\(\int x \sqrt {a+b x^3} (c+d x+e x^2+f x^3+g x^4) \, dx\) [447]

Optimal result

Integrand size = 33, antiderivative size = 667 \[ \int x \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {2 a (5 b d-2 a g) \sqrt {a+b x^3}}{45 b^2}+\frac {6 a e x \sqrt {a+b x^3}}{55 b}+\frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a (13 b c-4 a f) \sqrt {a+b x^3}}{91 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} (13 b c-4 a f) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{91 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{4/3} \left (182 a^{2/3} \sqrt [3]{b} e+55 \left (1-\sqrt {3}\right ) (13 b c-4 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{5005 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[Out]

2/45*a*(-2*a*g+5*b*d)*(b*x^3+a)^(1/2)/b^2+6/55*a*e*x*(b*x^3+a)^(1/2)/b+6/91*a*f*x^2*(b*x^3+a)^(1/2)/b+2/45*a*g
*x^3*(b*x^3+a)^(1/2)/b+2/45045*x*(3003*g*x^5+3465*f*x^4+4095*e*x^3+5005*d*x^2+6435*c*x)*(b*x^3+a)^(1/2)+6/91*a
*(-4*a*f+13*b*c)*(b*x^3+a)^(1/2)/b^(5/3)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))-3/91*3^(1/4)*a^(4/3)*(-4*a*f+13*b*c)*
(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^(
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)-2/5005*3^(3/4)*a^(4
/3)*(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)*(182*a^(2/3)*b^(1/3)*e+55*(-4*a*f+13*b*c)*(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^(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)

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1840, 1850, 1902, 1900, 267, 1892, 224, 1891} \[ \int x \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=-\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (182 a^{2/3} \sqrt [3]{b} e+55 \left (1-\sqrt {3}\right ) (13 b c-4 a f)\right )}{5005 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (13 b c-4 a f) E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{91 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {6 a \sqrt {a+b x^3} (13 b c-4 a f)}{91 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 a \sqrt {a+b x^3} (5 b d-2 a g)}{45 b^2}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {6 a e x \sqrt {a+b x^3}}{55 b}+\frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b} \]

[In]

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

[Out]

(2*a*(5*b*d - 2*a*g)*Sqrt[a + b*x^3])/(45*b^2) + (6*a*e*x*Sqrt[a + b*x^3])/(55*b) + (6*a*f*x^2*Sqrt[a + b*x^3]
)/(91*b) + (2*a*g*x^3*Sqrt[a + b*x^3])/(45*b) + (6*a*(13*b*c - 4*a*f)*Sqrt[a + b*x^3])/(91*b^(5/3)*((1 + Sqrt[
3])*a^(1/3) + b^(1/3)*x)) + (2*x*Sqrt[a + b*x^3]*(6435*c*x + 5005*d*x^2 + 4095*e*x^3 + 3465*f*x^4 + 3003*g*x^5
))/45045 - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(4/3)*(13*b*c - 4*a*f)*(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]])/(91*b^(5/3)*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*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^(4/3)*(182*a^(2/3)*
b^(1/3)*e + 55*(1 - Sqrt[3])*(13*b*c - 4*a*f))*(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]])/(5005*b^(5/3)*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 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 \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {1}{2} (3 a) \int \frac {x \left (\frac {2 c}{7}+\frac {2 d x}{9}+\frac {2 e x^2}{11}+\frac {2 f x^3}{13}+\frac {2 g x^4}{15}\right )}{\sqrt {a+b x^3}} \, dx \\ & = \frac {2 a g x^3 \sqrt {a+b x^3}}{45 b}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {a \int \frac {x \left (\frac {9 b c}{7}+\frac {1}{5} (5 b d-2 a g) x+\frac {9}{11} b e x^2+\frac {9}{13} b f x^3\right )}{\sqrt {a+b x^3}} \, dx}{3 b} \\ & = \frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {(2 a) \int \frac {x \left (\frac {9}{26} b (13 b c-4 a f)+\frac {7}{10} b (5 b d-2 a g) x+\frac {63}{22} b^2 e x^2\right )}{\sqrt {a+b x^3}} \, dx}{21 b^2} \\ & = \frac {6 a e x \sqrt {a+b x^3}}{55 b}+\frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {(4 a) \int \frac {-\frac {63}{22} a b^2 e+\frac {45}{52} b^2 (13 b c-4 a f) x+\frac {7}{4} b^2 (5 b d-2 a g) x^2}{\sqrt {a+b x^3}} \, dx}{105 b^3} \\ & = \frac {6 a e x \sqrt {a+b x^3}}{55 b}+\frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {(4 a) \int \frac {-\frac {63}{22} a b^2 e+\frac {45}{52} b^2 (13 b c-4 a f) x}{\sqrt {a+b x^3}} \, dx}{105 b^3}+\frac {(a (5 b d-2 a g)) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx}{15 b} \\ & = \frac {2 a (5 b d-2 a g) \sqrt {a+b x^3}}{45 b^2}+\frac {6 a e x \sqrt {a+b x^3}}{55 b}+\frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}+\frac {(3 a (13 b c-4 a f)) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{91 b^{4/3}}-\frac {\left (3 a^{4/3} \left (182 a^{2/3} \sqrt [3]{b} e+55 \left (1-\sqrt {3}\right ) (13 b c-4 a f)\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{5005 b^{4/3}} \\ & = \frac {2 a (5 b d-2 a g) \sqrt {a+b x^3}}{45 b^2}+\frac {6 a e x \sqrt {a+b x^3}}{55 b}+\frac {6 a f x^2 \sqrt {a+b x^3}}{91 b}+\frac {2 a g x^3 \sqrt {a+b x^3}}{45 b}+\frac {6 a (13 b c-4 a f) \sqrt {a+b x^3}}{91 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x \sqrt {a+b x^3} \left (6435 c x+5005 d x^2+4095 e x^3+3465 f x^4+3003 g x^5\right )}{45045}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} (13 b c-4 a f) \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 )}{91 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\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{4/3} \left (182 a^{2/3} \sqrt [3]{b} e+55 \left (1-\sqrt {3}\right ) (13 b c-4 a f)\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 )}{5005 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}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 10.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.21 \[ \int x \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {\sqrt {a+b x^3} \left (-4 \left (a+b x^3\right ) \sqrt {1+\frac {b x^3}{a}} \left (286 a g-b \left (715 d+585 e x+495 f x^2+429 g x^3\right )\right )-2340 a b e x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )+495 b (13 b c-4 a f) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{12870 b^2 \sqrt {1+\frac {b x^3}{a}}} \]

[In]

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

[Out]

(Sqrt[a + b*x^3]*(-4*(a + b*x^3)*Sqrt[1 + (b*x^3)/a]*(286*a*g - b*(715*d + 585*e*x + 495*f*x^2 + 429*g*x^3)) -
2340*a*b*e*x*Hypergeometric2F1[-1/2, 1/3, 4/3, -((b*x^3)/a)] + 495*b*(13*b*c - 4*a*f)*x^2*Hypergeometric2F1[-
1/2, 2/3, 5/3, -((b*x^3)/a)]))/(12870*b^2*Sqrt[1 + (b*x^3)/a])

Maple [A] (verified)

Time = 1.68 (sec) , antiderivative size = 829, normalized size of antiderivative = 1.24

method	result	size
risch	\(\text {Expression too large to display}\)	\(829\)
elliptic	\(\text {Expression too large to display}\)	\(884\)
default	\(\text {Expression too large to display}\)	\(1311\)

[In]

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

[Out]

-2/45045*(-3003*b^2*g*x^6-3465*b^2*f*x^5-4095*b^2*e*x^4-1001*a*b*g*x^3-5005*b^2*d*x^3-1485*a*b*f*x^2-6435*b^2*
c*x^2-2457*a*b*e*x+2002*a^2*g-5005*a*b*d)/b^2*(b*x^3+a)^(1/2)-3/5005*a/b*(-364/3*I*a*e*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*(220*a*f-715*b*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)*((-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 = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.22 \[ \int x \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=-\frac {2 \, {\left (4914 \, a^{2} \sqrt {b} e {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 1485 \, {\left (13 \, a b c - 4 \, a^{2} f\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (3003 \, b^{2} g x^{6} + 3465 \, b^{2} f x^{5} + 4095 \, b^{2} e x^{4} + 2457 \, a b e x + 1001 \, {\left (5 \, b^{2} d + a b g\right )} x^{3} + 5005 \, a b d - 2002 \, a^{2} g + 495 \, {\left (13 \, b^{2} c + 3 \, a b f\right )} x^{2}\right )} \sqrt {b x^{3} + a}\right )}}{45045 \, b^{2}} \]

[In]

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

[Out]

-2/45045*(4914*a^2*sqrt(b)*e*weierstrassPInverse(0, -4*a/b, x) + 1485*(13*a*b*c - 4*a^2*f)*sqrt(b)*weierstrass
Zeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) - (3003*b^2*g*x^6 + 3465*b^2*f*x^5 + 4095*b^2*e*x^4 + 2457*
a*b*e*x + 1001*(5*b^2*d + a*b*g)*x^3 + 5005*a*b*d - 2002*a^2*g + 495*(13*b^2*c + 3*a*b*f)*x^2)*sqrt(b*x^3 + a)
)/b^2

Sympy [A] (verification not implemented)

Time = 2.02 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.33 \[ \int x \sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {\sqrt {a} c x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {\sqrt {a} e x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {\sqrt {a} f x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + d \left (\begin {cases} \frac {\sqrt {a} x^{3}}{3} & \text {for}\: b = 0 \\\frac {2 \left (a + b x^{3}\right )^{\frac {3}{2}}}{9 b} & \text {otherwise} \end {cases}\right ) + g \left (\begin {cases} - \frac {4 a^{2} \sqrt {a + b x^{3}}}{45 b^{2}} + \frac {2 a x^{3} \sqrt {a + b x^{3}}}{45 b} + \frac {2 x^{6} \sqrt {a + b x^{3}}}{15} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

sqrt(a)*c*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)*e*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)*f*x**5*gamma(5/3)*hy
per((-1/2, 5/3), (8/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(8/3)) + d*Piecewise((sqrt(a)*x**3/3, Eq(b, 0)), (2
*(a + b*x**3)**(3/2)/(9*b), True)) + g*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]

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

[In]

integrate(x*(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, x)

Giac [F]

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

[In]

integrate(x*(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, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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