Integrand size = 32, antiderivative size = 590 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\sqrt {a+b x^3}} \, dx=\frac {2 e \sqrt {a+b x^3}}{3 b}+\frac {2 f x \sqrt {a+b x^3}}{5 b}+\frac {2 g x^2 \sqrt {a+b x^3}}{7 b}+\frac {2 (7 b d-4 a g) \sqrt {a+b x^3}}{7 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} (7 b d-4 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{7 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (7 \sqrt [3]{b} (5 b c-2 a f)-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (7 b d-4 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{35 \sqrt [4]{3} b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
2/3*e*(b*x^3+a)^(1/2)/b+2/5*f*x*(b*x^3+a)^(1/2)/b+2/7*g*x^2*(b*x^3+a)^(1/2 )/b+2/7*(-4*a*g+7*b*d)*(b*x^3+a)^(1/2)/b^(5/3)/((1+3^(1/2))*a^(1/3)+b^(1/3 )*x)-1/7*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*a^(1/3)*(-4*a*g+7*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)/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)+2/105*(1/2*6^(1/2)+ 1/2*2^(1/2))*(7*b^(1/3)*(-2*a*f+5*b*c)-5*(1-3^(1/2))*a^(1/3)*(-4*a*g+7*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)/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 = 10.10 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.23 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\sqrt {a+b x^3}} \, dx=\frac {4 \left (a+b x^3\right ) (35 e+3 x (7 f+5 g x))+42 (5 b c-2 a f) x \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a}\right )+15 (7 b d-4 a g) x^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )}{210 b \sqrt {a+b x^3}} \] Input:
Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/Sqrt[a + b*x^3],x]
Output:
(4*(a + b*x^3)*(35*e + 3*x*(7*f + 5*g*x)) + 42*(5*b*c - 2*a*f)*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x^3)/a)] + 15*(7*b*d - 4* a*g)*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^3)/a) ])/(210*b*Sqrt[a + b*x^3])
Time = 1.49 (sec) , antiderivative size = 594, 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 = {2427, 27, 2427, 27, 2425, 793, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4}{\sqrt {a+b x^3}} \, dx\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {2 \int \frac {7 b f x^3+7 b e x^2+(7 b d-4 a g) x+7 b c}{2 \sqrt {b x^3+a}}dx}{7 b}+\frac {2 g x^2 \sqrt {a+b x^3}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {7 b f x^3+7 b e x^2+(7 b d-4 a g) x+7 b c}{\sqrt {b x^3+a}}dx}{7 b}+\frac {2 g x^2 \sqrt {a+b x^3}}{7 b}\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle \frac {\frac {2 \int \frac {35 b^2 e x^2+5 b (7 b d-4 a g) x+7 b (5 b c-2 a f)}{2 \sqrt {b x^3+a}}dx}{5 b}+\frac {14}{5} f x \sqrt {a+b x^3}}{7 b}+\frac {2 g x^2 \sqrt {a+b x^3}}{7 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {35 b^2 e x^2+5 b (7 b d-4 a g) x+7 b (5 b c-2 a f)}{\sqrt {b x^3+a}}dx}{5 b}+\frac {14}{5} f x \sqrt {a+b x^3}}{7 b}+\frac {2 g x^2 \sqrt {a+b x^3}}{7 b}\) |
| \(\Big \downarrow \) 2425 |
| \(\displaystyle \frac {\frac {35 b^2 e \int \frac {x^2}{\sqrt {b x^3+a}}dx+\int \frac {7 b (5 b c-2 a f)+5 b (7 b d-4 a g) x}{\sqrt {b x^3+a}}dx}{5 b}+\frac {14}{5} f x \sqrt {a+b x^3}}{7 b}+\frac {2 g x^2 \sqrt {a+b x^3}}{7 b}\) |
| \(\Big \downarrow \) 793 |
| \(\displaystyle \frac {\frac {\int \frac {7 b (5 b c-2 a f)+5 b (7 b d-4 a g) x}{\sqrt {b x^3+a}}dx+\frac {70}{3} b e \sqrt {a+b x^3}}{5 b}+\frac {14}{5} f x \sqrt {a+b x^3}}{7 b}+\frac {2 g x^2 \sqrt {a+b x^3}}{7 b}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {\frac {b^{2/3} \left (7 \sqrt [3]{b} (5 b c-2 a f)-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (7 b d-4 a g)\right ) \int \frac {1}{\sqrt {b x^3+a}}dx+5 b^{2/3} (7 b d-4 a g) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx+\frac {70}{3} b e \sqrt {a+b x^3}}{5 b}+\frac {14}{5} f x \sqrt {a+b x^3}}{7 b}+\frac {2 g x^2 \sqrt {a+b x^3}}{7 b}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\frac {5 b^{2/3} (7 b d-4 a g) \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx+\frac {2 \sqrt {2+\sqrt {3}} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (7 \sqrt [3]{b} (5 b c-2 a f)-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (7 b d-4 a g)\right )}{\sqrt [4]{3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {70}{3} b e \sqrt {a+b x^3}}{5 b}+\frac {14}{5} f x \sqrt {a+b x^3}}{7 b}+\frac {2 g x^2 \sqrt {a+b x^3}}{7 b}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {2+\sqrt {3}} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (7 \sqrt [3]{b} (5 b c-2 a f)-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (7 b d-4 a g)\right )}{\sqrt [4]{3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+5 b^{2/3} (7 b d-4 a g) \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )+\frac {70}{3} b e \sqrt {a+b x^3}}{5 b}+\frac {14}{5} f x \sqrt {a+b x^3}}{7 b}+\frac {2 g x^2 \sqrt {a+b x^3}}{7 b}\) |
Input:
Int[(c + d*x + e*x^2 + f*x^3 + g*x^4)/Sqrt[a + b*x^3],x]
Output:
(2*g*x^2*Sqrt[a + b*x^3])/(7*b) + ((14*f*x*Sqrt[a + b*x^3])/5 + ((70*b*e*S qrt[a + b*x^3])/3 + 5*b^(2/3)*(7*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])) + (2*Sqrt[2 + Sqrt[3]]*b^(1/3)*(7*b^(1/3)*(5*b* c - 2*a*f) - 5*(1 - Sqrt[3])*a^(1/3)*(7*b*d - 4*a*g))*(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)*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]))/(5*b))/(7*b)
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n) ^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[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]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Coeff[Pq, x, n - 1] Int[x^(n - 1)*(a + b*x^n)^p, x], x] + Int[ExpandToSum[Pq - Coeff[Pq, x, n - 1]*x^(n - 1), x]*(a + b*x^n)^p, x] /; FreeQ[{a, b, p}, x] && PolyQ[P q, x] && IGtQ[n, 0] && Expon[Pq, x] == n - 1
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x ]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*x^(q - n + 1)*((a + b*x^n)^(p + 1)/(b*(q + n*p + 1))), x] + Simp[1/(b*(q + n*p + 1)) Int[ExpandToSum[b*(q + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] /; NeQ[q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ [p + (q + 1)/(2*n)])] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Time = 0.73 (sec) , antiderivative size = 785, normalized size of antiderivative = 1.33
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(785\) |
| risch | \(\text {Expression too large to display}\) | \(1043\) |
| default | \(\text {Expression too large to display}\) | \(1491\) |
Input:
int((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/7*g*x^2*(b*x^3+a)^(1/2)/b+2/5*f*x*(b*x^3+a)^(1/2)/b+2/3*e*(b*x^3+a)^(1/2 )/b-2/3*I*(c-2/5*a/b*f)*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))-2/3*I*(d-4/7*a/b*g)*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)))
Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.15 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\sqrt {a+b x^3}} \, dx=\frac {2 \, {\left (21 \, {\left (5 \, b c - 2 \, a f\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 15 \, {\left (7 \, b d - 4 \, a g\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (15 \, b g x^{2} + 21 \, b f x + 35 \, b e\right )} \sqrt {b x^{3} + a}\right )}}{105 \, b^{2}} \] Input:
integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^(1/2),x, algorithm="fricas")
Output:
2/105*(21*(5*b*c - 2*a*f)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) - 15*( 7*b*d - 4*a*g)*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, - 4*a/b, x)) + (15*b*g*x^2 + 21*b*f*x + 35*b*e)*sqrt(b*x^3 + a))/b^2
Time = 2.29 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.32 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\sqrt {a+b x^3}} \, dx=e \left (\begin {cases} \frac {x^{3}}{3 \sqrt {a}} & \text {for}\: b = 0 \\\frac {2 \sqrt {a + b x^{3}}}{3 b} & \text {otherwise} \end {cases}\right ) + \frac {c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} + \frac {d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {5}{3}\right )} + \frac {f x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {7}{3}\right )} + \frac {g x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {8}{3}\right )} \] Input:
integrate((g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a)**(1/2),x)
Output:
e*Piecewise((x**3/(3*sqrt(a)), Eq(b, 0)), (2*sqrt(a + b*x**3)/(3*b), True) ) + c*x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3* sqrt(a)*gamma(4/3)) + d*x**2*gamma(2/3)*hyper((1/2, 2/3), (5/3,), b*x**3*e xp_polar(I*pi)/a)/(3*sqrt(a)*gamma(5/3)) + f*x**4*gamma(4/3)*hyper((1/2, 4 /3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*sqrt(a)*gamma(7/3)) + g*x**5*gam ma(5/3)*hyper((1/2, 5/3), (8/3,), b*x**3*exp_polar(I*pi)/a)/(3*sqrt(a)*gam ma(8/3))
\[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\sqrt {a+b x^3}} \, dx=\int { \frac {g x^{4} + f x^{3} + e x^{2} + d x + c}{\sqrt {b x^{3} + a}} \,d x } \] Input:
integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^(1/2),x, algorithm="maxima")
Output:
integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)/sqrt(b*x^3 + a), x)
\[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\sqrt {a+b x^3}} \, dx=\int { \frac {g x^{4} + f x^{3} + e x^{2} + d x + c}{\sqrt {b x^{3} + a}} \,d x } \] Input:
integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^(1/2),x, algorithm="giac")
Output:
integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)/sqrt(b*x^3 + a), x)
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\sqrt {a+b x^3}} \, dx=\int \frac {g\,x^4+f\,x^3+e\,x^2+d\,x+c}{\sqrt {b\,x^3+a}} \,d x \] Input:
int((c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^3)^(1/2),x)
Output:
int((c + d*x + e*x^2 + f*x^3 + g*x^4)/(a + b*x^3)^(1/2), x)
\[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\sqrt {a+b x^3}} \, dx=\frac {70 \sqrt {b \,x^{3}+a}\, e +42 \sqrt {b \,x^{3}+a}\, f x +30 \sqrt {b \,x^{3}+a}\, g \,x^{2}-42 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) a f +105 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) b c -60 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b \,x^{3}+a}d x \right ) a g +105 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b \,x^{3}+a}d x \right ) b d}{105 b} \] Input:
int((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^(1/2),x)
Output:
(70*sqrt(a + b*x**3)*e + 42*sqrt(a + b*x**3)*f*x + 30*sqrt(a + b*x**3)*g*x **2 - 42*int(sqrt(a + b*x**3)/(a + b*x**3),x)*a*f + 105*int(sqrt(a + b*x** 3)/(a + b*x**3),x)*b*c - 60*int((sqrt(a + b*x**3)*x)/(a + b*x**3),x)*a*g + 105*int((sqrt(a + b*x**3)*x)/(a + b*x**3),x)*b*d)/(105*b)