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

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 1.90966, antiderivative size = 733, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {1826, 1836, 1888, 1594, 1886, 261, 1878, 218, 1877} \[ -\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) \left (1729 \sqrt [3]{b} (17 b c-8 a f)-1870 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (19 b d-10 a g)\right )}{1616615 b^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{24 a^2 \sqrt{a+b x^3} (19 b d-10 a g)}{1729 b^{8/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{12 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (19 b d-10 a g) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{4 a^2 e \sqrt{a+b x^3}}{45 b^2}+\frac{6 a x \sqrt{a+b x^3} (17 b c-8 a f)}{935 b^2}+\frac{6 a x^2 \sqrt{a+b x^3} (19 b d-10 a g)}{1729 b^2}+\frac{2 x^3 \sqrt{a+b x^3} \left (62985 c x+53295 d x^2+46189 e x^3+40755 f x^4+36465 g x^5\right )}{692835}+\frac{2 a e x^3 \sqrt{a+b x^3}}{45 b}+\frac{6 a f x^4 \sqrt{a+b x^3}}{187 b}+\frac{6 a g x^5 \sqrt{a+b x^3}}{247 b} \]

Antiderivative was successfully verified.

[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 1826

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 1836

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 1888

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]

Rule 1594

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 1886

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 261

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 1878

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 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((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]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 1877

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]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin{align*} \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 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]  time = 0.299319, size = 172, normalized size = 0.23 \[ \frac{2 \sqrt{a+b x^3} \left (-\left (a+b x^3\right ) \sqrt{\frac{b x^3}{a}+1} \left (a (92378 e+90 x (988 f+935 g x))-3 b x \left (62985 c+11 x \left (4845 d+13 x \left (323 e+285 f x+255 g x^2\right )\right )\right )\right )+11115 a x (8 a f-17 b c) \, _2F_1\left (-\frac{1}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )+8415 a x^2 (10 a g-19 b d) \, _2F_1\left (-\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )\right )}{2078505 b^2 \sqrt{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[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])

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Maple [B]  time = 0.019, size = 1674, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g*(2/19*x^8*(b*x^3+a)^(1/2)+6/247/b*a*x^5*(b*x^3+a)^(1/2)-60/1729*a^2/b^2*x^2*(b*x^3+a)^(1/2)-80/1729*I*a^3/b^
3*3^(1/2)*(-b^2*a)^(1/3)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^
(1/2)*((x-1/b*(-b^2*a)^(1/3))/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-b^2
*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-b^2*a)^(1
/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^
(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^
2*a)^(1/3)))^(1/2))+1/b*(-b^2*a)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*
a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(
-b^2*a)^(1/3)))^(1/2))))+f*(2/17*x^7*(b*x^3+a)^(1/2)+6/187/b*a*x^4*(b*x^3+a)^(1/2)-48/935*a^2/b^2*x*(b*x^3+a)^
(1/2)-32/935*I*a^3/b^3*3^(1/2)*(-b^2*a)^(1/3)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/
2)*b/(-b^2*a)^(1/3))^(1/2)*((x-1/b*(-b^2*a)^(1/3))/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/
2)*(-I*(x+1/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2)/(b*x^3+a)^(1/2)
*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1
/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2)))+e*(2/15*x^6*(b
*x^3+a)^(1/2)+2/45/b*a*x^3*(b*x^3+a)^(1/2)-4/45*a^2/b^2*(b*x^3+a)^(1/2))+d*(2/13*x^5*(b*x^3+a)^(1/2)+6/91/b*a*
x^2*(b*x^3+a)^(1/2)+8/91*I/b^2*a^2*3^(1/2)*(-b^2*a)^(1/3)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^
(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2)*((x-1/b*(-b^2*a)^(1/3))/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)
^(1/3)))^(1/2)*(-I*(x+1/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2)/(b*
x^3+a)^(1/2)*((-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-b^2*a
)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-
b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2))+1/b*(-b^2*a)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-b^
2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b
*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2))))+c*(2/11*x^4*(b*x^3+a)^(1/2)+6/55/b*a*x*(b*x^3+a)^(1/
2)+4/55*I/b^2*a^2*3^(1/2)*(-b^2*a)^(1/3)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/
(-b^2*a)^(1/3))^(1/2)*((x-1/b*(-b^2*a)^(1/3))/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2)*(-
I*(x+1/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*Elli
pticF(1/3*3^(1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2),(
I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt{b x^{3} + a} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (g x^{7} + f x^{6} + e x^{5} + d x^{4} + c x^{3}\right )} \sqrt{b x^{3} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

integral((g*x^7 + f*x^6 + e*x^5 + d*x^4 + c*x^3)*sqrt(b*x^3 + a), x)

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Sympy [A]  time = 5.10566, size = 238, normalized size = 0.32 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt{b x^{3} + a} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)