\(\int x^3 (a+b x^3)^{3/2} (c+d x+e x^2+f x^3+g x^4) \, dx\) [458]

Optimal result

Integrand size = 35, antiderivative size = 791 \[ \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 {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 (\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 )}{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}} \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 )}{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}} \]

[Out]

2/3900225*x^3*(b*x^3+a)^(3/2)*(156009*g*x^5+169575*f*x^4+185725*e*x^3+205275*d*x^2+229425*c*x)-4/105*a^3*e*(b*
x^3+a)^(1/2)/b^2+54/21505*a^2*(-8*a*f+23*b*c)*x*(b*x^3+a)^(1/2)/b^2+54/8645*a^2*(-2*a*g+5*b*d)*x^2*(b*x^3+a)^(
1/2)/b^2+2/105*a^2*e*x^3*(b*x^3+a)^(1/2)/b+54/4301*a^2*f*x^4*(b*x^3+a)^(1/2)/b+54/6175*a^2*g*x^5*(b*x^3+a)^(1/
2)/b+2/185910725*a*x^3*(3522519*g*x^5+4279275*f*x^4+5311735*e*x^3+6774075*d*x^2+8947575*c*x)*(b*x^3+a)^(1/2)-2
16/8645*a^3*(-2*a*g+5*b*d)*(b*x^3+a)^(1/2)/b^(8/3)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))+108/8645*3^(1/4)*a^(10/3)*(
-2*a*g+5*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)-36/37
182145*3^(3/4)*a^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)*(1729*b^(1/3)*(-8*a*f+23*b*c)-8602*a^(1/3)*(-2*a*g+5*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 = 1.50 (sec) , antiderivative size = 791, normalized size of antiderivative = 1.00, number of steps used = 14, 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 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\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 (\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 )}{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 {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 {4 a^3 e \sqrt {a+b x^3}}{105 b^2}+\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 {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 {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}} \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} (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 {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} \]

[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 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 \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] (verified)

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

Time = 10.56 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.23 \[ \int x^3 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=\frac {2 \sqrt {a+b x^3} \left (-\left (a+b x^3\right )^2 \sqrt {1+\frac {b x^3}{a}} \left (10 a (7429 e+21 x (380 f+391 g x))-b x \left (229425 c+17 x \left (12075 d+19 x \left (575 e+525 f x+483 g x^2\right )\right )\right )\right )+9975 a^2 (-23 b c+8 a f) x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{3},\frac {4}{3},-\frac {b x^3}{a}\right )+41055 a^2 (-5 b d+2 a g) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{3900225 b^2 \sqrt {1+\frac {b x^3}{a}}} \]

[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])

Maple [A] (verified)

Time = 1.78 (sec) , antiderivative size = 1161, normalized size of antiderivative = 1.47

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1161\)
risch	\(\text {Expression too large to display}\)	\(1198\)
default	\(\text {Expression too large to display}\)	\(1764\)

[In]

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

[Out]

2/25*g*b*x^11*(b*x^3+a)^(1/2)+2/23*b*f*x^10*(b*x^3+a)^(1/2)+2/21*b*e*x^9*(b*x^3+a)^(1/2)+2/19*(28/25*a*b*g+b^2
*d)/b*x^8*(b*x^3+a)^(1/2)+2/17*(26/23*a*f*b+b^2*c)/b*x^7*(b*x^3+a)^(1/2)+16/105*a*e*x^6*(b*x^3+a)^(1/2)+2/13*(
a^2*g+2*a*b*d-16/19*a/b*(28/25*a*b*g+b^2*d))/b*x^5*(b*x^3+a)^(1/2)+2/11*(a^2*f+2*a*b*c-14/17*a/b*(26/23*a*f*b+
b^2*c))/b*x^4*(b*x^3+a)^(1/2)+2/105*a^2*e*x^3*(b*x^3+a)^(1/2)/b+2/7*(a^2*d-10/13*a/b*(a^2*g+2*a*b*d-16/19*a/b*
(28/25*a*b*g+b^2*d)))/b*x^2*(b*x^3+a)^(1/2)+2/5*(a^2*c-8/11*a/b*(a^2*f+2*a*b*c-14/17*a/b*(26/23*a*f*b+b^2*c)))
/b*x*(b*x^3+a)^(1/2)-4/105*a^3*e*(b*x^3+a)^(1/2)/b^2+4/15*I*a/b^2*(a^2*c-8/11*a/b*(a^2*f+2*a*b*c-14/17*a/b*(26
/23*a*f*b+b^2*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)*Ell
ipticF(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^2*d
-10/13*a/b*(a^2*g+2*a*b*d-16/19*a/b*(28/25*a*b*g+b^2*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 = 262, normalized size of antiderivative = 0.33 \[ \int x^3 \left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right ) \, dx=-\frac {2 \, {\left (1400490 \, {\left (23 \, a^{3} b c - 8 \, a^{4} f\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 6967620 \, {\left (5 \, a^{3} b d - 2 \, a^{4} g\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (22309287 \, b^{4} g x^{11} + 24249225 \, b^{4} f x^{10} + 26558675 \, b^{4} e x^{9} + 42493880 \, a b^{3} e x^{6} + 1174173 \, {\left (25 \, b^{4} d + 28 \, a b^{3} g\right )} x^{8} + 5311735 \, a^{2} b^{2} e x^{3} + 1426425 \, {\left (23 \, b^{4} c + 26 \, a b^{3} f\right )} x^{7} + 90321 \, {\left (550 \, a b^{3} d + 27 \, a^{2} b^{2} g\right )} x^{5} - 10623470 \, a^{3} b e + 129675 \, {\left (460 \, a b^{3} c + 27 \, a^{2} b^{2} f\right )} x^{4} + 1741905 \, {\left (5 \, a^{2} b^{2} d - 2 \, a^{3} b g\right )} x^{2} + 700245 \, {\left (23 \, a^{2} b^{2} c - 8 \, a^{3} b f\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{557732175 \, b^{3}} \]

[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]

-2/557732175*(1400490*(23*a^3*b*c - 8*a^4*f)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) - 6967620*(5*a^3*b*d -
2*a^4*g)*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) - (22309287*b^4*g*x^11 + 242492
25*b^4*f*x^10 + 26558675*b^4*e*x^9 + 42493880*a*b^3*e*x^6 + 1174173*(25*b^4*d + 28*a*b^3*g)*x^8 + 5311735*a^2*
b^2*e*x^3 + 1426425*(23*b^4*c + 26*a*b^3*f)*x^7 + 90321*(550*a*b^3*d + 27*a^2*b^2*g)*x^5 - 10623470*a^3*b*e +
129675*(460*a*b^3*c + 27*a^2*b^2*f)*x^4 + 1741905*(5*a^2*b^2*d - 2*a^3*b*g)*x^2 + 700245*(23*a^2*b^2*c - 8*a^3
*b*f)*x)*sqrt(b*x^3 + a))/b^3

Sympy [A] (verification not implemented)

Time = 4.23 (sec) , antiderivative size = 512, normalized size of antiderivative = 0.65 \[ \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 {a^{\frac {3}{2}} 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 {a^{\frac {3}{2}} 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 {a^{\frac {3}{2}} 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 {a^{\frac {3}{2}} 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 )} + \frac {\sqrt {a} b c 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} b d 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 )} + \frac {\sqrt {a} b f x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {13}{3}\right )} + \frac {\sqrt {a} b g x^{11} \Gamma \left (\frac {11}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{3} \\ \frac {14}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {14}{3}\right )} + a 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 ) + b e \left (\begin {cases} \frac {16 a^{3} \sqrt {a + b x^{3}}}{315 b^{3}} - \frac {8 a^{2} x^{3} \sqrt {a + b x^{3}}}{315 b^{2}} + \frac {2 a x^{6} \sqrt {a + b x^{3}}}{105 b} + \frac {2 x^{9} \sqrt {a + b x^{3}}}{21} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{9}}{9} & \text {otherwise} \end {cases}\right ) \]

[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))

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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