3.5.59 \(\int x^2 (a+b x^3)^{3/2} (c+d x+e x^2+f x^3+g x^4) \, dx\) [459]

Optimal. Leaf size=742 \[ \frac {2 a^2 (7 b c-2 a f) \sqrt {a+b x^3}}{105 b^2}+\frac {54 a^2 (23 b d-8 a g) x \sqrt {a+b x^3}}{21505 b^2}+\frac {54 a^2 e x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 f x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 g x^4 \sqrt {a+b x^3}}{4301 b}-\frac {216 a^3 e \sqrt {a+b x^3}}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac {2 a x^2 \sqrt {a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac {108 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{1729 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {36\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (43010 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-1729 (23 b d-8 a g)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{37182145 b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[Out]

2/780045*x^2*(b*x^3+a)^(3/2)*(33915*g*x^5+37145*f*x^4+41055*e*x^3+45885*d*x^2+52003*c*x)+2/105*a^2*(-2*a*f+7*b
*c)*(b*x^3+a)^(1/2)/b^2+54/21505*a^2*(-8*a*g+23*b*d)*x*(b*x^3+a)^(1/2)/b^2+54/1729*a^2*e*x^2*(b*x^3+a)^(1/2)/b
+2/105*a^2*f*x^3*(b*x^3+a)^(1/2)/b+54/4301*a^2*g*x^4*(b*x^3+a)^(1/2)/b+2/111546435*a*x^2*(2567565*g*x^5+318704
1*f*x^4+4064445*e*x^3+5368545*d*x^2+7436429*c*x)*(b*x^3+a)^(1/2)-216/1729*a^3*e*(b*x^3+a)^(1/2)/b^(5/3)/(b^(1/
3)*x+a^(1/3)*(1+3^(1/2)))+108/1729*3^(1/4)*a^(10/3)*e*(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)+36/37182145*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)*(13832*a*g-39767*b*d+43010*a^(1/3)*b^(2/3)*e*(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^(7/3)/(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]
time = 1.04, antiderivative size = 742, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {1840, 1850, 1902, 1608, 1900, 267, 1892, 224, 1891} \begin {gather*} \frac {108 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} e \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 (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{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 {216 a^3 e \sqrt {a+b x^3}}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 a^2 \sqrt {a+b x^3} (7 b c-2 a f)}{105 b^2}+\frac {54 a^2 x \sqrt {a+b x^3} (23 b d-8 a g)}{21505 b^2}+\frac {54 a^2 e x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 f x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 g x^4 \sqrt {a+b x^3}}{4301 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}} F\left (\text {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 (43010 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-1729 (23 b d-8 a g)\right )}{37182145 b^{7/3} \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^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac {2 a x^2 \sqrt {a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*(7*b*c - 2*a*f)*Sqrt[a + b*x^3])/(105*b^2) + (54*a^2*(23*b*d - 8*a*g)*x*Sqrt[a + b*x^3])/(21505*b^2) +
(54*a^2*e*x^2*Sqrt[a + b*x^3])/(1729*b) + (2*a^2*f*x^3*Sqrt[a + b*x^3])/(105*b) + (54*a^2*g*x^4*Sqrt[a + b*x^3
])/(4301*b) - (216*a^3*e*Sqrt[a + b*x^3])/(1729*b^(5/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) + (2*x^2*(a + b*x
^3)^(3/2)*(52003*c*x + 45885*d*x^2 + 41055*e*x^3 + 37145*f*x^4 + 33915*g*x^5))/780045 + (2*a*x^2*Sqrt[a + b*x^
3]*(7436429*c*x + 5368545*d*x^2 + 4064445*e*x^3 + 3187041*f*x^4 + 2567565*g*x^5))/111546435 + (108*3^(1/4)*Sqr
t[2 - Sqrt[3]]*a^(10/3)*e*(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^(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]) + (36*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^3*(43010*(1 - Sqrt[3])*a^(1/3)*b^(2/3)*e - 1729*(23*b*d
- 8*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*S
qrt[3]])/(37182145*b^(7/3)*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*} \int x^2 \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^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac {1}{2} (9 a) \int x^2 \sqrt {a+b x^3} \left (\frac {2 c}{15}+\frac {2 d x}{17}+\frac {2 e x^2}{19}+\frac {2 f x^3}{21}+\frac {2 g x^4}{23}\right ) \, dx\\ &=\frac {2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac {2 a x^2 \sqrt {a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac {1}{4} \left (27 a^2\right ) \int \frac {x^2 \left (\frac {4 c}{135}+\frac {4 d x}{187}+\frac {4 e x^2}{247}+\frac {4 f x^3}{315}+\frac {4 g x^4}{391}\right )}{\sqrt {a+b x^3}} \, dx\\ &=\frac {54 a^2 g x^4 \sqrt {a+b x^3}}{4301 b}+\frac {2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac {2 a x^2 \sqrt {a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac {\left (27 a^2\right ) \int \frac {x^2 \left (\frac {22 b c}{135}+\frac {2}{391} (23 b d-8 a g) x+\frac {22}{247} b e x^2+\frac {22}{315} b f x^3\right )}{\sqrt {a+b x^3}} \, dx}{22 b}\\ &=\frac {2 a^2 f x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 g x^4 \sqrt {a+b x^3}}{4301 b}+\frac {2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac {2 a x^2 \sqrt {a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac {\left (3 a^2\right ) \int \frac {x^2 \left (\frac {11}{105} b (7 b c-2 a f)+\frac {9}{391} b (23 b d-8 a g) x+\frac {99}{247} b^2 e x^2\right )}{\sqrt {a+b x^3}} \, dx}{11 b^2}\\ &=\frac {54 a^2 e x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 f x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 g x^4 \sqrt {a+b x^3}}{4301 b}+\frac {2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac {2 a x^2 \sqrt {a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac {\left (6 a^2\right ) \int \frac {-\frac {198}{247} a b^2 e x+\frac {11}{30} b^2 (7 b c-2 a f) x^2+\frac {63}{782} b^2 (23 b d-8 a g) x^3}{\sqrt {a+b x^3}} \, dx}{77 b^3}\\ &=\frac {54 a^2 e x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 f x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 g x^4 \sqrt {a+b x^3}}{4301 b}+\frac {2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac {2 a x^2 \sqrt {a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac {\left (6 a^2\right ) \int \frac {x \left (-\frac {198}{247} a b^2 e+\frac {11}{30} b^2 (7 b c-2 a f) x+\frac {63}{782} b^2 (23 b d-8 a g) x^2\right )}{\sqrt {a+b x^3}} \, dx}{77 b^3}\\ &=\frac {54 a^2 (23 b d-8 a g) x \sqrt {a+b x^3}}{21505 b^2}+\frac {54 a^2 e x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 f x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 g x^4 \sqrt {a+b x^3}}{4301 b}+\frac {2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac {2 a x^2 \sqrt {a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac {\left (12 a^2\right ) \int \frac {-\frac {63}{782} a b^2 (23 b d-8 a g)-\frac {495}{247} a b^3 e x+\frac {11}{12} b^3 (7 b c-2 a f) x^2}{\sqrt {a+b x^3}} \, dx}{385 b^4}\\ &=\frac {54 a^2 (23 b d-8 a g) x \sqrt {a+b x^3}}{21505 b^2}+\frac {54 a^2 e x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 f x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 g x^4 \sqrt {a+b x^3}}{4301 b}+\frac {2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac {2 a x^2 \sqrt {a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac {\left (12 a^2\right ) \int \frac {-\frac {63}{782} a b^2 (23 b d-8 a g)-\frac {495}{247} a b^3 e x}{\sqrt {a+b x^3}} \, dx}{385 b^4}+\frac {\left (a^2 (7 b c-2 a f)\right ) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx}{35 b}\\ &=\frac {2 a^2 (7 b c-2 a f) \sqrt {a+b x^3}}{105 b^2}+\frac {54 a^2 (23 b d-8 a g) x \sqrt {a+b x^3}}{21505 b^2}+\frac {54 a^2 e x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 f x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 g x^4 \sqrt {a+b x^3}}{4301 b}+\frac {2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac {2 a x^2 \sqrt {a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}-\frac {\left (108 a^3 e\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{1729 b^{4/3}}-\frac {\left (54 a^3 \left (39767 b d-43010 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-13832 a g\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{37182145 b^2}\\ &=\frac {2 a^2 (7 b c-2 a f) \sqrt {a+b x^3}}{105 b^2}+\frac {54 a^2 (23 b d-8 a g) x \sqrt {a+b x^3}}{21505 b^2}+\frac {54 a^2 e x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 f x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 g x^4 \sqrt {a+b x^3}}{4301 b}-\frac {216 a^3 e \sqrt {a+b x^3}}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac {2 a x^2 \sqrt {a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435}+\frac {108 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} e \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^{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 {36\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (39767 b d-43010 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-13832 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^{7/3} \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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.12, size = 162, normalized size = 0.22 \begin {gather*} \frac {2 \left (\left (a+b x^3\right )^3 (52003 b c-38 a (391 f+420 g x)+5 b x (9177 d+17 x (483 e+19 x (23 f+21 g x))))+1995 a^3 (-23 b d+8 a g) x \sqrt {1+\frac {b x^3}{a}} \, _2F_1\left (-\frac {3}{2},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )-41055 a^3 b e x^2 \sqrt {1+\frac {b x^3}{a}} \, _2F_1\left (-\frac {3}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )\right )}{780045 b^2 \sqrt {a+b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a + b*x^3)^3*(52003*b*c - 38*a*(391*f + 420*g*x) + 5*b*x*(9177*d + 17*x*(483*e + 19*x*(23*f + 21*g*x))))
+ 1995*a^3*(-23*b*d + 8*a*g)*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)] - 41055*a^3
*b*e*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)]))/(780045*b^2*Sqrt[a + b*x^3])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1268 vs. \(2 (579 ) = 1158\).
time = 0.38, size = 1269, normalized size = 1.71

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1103\)
risch	\(\text {Expression too large to display}\)	\(1175\)
default	\(\text {Expression too large to display}\)	\(1269\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g*(2/23*b*x^10*(b*x^3+a)^(1/2)+52/391*a*x^7*(b*x^3+a)^(1/2)+54/4301*a^2*x^4*(b*x^3+a)^(1/2)/b-432/21505*a^3*x*
(b*x^3+a)^(1/2)/b^2-288/21505*I*a^4/b^3*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*
b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a
*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2
)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(
-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)
))+f*(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))+e*(2/19*b*x^8*(b*x^3+a)^(1/2)+44/247*a*x^5*(b*x^3+a)^(1/2)+54/1729*a^2*x^2*(b*x^3+a)^(1/2)/b+72
/1729*I/b^2*a^3*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))))+d*(2/17*b*x^7*(b*x^3+a)^(1/2)+40/187*a*x^4*(b*x^3+a)^(1/2)+54/935*a^2*x
*(b*x^3+a)^(1/2)/b+36/935*I/b^2*a^3*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)
^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2
)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b
*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b
^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)))+2
/15*c*(b*x^3+a)^(5/2)/b

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(b*x^3 + a)^(5/2)*c/b + integrate((b*g*x^9 + b*f*x^8 + b*x^7*e + a*f*x^5 + (b*d + a*g)*x^6 + a*x^4*e + a*
d*x^3)*sqrt(b*x^3 + a), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 237, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (6967620 \, a^{3} b^{\frac {3}{2}} e {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - 280098 \, {\left (23 \, a^{3} b d - 8 \, a^{4} g\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (4849845 \, b^{4} g x^{10} + 5311735 \, b^{4} f x^{9} + 5870865 \, b^{4} e x^{8} + 9935310 \, a b^{3} e x^{5} + 285285 \, {\left (23 \, b^{4} d + 26 \, a b^{3} g\right )} x^{7} + 1741905 \, a^{2} b^{2} e x^{2} + 1062347 \, {\left (7 \, b^{4} c + 8 \, a b^{3} f\right )} x^{6} + 7436429 \, a^{2} b^{2} c - 2124694 \, a^{3} b f + 25935 \, {\left (460 \, a b^{3} d + 27 \, a^{2} b^{2} g\right )} x^{4} + 1062347 \, {\left (14 \, a b^{3} c + a^{2} b^{2} f\right )} x^{3} + 140049 \, {\left (23 \, a^{2} b^{2} d - 8 \, a^{3} b g\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{111546435 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/111546435*(6967620*a^3*b^(3/2)*e*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) - 280098*(23*
a^3*b*d - 8*a^4*g)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) + (4849845*b^4*g*x^10 + 5311735*b^4*f*x^9 + 58708
65*b^4*e*x^8 + 9935310*a*b^3*e*x^5 + 285285*(23*b^4*d + 26*a*b^3*g)*x^7 + 1741905*a^2*b^2*e*x^2 + 1062347*(7*b
^4*c + 8*a*b^3*f)*x^6 + 7436429*a^2*b^2*c - 2124694*a^3*b*f + 25935*(460*a*b^3*d + 27*a^2*b^2*g)*x^4 + 1062347
*(14*a*b^3*c + a^2*b^2*f)*x^3 + 140049*(23*a^2*b^2*d - 8*a^3*b*g)*x)*sqrt(b*x^3 + a))/b^3

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Sympy [A]
time = 4.29, size = 525, normalized size = 0.71 \begin {gather*} \frac {a^{\frac {3}{2}} d 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}} e 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}} g 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^{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 e 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 g 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 )} + a c \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 ) + a f \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 c \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 f \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(3/2)*d*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)*e*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)*g*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**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*e*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*g*x**10*gamma(10/3)*hyper((-1/2, 10/3), (13/3,)
, b*x**3*exp_polar(I*pi)/a)/(3*gamma(13/3)) + a*c*Piecewise((sqrt(a)*x**3/3, Eq(b, 0)), (2*(a + b*x**3)**(3/2)
/(9*b), True)) + a*f*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*c*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*f*Piece
wise((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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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