamma(1/3)*hyper((1/3, 5/2), (4 /3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(5/2)*gamma(4/3)) + d*x**2*gamma(2/3 )*hyper((2/3, 5/2), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(5/2)*gamma(5/ 3)) + f*x**4*gamma(4/3)*hyper((4/3, 5/2), (7/3,), b*x**3*exp_polar(I*pi)/a )/(3*a**(5/2)*gamma(7/3)) + g*x**5*gamma(5/3)*hyper((5/3, 5/2), (8/3,), b* x**3*exp_polar(I*pi)/a)/(3*a**(5/2)*gamma(8/3))
\[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{5/2}} \, dx=\int { \frac {g x^{4} + f x^{3} + e x^{2} + d x + c}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^(5/2),x, algorithm="maxima")
Output:
integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^3 + a)^(5/2), x)
\[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{5/2}} \, dx=\int { \frac {g x^{4} + f x^{3} + e x^{2} + d x + c}{{\left (b x^{3} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^(5/2),x, algorithm="giac")
Output:
integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)/(b*x^3 + a)^(5/2), x)
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{5/2}} \, dx=\int \frac {g\,x^4+f\,x^3+e\,x^2+d\,x+c}{{\left (b\,x^3+a\right )}^{5/2}} \,d x \] Input:
int((c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^3)^(5/2),x)
Output:
int((c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^3)^(5/2), x)
\[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^3\right )^{5/2}} \, dx=\frac {-70 \sqrt {b \,x^{3}+a}\, e -90 \sqrt {b \,x^{3}+a}\, f x -126 \sqrt {b \,x^{3}+a}\, g \,x^{2}+90 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right ) a^{3} f +315 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right ) a^{2} b c +180 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right ) a^{2} b f \,x^{3}+630 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right ) a \,b^{2} c \,x^{3}+90 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right ) a \,b^{2} f \,x^{6}+315 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right ) b^{3} c \,x^{6}+252 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right ) a^{3} g +315 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right ) a^{2} b d +504 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right ) a^{2} b g \,x^{3}+630 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right ) a \,b^{2} d \,x^{3}+252 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right ) a \,b^{2} g \,x^{6}+315 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b^{3} x^{9}+3 a \,b^{2} x^{6}+3 a^{2} b \,x^{3}+a^{3}}d x \right ) b^{3} d \,x^{6}}{315 b \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )} \] Input:
int((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^(5/2),x)
Output:
( - 70*sqrt(a + b*x**3)*e - 90*sqrt(a + b*x**3)*f*x - 126*sqrt(a + b*x**3) *g*x**2 + 90*int(sqrt(a + b*x**3)/(a**3 + 3*a**2*b*x**3 + 3*a*b**2*x**6 + b**3*x**9),x)*a**3*f + 315*int(sqrt(a + b*x**3)/(a**3 + 3*a**2*b*x**3 + 3* a*b**2*x**6 + b**3*x**9),x)*a**2*b*c + 180*int(sqrt(a + b*x**3)/(a**3 + 3* a**2*b*x**3 + 3*a*b**2*x**6 + b**3*x**9),x)*a**2*b*f*x**3 + 630*int(sqrt(a + b*x**3)/(a**3 + 3*a**2*b*x**3 + 3*a*b**2*x**6 + b**3*x**9),x)*a*b**2*c* x**3 + 90*int(sqrt(a + b*x**3)/(a**3 + 3*a**2*b*x**3 + 3*a*b**2*x**6 + b** 3*x**9),x)*a*b**2*f*x**6 + 315*int(sqrt(a + b*x**3)/(a**3 + 3*a**2*b*x**3 + 3*a*b**2*x**6 + b**3*x**9),x)*b**3*c*x**6 + 252*int((sqrt(a + b*x**3)*x) /(a**3 + 3*a**2*b*x**3 + 3*a*b**2*x**6 + b**3*x**9),x)*a**3*g + 315*int((s qrt(a + b*x**3)*x)/(a**3 + 3*a**2*b*x**3 + 3*a*b**2*x**6 + b**3*x**9),x)*a **2*b*d + 504*int((sqrt(a + b*x**3)*x)/(a**3 + 3*a**2*b*x**3 + 3*a*b**2*x* *6 + b**3*x**9),x)*a**2*b*g*x**3 + 630*int((sqrt(a + b*x**3)*x)/(a**3 + 3* a**2*b*x**3 + 3*a*b**2*x**6 + b**3*x**9),x)*a*b**2*d*x**3 + 252*int((sqrt( a + b*x**3)*x)/(a**3 + 3*a**2*b*x**3 + 3*a*b**2*x**6 + b**3*x**9),x)*a*b** 2*g*x**6 + 315*int((sqrt(a + b*x**3)*x)/(a**3 + 3*a**2*b*x**3 + 3*a*b**2*x **6 + b**3*x**9),x)*b**3*d*x**6)/(315*b*(a**2 + 2*a*b*x**3 + b**2*x**6))