Integrand size = 32, antiderivative size = 556 \[ \int \sqrt {a+b x^3} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {54 a^2 d \sqrt {a+b x^3}}{91 b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {18 a \left (91 c x+55 d x^2\right ) \sqrt {a+b x^3}}{5005}+\frac {2}{143} \left (13 c x+11 d x^2\right ) \left (a+b x^3\right )^{3/2}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{91 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (91 \sqrt [3]{b} c-55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{5005 b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
54/91*a^2*d*(b*x^3+a)^(1/2)/b^(2/3)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)+18/500 5*a*(55*d*x^2+91*c*x)*(b*x^3+a)^(1/2)+2/143*(11*d*x^2+13*c*x)*(b*x^3+a)^(3 /2)-27/91*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*a^(7/3)*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^(2/3)/(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)+18/5005*3^(3/4)*(1/2*6^(1/2)+1/ 2*2^(1/2))*a^2*(91*b^(1/3)*c-55*(1-3^(1/2))*a^(1/3)*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)/b^(2/3)/(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 = 6.79 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.14 \[ \int \sqrt {a+b x^3} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {a x \sqrt {a+b x^3} \left (2 c \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )+d x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{2 \sqrt {1+\frac {b x^3}{a}}} \] Input:
Integrate[Sqrt[a + b*x^3]*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x]
Output:
(a*x*Sqrt[a + b*x^3]*(2*c*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)] + d*x*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)]))/(2*Sqrt[1 + (b*x^3 )/a])
Time = 1.16 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2391, 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.
| \(\displaystyle \int \sqrt {a+b x^3} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2391 |
| \(\displaystyle \int \left (a+b x^3\right )^{3/2} (c+d x)dx\) |
| \(\Big \downarrow \) 2392 |
| \(\displaystyle \frac {9}{2} a \int \frac {2}{143} (13 c+11 d x) \sqrt {b x^3+a}dx+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (13 c x+11 d x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {9}{143} a \int (13 c+11 d x) \sqrt {b x^3+a}dx+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (13 c x+11 d x^2\right )\) |
| \(\Big \downarrow \) 2392 |
| \(\displaystyle \frac {9}{143} a \left (\frac {3}{2} a \int \frac {2 (91 c+55 d x)}{35 \sqrt {b x^3+a}}dx+\frac {2}{35} \sqrt {a+b x^3} \left (91 c x+55 d x^2\right )\right )+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (13 c x+11 d x^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {9}{143} a \left (\frac {3}{35} a \int \frac {91 c+55 d x}{\sqrt {b x^3+a}}dx+\frac {2}{35} \sqrt {a+b x^3} \left (91 c x+55 d x^2\right )\right )+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (13 c x+11 d x^2\right )\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \frac {9}{143} a \left (\frac {3}{35} a \left (\left (91 c-\frac {55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt {b x^3+a}}dx+\frac {55 d \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}\right )+\frac {2}{35} \sqrt {a+b x^3} \left (91 c x+55 d x^2\right )\right )+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (13 c x+11 d x^2\right )\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {9}{143} a \left (\frac {3}{35} a \left (\frac {55 d \int \frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {b x^3+a}}dx}{\sqrt [3]{b}}+\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (91 c-\frac {55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )+\frac {2}{35} \sqrt {a+b x^3} \left (91 c x+55 d x^2\right )\right )+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (13 c x+11 d x^2\right )\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \frac {9}{143} a \left (\frac {3}{35} a \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (91 c-\frac {55 \left (1-\sqrt {3}\right ) \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {55 d \left (\frac {2 \sqrt {a+b x^3}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\right )}{\sqrt [3]{b}}\right )+\frac {2}{35} \sqrt {a+b x^3} \left (91 c x+55 d x^2\right )\right )+\frac {2}{143} \left (a+b x^3\right )^{3/2} \left (13 c x+11 d x^2\right )\) |
Input:
Int[Sqrt[a + b*x^3]*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x]
Output:
(2*(13*c*x + 11*d*x^2)*(a + b*x^3)^(3/2))/143 + (9*a*((2*(91*c*x + 55*d*x^ 2)*Sqrt[a + b*x^3])/35 + (3*a*((55*d*((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*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)*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
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 = 0.66 (sec) , antiderivative size = 762, normalized size of antiderivative = 1.37
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(762\) |
| elliptic | \(\text {Expression too large to display}\) | \(788\) |
| default | \(\text {Expression too large to display}\) | \(1546\) |
Input:
int((b*x^3+a)^(1/2)*(b*d*x^4+b*c*x^3+a*d*x+a*c),x,method=_RETURNVERBOSE)
Output:
2/5005*x*(385*b*d*x^4+455*b*c*x^3+880*a*d*x+1274*a*c)*(b*x^3+a)^(1/2)+27/5 005*a^2*(-182/3*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))-110/3*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))))
Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.17 \[ \int \sqrt {a+b x^3} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {2 \, {\left (2457 \, a^{2} \sqrt {b} c {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 1485 \, a^{2} \sqrt {b} d {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (385 \, b^{2} d x^{5} + 455 \, b^{2} c x^{4} + 880 \, a b d x^{2} + 1274 \, a b c x\right )} \sqrt {b x^{3} + a}\right )}}{5005 \, b} \] Input:
integrate((b*x^3+a)^(1/2)*(b*d*x^4+b*c*x^3+a*d*x+a*c),x, algorithm="fricas ")
Output:
2/5005*(2457*a^2*sqrt(b)*c*weierstrassPInverse(0, -4*a/b, x) - 1485*a^2*sq rt(b)*d*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (3 85*b^2*d*x^5 + 455*b^2*c*x^4 + 880*a*b*d*x^2 + 1274*a*b*c*x)*sqrt(b*x^3 + a))/b
Time = 2.83 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.31 \[ \int \sqrt {a+b x^3} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {a^{\frac {3}{2}} c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a^{\frac {3}{2}} d x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {\sqrt {a} b c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {\sqrt {a} b d x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} \] Input:
integrate((b*x**3+a)**(1/2)*(b*d*x**4+b*c*x**3+a*d*x+a*c),x)
Output:
a**(3/2)*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**(3/2)*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)) + sqrt(a)*b*c*x**4*gamma(4/3)*hyp er((-1/2, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3)) + sqrt(a) *b*d*x**5*gamma(5/3)*hyper((-1/2, 5/3), (8/3,), b*x**3*exp_polar(I*pi)/a)/ (3*gamma(8/3))
\[ \int \sqrt {a+b x^3} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\int { {\left (b d x^{4} + b c x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(b*d*x^4+b*c*x^3+a*d*x+a*c),x, algorithm="maxima ")
Output:
integrate((b*d*x^4 + b*c*x^3 + a*d*x + a*c)*sqrt(b*x^3 + a), x)
\[ \int \sqrt {a+b x^3} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\int { {\left (b d x^{4} + b c x^{3} + a d x + a c\right )} \sqrt {b x^{3} + a} \,d x } \] Input:
integrate((b*x^3+a)^(1/2)*(b*d*x^4+b*c*x^3+a*d*x+a*c),x, algorithm="giac")
Output:
integrate((b*d*x^4 + b*c*x^3 + a*d*x + a*c)*sqrt(b*x^3 + a), x)
Timed out. \[ \int \sqrt {a+b x^3} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\int \sqrt {b\,x^3+a}\,\left (b\,d\,x^4+b\,c\,x^3+a\,d\,x+a\,c\right ) \,d x \] Input:
int((a + b*x^3)^(1/2)*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x)
Output:
int((a + b*x^3)^(1/2)*(a*c + a*d*x + b*c*x^3 + b*d*x^4), x)
\[ \int \sqrt {a+b x^3} \left (a c+a d x+b c x^3+b d x^4\right ) \, dx=\frac {28 \sqrt {b \,x^{3}+a}\, a c x}{55}+\frac {32 \sqrt {b \,x^{3}+a}\, a d \,x^{2}}{91}+\frac {2 \sqrt {b \,x^{3}+a}\, b c \,x^{4}}{11}+\frac {2 \sqrt {b \,x^{3}+a}\, b d \,x^{5}}{13}+\frac {27 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \right ) a^{2} c}{55}+\frac {27 \left (\int \frac {\sqrt {b \,x^{3}+a}\, x}{b \,x^{3}+a}d x \right ) a^{2} d}{91} \] Input:
int((b*x^3+a)^(1/2)*(b*d*x^4+b*c*x^3+a*d*x+a*c),x)
Output:
(2548*sqrt(a + b*x**3)*a*c*x + 1760*sqrt(a + b*x**3)*a*d*x**2 + 910*sqrt(a + b*x**3)*b*c*x**4 + 770*sqrt(a + b*x**3)*b*d*x**5 + 2457*int(sqrt(a + b* x**3)/(a + b*x**3),x)*a**2*c + 1485*int((sqrt(a + b*x**3)*x)/(a + b*x**3), x)*a**2*d)/5005