\(\int x^3 \sqrt {a+b x^3} (c+d x+e x^2+f x^3+g x^4) \, dx\) [445]

   Optimal result
   Rubi [A] (verified)
   Mathematica [C] (verified)
   Maple [A] (verified)
   Fricas [C] (verification not implemented)
   Sympy [A] (verification not implemented)
   Maxima [F]
   Giac [F]
   Mupad [F(-1)]

Optimal result

Integrand size = 35, antiderivative size = 733 \[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=-\frac {4 a^2 e \sqrt {a+b x^3}}{45 b^2}+\frac {6 a (17 b c-8 a f) x \sqrt {a+b x^3}}{935 b^2}+\frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}-\frac {24 a^2 (19 b d-10 a g) \sqrt {a+b x^3}}{1729 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} (19 b d-10 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{1729 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {4\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (1729 \sqrt [3]{b} (17 b c-8 a f)-1870 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-10 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{1616615 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[Out]

-4/45*a^2*e*(b*x^3+a)^(1/2)/b^2+6/935*a*(-8*a*f+17*b*c)*x*(b*x^3+a)^(1/2)/b^2+6/1729*a*(-10*a*g+19*b*d)*x^2*(b
*x^3+a)^(1/2)/b^2+2/45*a*e*x^3*(b*x^3+a)^(1/2)/b+6/187*a*f*x^4*(b*x^3+a)^(1/2)/b+6/247*a*g*x^5*(b*x^3+a)^(1/2)
/b+2/692835*x^3*(36465*g*x^5+40755*f*x^4+46189*e*x^3+53295*d*x^2+62985*c*x)*(b*x^3+a)^(1/2)-24/1729*a^2*(-10*a
*g+19*b*d)*(b*x^3+a)^(1/2)/b^(8/3)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))+12/1729*3^(1/4)*a^(7/3)*(-10*a*g+19*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)/b^(8/
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)-4/1616615*3^(3/4)*a^2
*(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)*
(1729*b^(1/3)*(-8*a*f+17*b*c)-1870*a^(1/3)*(-10*a*g+19*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)/b^(8/3)/(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)

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {1840, 1850, 1902, 1608, 1900, 267, 1892, 224, 1891} \[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (19 b d-10 a g) E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {24 a^2 \sqrt {a+b x^3} (19 b d-10 a g)}{1729 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {4 a^2 e \sqrt {a+b x^3}}{45 b^2}-\frac {4\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (1729 \sqrt [3]{b} (17 b c-8 a f)-1870 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-10 a g)\right )}{1616615 b^{8/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {6 a x \sqrt {a+b x^3} (17 b c-8 a f)}{935 b^2}+\frac {6 a x^2 \sqrt {a+b x^3} (19 b d-10 a g)}{1729 b^2}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b} \]

[In]

Int[x^3*Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4),x]

[Out]

(-4*a^2*e*Sqrt[a + b*x^3])/(45*b^2) + (6*a*(17*b*c - 8*a*f)*x*Sqrt[a + b*x^3])/(935*b^2) + (6*a*(19*b*d - 10*a
*g)*x^2*Sqrt[a + b*x^3])/(1729*b^2) + (2*a*e*x^3*Sqrt[a + b*x^3])/(45*b) + (6*a*f*x^4*Sqrt[a + b*x^3])/(187*b)
 + (6*a*g*x^5*Sqrt[a + b*x^3])/(247*b) - (24*a^2*(19*b*d - 10*a*g)*Sqrt[a + b*x^3])/(1729*b^(8/3)*((1 + Sqrt[3
])*a^(1/3) + b^(1/3)*x)) + (2*x^3*Sqrt[a + b*x^3]*(62985*c*x + 53295*d*x^2 + 46189*e*x^3 + 40755*f*x^4 + 36465
*g*x^5))/692835 + (12*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(7/3)*(19*b*d - 10*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]*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]])/(1729*b^(8/3)*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]) - (4*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*(1729*
b^(1/3)*(17*b*c - 8*a*f) - 1870*(1 - Sqrt[3])*a^(1/3)*(19*b*d - 10*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]])/(1616615*b^(8/3)*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])

Rule 224

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]*Sq
rt[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]

Rule 267

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]

Rule 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1840

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(
c*x)^m*(a + b*x^n)^p*Sum[Coeff[Pq, x, i]*(x^(i + 1)/(m + n*p + i + 1)), {i, 0, q}], x] + Dist[a*n*p, Int[(c*x)
^m*(a + b*x^n)^(p - 1)*Sum[Coeff[Pq, x, i]*(x^i/(m + n*p + i + 1)), {i, 0, q}], x], x]] /; FreeQ[{a, b, c, m},
 x] && PolyQ[Pq, x] && IGtQ[(n - 1)/2, 0] && GtQ[p, 0]

Rule 1850

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[Pqq*(c*x)^(m + q - n + 1)*((a + b*x^n)^(p + 1)
/(b*c^(q - n + 1)*(m + q + n*p + 1))), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]

Rule 1891

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[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
] - Simp[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] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rule 1892

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a,
 3]]}, Dist[(c*r - (1 - Sqrt[3])*d*s)/r, Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/r, Int[((1 - Sqrt[3])*s + r*x)
/Sqrt[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]

Rule 1900

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[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] && Pol
yQ[Pq, x] && IGtQ[n, 0] && Expon[Pq, x] == n - 1

Rule 1902

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, D
ist[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] + Simp[Pqq*x^(q - n + 1)*((a + b*x^n)^(p + 1)/(b*(q + n*p + 1))), 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] && IG
tQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {1}{2} (3 a) \int \frac {x^3 \left (\frac {2 c}{11}+\frac {2 d x}{13}+\frac {2 e x^2}{15}+\frac {2 f x^3}{17}+\frac {2 g x^4}{19}\right )}{\sqrt {a+b x^3}} \, dx \\ & = \frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(3 a) \int \frac {x^3 \left (\frac {13 b c}{11}+\frac {1}{19} (19 b d-10 a g) x+\frac {13}{15} b e x^2+\frac {13}{17} b f x^3\right )}{\sqrt {a+b x^3}} \, dx}{13 b} \\ & = \frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(6 a) \int \frac {x^3 \left (\frac {13}{34} b (17 b c-8 a f)+\frac {11}{38} b (19 b d-10 a g) x+\frac {143}{30} b^2 e x^2\right )}{\sqrt {a+b x^3}} \, dx}{143 b^2} \\ & = \frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(4 a) \int \frac {-\frac {143}{10} a b^2 e x^2+\frac {117}{68} b^2 (17 b c-8 a f) x^3+\frac {99}{76} b^2 (19 b d-10 a g) x^4}{\sqrt {a+b x^3}} \, dx}{429 b^3} \\ & = \frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(4 a) \int \frac {x^2 \left (-\frac {143}{10} a b^2 e+\frac {117}{68} b^2 (17 b c-8 a f) x+\frac {99}{76} b^2 (19 b d-10 a g) x^2\right )}{\sqrt {a+b x^3}} \, dx}{429 b^3} \\ & = \frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(8 a) \int \frac {-\frac {99}{38} a b^2 (19 b d-10 a g) x-\frac {1001}{20} a b^3 e x^2+\frac {819}{136} b^3 (17 b c-8 a f) x^3}{\sqrt {a+b x^3}} \, dx}{3003 b^4} \\ & = \frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(8 a) \int \frac {x \left (-\frac {99}{38} a b^2 (19 b d-10 a g)-\frac {1001}{20} a b^3 e x+\frac {819}{136} b^3 (17 b c-8 a f) x^2\right )}{\sqrt {a+b x^3}} \, dx}{3003 b^4} \\ & = \frac {6 a (17 b c-8 a f) x \sqrt {a+b x^3}}{935 b^2}+\frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(16 a) \int \frac {-\frac {819}{136} a b^3 (17 b c-8 a f)-\frac {495}{76} a b^3 (19 b d-10 a g) x-\frac {1001}{8} a b^4 e x^2}{\sqrt {a+b x^3}} \, dx}{15015 b^5} \\ & = \frac {6 a (17 b c-8 a f) x \sqrt {a+b x^3}}{935 b^2}+\frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {(16 a) \int \frac {-\frac {819}{136} a b^3 (17 b c-8 a f)-\frac {495}{76} a b^3 (19 b d-10 a g) x}{\sqrt {a+b x^3}} \, dx}{15015 b^5}-\frac {\left (2 a^2 e\right ) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx}{15 b} \\ & = -\frac {4 a^2 e \sqrt {a+b x^3}}{45 b^2}+\frac {6 a (17 b c-8 a f) x \sqrt {a+b x^3}}{935 b^2}+\frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}-\frac {\left (12 a^2 (19 b d-10 a g)\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{1729 b^{7/3}}-\frac {\left (6 a^2 \left (1729 \sqrt [3]{b} (17 b c-8 a f)-1870 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-10 a g)\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{1616615 b^{7/3}} \\ & = -\frac {4 a^2 e \sqrt {a+b x^3}}{45 b^2}+\frac {6 a (17 b c-8 a f) x \sqrt {a+b x^3}}{935 b^2}+\frac {6 a (19 b d-10 a g) x^2 \sqrt {a+b x^3}}{1729 b^2}+\frac {2 a e x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a f x^4 \sqrt {a+b x^3}}{187 b}+\frac {6 a g x^5 \sqrt {a+b x^3}}{247 b}-\frac {24 a^2 (19 b d-10 a g) \sqrt {a+b x^3}}{1729 b^{8/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x^3 \sqrt {a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} (19 b d-10 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 (\sin ^{-1}\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 )}{1729 b^{8/3} \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 {4\ 3^{3/4} \sqrt {2+\sqrt {3}} a^2 \left (1729 \sqrt [3]{b} (17 b c-8 a f)-1870 \left (1-\sqrt {3}\right ) \sqrt [3]{a} (19 b d-10 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}} F\left (\sin ^{-1}\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 )}{1616615 b^{8/3} \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}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 9.91 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.23 \[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {2 \sqrt {a+b x^3} \left (-\left (\left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \left (a (92378 e+90 x (988 f+935 g x))-3 b x \left (62985 c+11 x \left (4845 d+13 x \left (323 e+285 f x+255 g x^2\right )\right )\right )\right )\right )+11115 a (-17 b c+8 a f) x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )+8415 a (-19 b d+10 a g) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{2078505 b^2 \sqrt {1+\frac {b x^3}{a}}} \]

[In]

Integrate[x^3*Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4),x]

[Out]

(2*Sqrt[a + b*x^3]*(-((a + b*x^3)*Sqrt[1 + (b*x^3)/a]*(a*(92378*e + 90*x*(988*f + 935*g*x)) - 3*b*x*(62985*c +
 11*x*(4845*d + 13*x*(323*e + 285*f*x + 255*g*x^2))))) + 11115*a*(-17*b*c + 8*a*f)*x*Hypergeometric2F1[-1/2, 1
/3, 4/3, -((b*x^3)/a)] + 8415*a*(-19*b*d + 10*a*g)*x^2*Hypergeometric2F1[-1/2, 2/3, 5/3, -((b*x^3)/a)]))/(2078
505*b^2*Sqrt[1 + (b*x^3)/a])

Maple [A] (verified)

Time = 1.76 (sec) , antiderivative size = 956, normalized size of antiderivative = 1.30

method result size
elliptic \(\text {Expression too large to display}\) \(956\)
risch \(\text {Expression too large to display}\) \(1138\)
default \(\text {Expression too large to display}\) \(1674\)

[In]

int(x^3*(g*x^4+f*x^3+e*x^2+d*x+c)*(b*x^3+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/19*g*x^8*(b*x^3+a)^(1/2)+2/17*f*x^7*(b*x^3+a)^(1/2)+2/15*e*x^6*(b*x^3+a)^(1/2)+2/13*(3/19*a*g+b*d)/b*x^5*(b*
x^3+a)^(1/2)+2/11*(3/17*a*f+b*c)/b*x^4*(b*x^3+a)^(1/2)+2/45*a*e*x^3*(b*x^3+a)^(1/2)/b+2/7*(a*d-10/13*a/b*(3/19
*a*g+b*d))/b*x^2*(b*x^3+a)^(1/2)+2/5*(a*c-8/11*a/b*(3/17*a*f+b*c))/b*x*(b*x^3+a)^(1/2)-4/45*a^2*e*(b*x^3+a)^(1
/2)/b^2+4/15*I*a/b^2*(a*c-8/11*a/b*(3/17*a*f+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)*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))+8/21*I*a/b^2*(a*d-10/13*a/b*(3/19*a*g+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))*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)))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.28 \[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=-\frac {2 \, {\left (93366 \, {\left (17 \, a^{2} b c - 8 \, a^{3} f\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 100980 \, {\left (19 \, a^{2} b d - 10 \, a^{3} g\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (765765 \, b^{3} g x^{8} + 855855 \, b^{3} f x^{7} + 969969 \, b^{3} e x^{6} + 323323 \, a b^{2} e x^{3} + 58905 \, {\left (19 \, b^{3} d + 3 \, a b^{2} g\right )} x^{5} + 77805 \, {\left (17 \, b^{3} c + 3 \, a b^{2} f\right )} x^{4} - 646646 \, a^{2} b e + 25245 \, {\left (19 \, a b^{2} d - 10 \, a^{2} b g\right )} x^{2} + 46683 \, {\left (17 \, a b^{2} c - 8 \, a^{2} b f\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{14549535 \, b^{3}} \]

[In]

integrate(x^3*(g*x^4+f*x^3+e*x^2+d*x+c)*(b*x^3+a)^(1/2),x, algorithm="fricas")

[Out]

-2/14549535*(93366*(17*a^2*b*c - 8*a^3*f)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) - 100980*(19*a^2*b*d - 10*
a^3*g)*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) - (765765*b^3*g*x^8 + 855855*b^3*
f*x^7 + 969969*b^3*e*x^6 + 323323*a*b^2*e*x^3 + 58905*(19*b^3*d + 3*a*b^2*g)*x^5 + 77805*(17*b^3*c + 3*a*b^2*f
)*x^4 - 646646*a^2*b*e + 25245*(19*a*b^2*d - 10*a^2*b*g)*x^2 + 46683*(17*a*b^2*c - 8*a^2*b*f)*x)*sqrt(b*x^3 +
a))/b^3

Sympy [A] (verification not implemented)

Time = 2.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.32 \[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {\sqrt {a} c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {\sqrt {a} d x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {\sqrt {a} f x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} + \frac {\sqrt {a} g x^{8} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} + e \left (\begin {cases} - \frac {4 a^{2} \sqrt {a + b x^{3}}}{45 b^{2}} + \frac {2 a x^{3} \sqrt {a + b x^{3}}}{45 b} + \frac {2 x^{6} \sqrt {a + b x^{3}}}{15} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate(x**3*(g*x**4+f*x**3+e*x**2+d*x+c)*(b*x**3+a)**(1/2),x)

[Out]

sqrt(a)*c*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3)) + sqrt(a)*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)) + sqrt(a)*f*x**7*gamma(7/3)*hy
per((-1/2, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3)) + sqrt(a)*g*x**8*gamma(8/3)*hyper((-1/2, 8
/3), (11/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(11/3)) + e*Piecewise((-4*a**2*sqrt(a + b*x**3)/(45*b**2) + 2*
a*x**3*sqrt(a + b*x**3)/(45*b) + 2*x**6*sqrt(a + b*x**3)/15, Ne(b, 0)), (sqrt(a)*x**6/6, True))

Maxima [F]

\[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int { {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a} x^{3} \,d x } \]

[In]

integrate(x^3*(g*x^4+f*x^3+e*x^2+d*x+c)*(b*x^3+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*sqrt(b*x^3 + a)*x^3, x)

Giac [F]

\[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int { {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a} x^{3} \,d x } \]

[In]

integrate(x^3*(g*x^4+f*x^3+e*x^2+d*x+c)*(b*x^3+a)^(1/2),x, algorithm="giac")

[Out]

integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*sqrt(b*x^3 + a)*x^3, x)

Mupad [F(-1)]

Timed out. \[ \int x^3 \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\int x^3\,\sqrt {b\,x^3+a}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]

[In]

int(x^3*(a + b*x^3)^(1/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4),x)

[Out]

int(x^3*(a + b*x^3)^(1/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4), x)