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

Optimal. Leaf size=791 \[ -\frac{36\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right ) \left (1729 \sqrt [3]{b} (23 b c-8 a f)-8602 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (5 b d-2 a g)\right )}{37182145 b^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{54 a^2 x \sqrt{a+b x^3} (23 b c-8 a f)}{21505 b^2}+\frac{54 a^2 x^2 \sqrt{a+b x^3} (5 b d-2 a g)}{8645 b^2}-\frac{216 a^3 \sqrt{a+b x^3} (5 b d-2 a g)}{8645 b^{8/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{108 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 b d-2 a g) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{8645 b^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{4 a^3 e \sqrt{a+b x^3}}{105 b^2}+\frac{2 a^2 e x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 f x^4 \sqrt{a+b x^3}}{4301 b}+\frac{54 a^2 g x^5 \sqrt{a+b x^3}}{6175 b}+\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725} \]

[Out]

(-4*a^3*e*Sqrt[a + b*x^3])/(105*b^2) + (54*a^2*(23*b*c - 8*a*f)*x*Sqrt[a + b*x^3])/(21505*b^2) + (54*a^2*(5*b*
d - 2*a*g)*x^2*Sqrt[a + b*x^3])/(8645*b^2) + (2*a^2*e*x^3*Sqrt[a + b*x^3])/(105*b) + (54*a^2*f*x^4*Sqrt[a + b*
x^3])/(4301*b) + (54*a^2*g*x^5*Sqrt[a + b*x^3])/(6175*b) - (216*a^3*(5*b*d - 2*a*g)*Sqrt[a + b*x^3])/(8645*b^(
8/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) + (2*x^3*(a + b*x^3)^(3/2)*(229425*c*x + 205275*d*x^2 + 185725*e*x^3
+ 169575*f*x^4 + 156009*g*x^5))/3900225 + (2*a*x^3*Sqrt[a + b*x^3]*(8947575*c*x + 6774075*d*x^2 + 5311735*e*x
^3 + 4279275*f*x^4 + 3522519*g*x^5))/185910725 + (108*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(10/3)*(5*b*d - 2*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]*Ellip
ticE[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]])/(8645*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]) - (36*3^(3
/4)*Sqrt[2 + Sqrt[3]]*a^3*(1729*b^(1/3)*(23*b*c - 8*a*f) - 8602*(1 - Sqrt[3])*a^(1/3)*(5*b*d - 2*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]*Ellipti
cF[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]])/(37182145
*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])

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Rubi [A]  time = 2.11123, antiderivative size = 791, normalized size of antiderivative = 1., number of steps used = 14, 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} \[ -\frac{36\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) \left (1729 \sqrt [3]{b} (23 b c-8 a f)-8602 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (5 b d-2 a g)\right )}{37182145 b^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{54 a^2 x \sqrt{a+b x^3} (23 b c-8 a f)}{21505 b^2}+\frac{54 a^2 x^2 \sqrt{a+b x^3} (5 b d-2 a g)}{8645 b^2}-\frac{216 a^3 \sqrt{a+b x^3} (5 b d-2 a g)}{8645 b^{8/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{108 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (5 b d-2 a g) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{8645 b^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{4 a^3 e \sqrt{a+b x^3}}{105 b^2}+\frac{2 a^2 e x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 f x^4 \sqrt{a+b x^3}}{4301 b}+\frac{54 a^2 g x^5 \sqrt{a+b x^3}}{6175 b}+\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a^3*e*Sqrt[a + b*x^3])/(105*b^2) + (54*a^2*(23*b*c - 8*a*f)*x*Sqrt[a + b*x^3])/(21505*b^2) + (54*a^2*(5*b*
d - 2*a*g)*x^2*Sqrt[a + b*x^3])/(8645*b^2) + (2*a^2*e*x^3*Sqrt[a + b*x^3])/(105*b) + (54*a^2*f*x^4*Sqrt[a + b*
x^3])/(4301*b) + (54*a^2*g*x^5*Sqrt[a + b*x^3])/(6175*b) - (216*a^3*(5*b*d - 2*a*g)*Sqrt[a + b*x^3])/(8645*b^(
8/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) + (2*x^3*(a + b*x^3)^(3/2)*(229425*c*x + 205275*d*x^2 + 185725*e*x^3
+ 169575*f*x^4 + 156009*g*x^5))/3900225 + (2*a*x^3*Sqrt[a + b*x^3]*(8947575*c*x + 6774075*d*x^2 + 5311735*e*x
^3 + 4279275*f*x^4 + 3522519*g*x^5))/185910725 + (108*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(10/3)*(5*b*d - 2*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]*Ellip
ticE[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]])/(8645*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]) - (36*3^(3
/4)*Sqrt[2 + Sqrt[3]]*a^3*(1729*b^(1/3)*(23*b*c - 8*a*f) - 8602*(1 - Sqrt[3])*a^(1/3)*(5*b*d - 2*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]*Ellipti
cF[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]])/(37182145
*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 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx &=\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{1}{2} (9 a) \int x^3 \sqrt{a+b x^3} \left (\frac{2 c}{17}+\frac{2 d x}{19}+\frac{2 e x^2}{21}+\frac{2 f x^3}{23}+\frac{2 g x^4}{25}\right ) \, dx\\ &=\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725}+\frac{1}{4} \left (27 a^2\right ) \int \frac{x^3 \left (\frac{4 c}{187}+\frac{4 d x}{247}+\frac{4 e x^2}{315}+\frac{4 f x^3}{391}+\frac{4 g x^4}{475}\right )}{\sqrt{a+b x^3}} \, dx\\ &=\frac{54 a^2 g x^5 \sqrt{a+b x^3}}{6175 b}+\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725}+\frac{\left (27 a^2\right ) \int \frac{x^3 \left (\frac{26 b c}{187}+\frac{2}{95} (5 b d-2 a g) x+\frac{26}{315} b e x^2+\frac{26}{391} b f x^3\right )}{\sqrt{a+b x^3}} \, dx}{26 b}\\ &=\frac{54 a^2 f x^4 \sqrt{a+b x^3}}{4301 b}+\frac{54 a^2 g x^5 \sqrt{a+b x^3}}{6175 b}+\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725}+\frac{\left (27 a^2\right ) \int \frac{x^3 \left (\frac{13}{391} b (23 b c-8 a f)+\frac{11}{95} b (5 b d-2 a g) x+\frac{143}{315} b^2 e x^2\right )}{\sqrt{a+b x^3}} \, dx}{143 b^2}\\ &=\frac{2 a^2 e x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 f x^4 \sqrt{a+b x^3}}{4301 b}+\frac{54 a^2 g x^5 \sqrt{a+b x^3}}{6175 b}+\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725}+\frac{\left (6 a^2\right ) \int \frac{-\frac{143}{105} a b^2 e x^2+\frac{117}{782} b^2 (23 b c-8 a f) x^3+\frac{99}{190} b^2 (5 b d-2 a g) x^4}{\sqrt{a+b x^3}} \, dx}{143 b^3}\\ &=\frac{2 a^2 e x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 f x^4 \sqrt{a+b x^3}}{4301 b}+\frac{54 a^2 g x^5 \sqrt{a+b x^3}}{6175 b}+\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725}+\frac{\left (6 a^2\right ) \int \frac{x^2 \left (-\frac{143}{105} a b^2 e+\frac{117}{782} b^2 (23 b c-8 a f) x+\frac{99}{190} b^2 (5 b d-2 a g) x^2\right )}{\sqrt{a+b x^3}} \, dx}{143 b^3}\\ &=\frac{54 a^2 (5 b d-2 a g) x^2 \sqrt{a+b x^3}}{8645 b^2}+\frac{2 a^2 e x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 f x^4 \sqrt{a+b x^3}}{4301 b}+\frac{54 a^2 g x^5 \sqrt{a+b x^3}}{6175 b}+\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725}+\frac{\left (12 a^2\right ) \int \frac{-\frac{99}{95} a b^2 (5 b d-2 a g) x-\frac{143}{30} a b^3 e x^2+\frac{819 b^3 (23 b c-8 a f) x^3}{1564}}{\sqrt{a+b x^3}} \, dx}{1001 b^4}\\ &=\frac{54 a^2 (5 b d-2 a g) x^2 \sqrt{a+b x^3}}{8645 b^2}+\frac{2 a^2 e x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 f x^4 \sqrt{a+b x^3}}{4301 b}+\frac{54 a^2 g x^5 \sqrt{a+b x^3}}{6175 b}+\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725}+\frac{\left (12 a^2\right ) \int \frac{x \left (-\frac{99}{95} a b^2 (5 b d-2 a g)-\frac{143}{30} a b^3 e x+\frac{819 b^3 (23 b c-8 a f) x^2}{1564}\right )}{\sqrt{a+b x^3}} \, dx}{1001 b^4}\\ &=\frac{54 a^2 (23 b c-8 a f) x \sqrt{a+b x^3}}{21505 b^2}+\frac{54 a^2 (5 b d-2 a g) x^2 \sqrt{a+b x^3}}{8645 b^2}+\frac{2 a^2 e x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 f x^4 \sqrt{a+b x^3}}{4301 b}+\frac{54 a^2 g x^5 \sqrt{a+b x^3}}{6175 b}+\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725}+\frac{\left (24 a^2\right ) \int \frac{-\frac{819 a b^3 (23 b c-8 a f)}{1564}-\frac{99}{38} a b^3 (5 b d-2 a g) x-\frac{143}{12} a b^4 e x^2}{\sqrt{a+b x^3}} \, dx}{5005 b^5}\\ &=\frac{54 a^2 (23 b c-8 a f) x \sqrt{a+b x^3}}{21505 b^2}+\frac{54 a^2 (5 b d-2 a g) x^2 \sqrt{a+b x^3}}{8645 b^2}+\frac{2 a^2 e x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 f x^4 \sqrt{a+b x^3}}{4301 b}+\frac{54 a^2 g x^5 \sqrt{a+b x^3}}{6175 b}+\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725}+\frac{\left (24 a^2\right ) \int \frac{-\frac{819 a b^3 (23 b c-8 a f)}{1564}-\frac{99}{38} a b^3 (5 b d-2 a g) x}{\sqrt{a+b x^3}} \, dx}{5005 b^5}-\frac{\left (2 a^3 e\right ) \int \frac{x^2}{\sqrt{a+b x^3}} \, dx}{35 b}\\ &=-\frac{4 a^3 e \sqrt{a+b x^3}}{105 b^2}+\frac{54 a^2 (23 b c-8 a f) x \sqrt{a+b x^3}}{21505 b^2}+\frac{54 a^2 (5 b d-2 a g) x^2 \sqrt{a+b x^3}}{8645 b^2}+\frac{2 a^2 e x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 f x^4 \sqrt{a+b x^3}}{4301 b}+\frac{54 a^2 g x^5 \sqrt{a+b x^3}}{6175 b}+\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725}-\frac{\left (108 a^3 (5 b d-2 a g)\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{8645 b^{7/3}}-\frac{\left (54 a^3 \left (1729 \sqrt [3]{b} (23 b c-8 a f)-8602 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (5 b d-2 a g)\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{37182145 b^{7/3}}\\ &=-\frac{4 a^3 e \sqrt{a+b x^3}}{105 b^2}+\frac{54 a^2 (23 b c-8 a f) x \sqrt{a+b x^3}}{21505 b^2}+\frac{54 a^2 (5 b d-2 a g) x^2 \sqrt{a+b x^3}}{8645 b^2}+\frac{2 a^2 e x^3 \sqrt{a+b x^3}}{105 b}+\frac{54 a^2 f x^4 \sqrt{a+b x^3}}{4301 b}+\frac{54 a^2 g x^5 \sqrt{a+b x^3}}{6175 b}-\frac{216 a^3 (5 b d-2 a g) \sqrt{a+b x^3}}{8645 b^{8/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 x^3 \left (a+b x^3\right )^{3/2} \left (229425 c x+205275 d x^2+185725 e x^3+169575 f x^4+156009 g x^5\right )}{3900225}+\frac{2 a x^3 \sqrt{a+b x^3} \left (8947575 c x+6774075 d x^2+5311735 e x^3+4279275 f x^4+3522519 g x^5\right )}{185910725}+\frac{108 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} (5 b d-2 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 )}{8645 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{36\ 3^{3/4} \sqrt{2+\sqrt{3}} a^3 \left (1729 \sqrt [3]{b} (23 b c-8 a f)-8602 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (5 b d-2 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 )}{37182145 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 = 0.52666, size = 179, normalized size = 0.23 \[ \frac{2 \sqrt{a+b x^3} \left (9975 a^2 x (8 a f-23 b c) \, _2F_1\left (-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )+41055 a^2 x^2 (2 a g-5 b d) \, _2F_1\left (-\frac{3}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )-\left (a+b x^3\right )^2 \sqrt{\frac{b x^3}{a}+1} \left (10 a (7429 e+21 x (380 f+391 g x))-b x \left (229425 c+17 x \left (12075 d+19 x \left (575 e+525 f x+483 g x^2\right )\right )\right )\right )\right )}{3900225 b^2 \sqrt{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + b*x^3]*(-((a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*(10*a*(7429*e + 21*x*(380*f + 391*g*x)) - b*x*(229425*
c + 17*x*(12075*d + 19*x*(575*e + 525*f*x + 483*g*x^2))))) + 9975*a^2*(-23*b*c + 8*a*f)*x*Hypergeometric2F1[-3
/2, 1/3, 4/3, -((b*x^3)/a)] + 41055*a^2*(-5*b*d + 2*a*g)*x^2*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)]))
/(3900225*b^2*Sqrt[1 + (b*x^3)/a])

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Maple [B]  time = 0.017, size = 1764, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g*(2/25*b*x^11*(b*x^3+a)^(1/2)+56/475*a*x^8*(b*x^3+a)^(1/2)+54/6175/b*a^2*x^5*(b*x^3+a)^(1/2)-108/8645*a^3/b^2
*x^2*(b*x^3+a)^(1/2)-144/8645*I*a^4/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*(2/23*b*x^10*(b*x^3+a)^(1/2)+52/391*a*x^7*(b*
x^3+a)^(1/2)+54/4301/b*a^2*x^4*(b*x^3+a)^(1/2)-432/21505*a^3/b^2*x*(b*x^3+a)^(1/2)-288/21505*I*a^4/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*(2/21*b*x^9*(b*x^3+a)^(1/2)+16/105*a*x^6*(
b*x^3+a)^(1/2)+2/105/b*a^2*x^3*(b*x^3+a)^(1/2)-4/105*a^3/b^2*(b*x^3+a)^(1/2))+d*(2/19*b*x^8*(b*x^3+a)^(1/2)+44
/247*a*x^5*(b*x^3+a)^(1/2)+54/1729/b*a^2*x^2*(b*x^3+a)^(1/2)+72/1729*I/b^2*a^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))))+c*(2/17*
b*x^7*(b*x^3+a)^(1/2)+40/187*a*x^4*(b*x^3+a)^(1/2)+54/935/b*a^2*x*(b*x^3+a)^(1/2)+36/935*I/b^2*a^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)))

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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 )}{\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b g x^{10} + b f x^{9} + b e x^{8} +{\left (b d + a g\right )} x^{7} + a e x^{5} +{\left (b c + a f\right )} x^{6} + a d x^{4} + a 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*(b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="fricas")

[Out]

integral((b*g*x^10 + b*f*x^9 + b*e*x^8 + (b*d + a*g)*x^7 + a*e*x^5 + (b*c + a*f)*x^6 + a*d*x^4 + a*c*x^3)*sqrt
(b*x^3 + a), x)

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Sympy [A]  time = 10.7682, size = 512, normalized size = 0.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(3/2)*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)) + a**(3/2)*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)) + a**(3/2)*f*x**7*gamma(7/3)
*hyper((-1/2, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3)) + a**(3/2)*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)) + sqrt(a)*b*c*x**7*gamma(7/3)*hyper((-1/2, 7/3), (
10/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(10/3)) + sqrt(a)*b*d*x**8*gamma(8/3)*hyper((-1/2, 8/3), (11/3,), b*
x**3*exp_polar(I*pi)/a)/(3*gamma(11/3)) + sqrt(a)*b*f*x**10*gamma(10/3)*hyper((-1/2, 10/3), (13/3,), b*x**3*ex
p_polar(I*pi)/a)/(3*gamma(13/3)) + sqrt(a)*b*g*x**11*gamma(11/3)*hyper((-1/2, 11/3), (14/3,), b*x**3*exp_polar
(I*pi)/a)/(3*gamma(14/3)) + a*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)) + b*e*Piecewise((16*a**3*sqrt(a + b*x**3)/
(315*b**3) - 8*a**2*x**3*sqrt(a + b*x**3)/(315*b**2) + 2*a*x**6*sqrt(a + b*x**3)/(105*b) + 2*x**9*sqrt(a + b*x
**3)/21, Ne(b, 0)), (sqrt(a)*x**9/9, 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 )}{\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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