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

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

[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 = 1.55, 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 = {1826, 1836, 1888, 1594, 1886, 261, 1878, 218, 1877} \[ \frac {2 a^2 \sqrt {a+b x^3} (7 b c-2 a f)}{105 b^2}+\frac {36\ 3^{3/4} \sqrt {2+\sqrt {3}} a^3 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (43010 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e-1729 (23 b d-8 a g)\right ) F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{37182145 b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {54 a^2 x \sqrt {a+b x^3} (23 b d-8 a g)}{21505 b^2}-\frac {216 a^3 e \sqrt {a+b x^3}}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {108 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{10/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{1729 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {54 a^2 e x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 f x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 g x^4 \sqrt {a+b x^3}}{4301 b}+\frac {2 x^2 \left (a+b x^3\right )^{3/2} \left (52003 c x+45885 d x^2+41055 e x^3+37145 f x^4+33915 g x^5\right )}{780045}+\frac {2 a x^2 \sqrt {a+b x^3} \left (7436429 c x+5368545 d x^2+4064445 e x^3+3187041 f x^4+2567565 g x^5\right )}{111546435} \]

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 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 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 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 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 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]

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 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 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]

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]  time = 0.41, size = 162, normalized size = 0.22 \[ \frac {2 \left (1995 a^3 x \sqrt {\frac {b x^3}{a}+1} (8 a g-23 b d) \, _2F_1\left (-\frac {3}{2},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )-41055 a^3 b e x^2 \sqrt {\frac {b x^3}{a}+1} \, _2F_1\left (-\frac {3}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )+\left (a+b x^3\right )^3 (-38 a (391 f+420 g x)+52003 b c+5 b x (9177 d+17 x (483 e+19 x (23 f+21 g x))))\right )}{780045 b^2 \sqrt {a+b x^3}} \]

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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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b g x^{9} + b f x^{8} + b e x^{7} + {\left (b d + a g\right )} x^{6} + a e x^{4} + {\left (b c + a f\right )} x^{5} + a d x^{3} + a c x^{2}\right )} \sqrt {b x^{3} + a}, x\right ) \]

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]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} x^{2}\,{d x} \]

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

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maple [B]  time = 0.06, size = 1269, normalized size = 1.71 \[ \text {result too large to display} \]

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)

[Out]

g*(2/23*(b*x^3+a)^(1/2)*b*x^10+52/391*(b*x^3+a)^(1/2)*a*x^7+54/4301*(b*x^3+a)^(1/2)*a^2/b*x^4-432/21505*(b*x^3
+a)^(1/2)*a^3/b^2*x-288/21505*I*a^4/b^3*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(-a*b^
2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2)*((x-(-a*b^2)^(1/3)/b)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2
)^(1/3)/b))^(1/2)*(-I*(x+1/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2)/
(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b
^2)^(1/3)*b)^(1/2),(I*3^(1/2)*(-a*b^2)^(1/3)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)/b)^(1/2)))
+f*(2/21*(b*x^3+a)^(1/2)*b*x^9+16/105*(b*x^3+a)^(1/2)*a*x^6+2/105*(b*x^3+a)^(1/2)*a^2/b*x^3-4/105*(b*x^3+a)^(1
/2)*a^3/b^2)+e*(2/19*(b*x^3+a)^(1/2)*b*x^8+44/247*(b*x^3+a)^(1/2)*a*x^5+54/1729*(b*x^3+a)^(1/2)*a^2/b*x^2+72/1
729*I*a^3/b^2*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^
2)^(1/3)*b)^(1/2)*((x-(-a*b^2)^(1/3)/b)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b))^(1/2)*(-I*(x+1
/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2)/(b*x^3+a)^(1/2)*((-3/2*(-a
*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*EllipticE(1/3*3^(1/2)*(I*(x+1/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(
-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2),(I*3^(1/2)*(-a*b^2)^(1/3)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2
)*(-a*b^2)^(1/3)/b)/b)^(1/2))+(-a*b^2)^(1/3)/b*EllipticF(1/3*3^(1/2)*(I*(x+1/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*
(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2),(I*3^(1/2)*(-a*b^2)^(1/3)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/
2)*(-a*b^2)^(1/3)/b)/b)^(1/2))))+d*(2/17*(b*x^3+a)^(1/2)*b*x^7+40/187*(b*x^3+a)^(1/2)*a*x^4+54/935*(b*x^3+a)^(
1/2)*a^2/b*x+36/935*I*a^3/b^2*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b
)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2)*((x-(-a*b^2)^(1/3)/b)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)
)^(1/2)*(-I*(x+1/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*b)^(1/2)/(b*x^3+a)^
(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2*(-a*b^2)^(1/3)/b-1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)*3^(1/2)/(-a*b^2)^(1/3)*
b)^(1/2),(I*3^(1/2)*(-a*b^2)^(1/3)/(-3/2*(-a*b^2)^(1/3)/b+1/2*I*3^(1/2)*(-a*b^2)^(1/3)/b)/b)^(1/2)))+2/15*c/b*
(b*x^3+a)^(5/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} c}{15 \, b} + \int {\left (b g x^{9} + b f x^{8} + b e x^{7} + a f x^{5} + {\left (b d + a g\right )} x^{6} + a e x^{4} + a d x^{3}\right )} \sqrt {b x^{3} + a}\,{d x} \]

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

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

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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sympy [A]  time = 11.69, size = 525, normalized size = 0.71 \[ \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 ) \]

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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