3.5.60 \(\int x (a+b x^3)^{3/2} (c+d x+e x^2+f x^3+g x^4) \, dx\) [460]

Optimal. Leaf size=723 \[ \frac {2 a^2 (7 b d-2 a g) \sqrt {a+b x^3}}{105 b^2}+\frac {54 a^2 e x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 (19 b c-4 a f) \sqrt {a+b x^3}}{1729 b^{5/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} (19 b c-4 a f) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{1729 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{7/3} \left (3458 a^{2/3} \sqrt [3]{b} e+935 \left (1-\sqrt {3}\right ) (19 b c-4 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{1616615 b^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[Out]

2/440895*x*(b*x^3+a)^(3/2)*(20995*g*x^5+23205*f*x^4+25935*e*x^3+29393*d*x^2+33915*c*x)+2/105*a^2*(-2*a*g+7*b*d
)*(b*x^3+a)^(1/2)/b^2+54/935*a^2*e*x*(b*x^3+a)^(1/2)/b+54/1729*a^2*f*x^2*(b*x^3+a)^(1/2)/b+2/105*a^2*g*x^3*(b*
x^3+a)^(1/2)/b+2/4849845*a*x*(138567*g*x^5+176715*f*x^4+233415*e*x^3+323323*d*x^2+479655*c*x)*(b*x^3+a)^(1/2)+
54/1729*a^2*(-4*a*f+19*b*c)*(b*x^3+a)^(1/2)/b^(5/3)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))-27/1729*3^(1/4)*a^(7/3)*(-
4*a*f+19*b*c)*(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)-18/16
16615*3^(3/4)*a^(7/3)*(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)*(3458*a^(2/3)*b^(1/3)*e+935*(-4*a*f+19*b*c)*(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^(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)

________________________________________________________________________________________

Rubi [A]
time = 0.82, antiderivative size = 723, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1840, 1850, 1902, 1900, 267, 1892, 224, 1891} \begin {gather*} -\frac {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} 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 (3458 a^{2/3} \sqrt [3]{b} e+935 \left (1-\sqrt {3}\right ) (19 b c-4 a f)\right )}{1616615 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 {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (19 b c-4 a f) 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 {54 a^2 \sqrt {a+b x^3} (19 b c-4 a f)}{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 d-2 a g)}{105 b^2}+\frac {54 a^2 e x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(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*d - 2*a*g)*Sqrt[a + b*x^3])/(105*b^2) + (54*a^2*e*x*Sqrt[a + b*x^3])/(935*b) + (54*a^2*f*x^2*Sqrt[
a + b*x^3])/(1729*b) + (2*a^2*g*x^3*Sqrt[a + b*x^3])/(105*b) + (54*a^2*(19*b*c - 4*a*f)*Sqrt[a + b*x^3])/(1729
*b^(5/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) + (2*x*(a + b*x^3)^(3/2)*(33915*c*x + 29393*d*x^2 + 25935*e*x^3
+ 23205*f*x^4 + 20995*g*x^5))/440895 + (2*a*x*Sqrt[a + b*x^3]*(479655*c*x + 323323*d*x^2 + 233415*e*x^3 + 1767
15*f*x^4 + 138567*g*x^5))/4849845 - (27*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(7/3)*(19*b*c - 4*a*f)*(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]) - (18*3^(3/4)*Sqrt[2 + S
qrt[3]]*a^(7/3)*(3458*a^(2/3)*b^(1/3)*e + 935*(1 - Sqrt[3])*(19*b*c - 4*a*f))*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2
/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*
a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(1616615*b^(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])

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 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 \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 \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {1}{2} (9 a) \int x \sqrt {a+b x^3} \left (\frac {2 c}{13}+\frac {2 d x}{15}+\frac {2 e x^2}{17}+\frac {2 f x^3}{19}+\frac {2 g x^4}{21}\right ) \, dx\\ &=\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac {1}{4} \left (27 a^2\right ) \int \frac {x \left (\frac {4 c}{91}+\frac {4 d x}{135}+\frac {4 e x^2}{187}+\frac {4 f x^3}{247}+\frac {4 g x^4}{315}\right )}{\sqrt {a+b x^3}} \, dx\\ &=\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac {\left (3 a^2\right ) \int \frac {x \left (\frac {18 b c}{91}+\frac {2}{105} (7 b d-2 a g) x+\frac {18}{187} b e x^2+\frac {18}{247} b f x^3\right )}{\sqrt {a+b x^3}} \, dx}{2 b}\\ &=\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac {\left (3 a^2\right ) \int \frac {x \left (\frac {9}{247} b (19 b c-4 a f)+\frac {1}{15} b (7 b d-2 a g) x+\frac {63}{187} b^2 e x^2\right )}{\sqrt {a+b x^3}} \, dx}{7 b^2}\\ &=\frac {54 a^2 e x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac {\left (6 a^2\right ) \int \frac {-\frac {63}{187} a b^2 e+\frac {45}{494} b^2 (19 b c-4 a f) x+\frac {1}{6} b^2 (7 b d-2 a g) x^2}{\sqrt {a+b x^3}} \, dx}{35 b^3}\\ &=\frac {54 a^2 e x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac {\left (6 a^2\right ) \int \frac {-\frac {63}{187} a b^2 e+\frac {45}{494} b^2 (19 b c-4 a f) x}{\sqrt {a+b x^3}} \, dx}{35 b^3}+\frac {\left (a^2 (7 b d-2 a g)\right ) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx}{35 b}\\ &=\frac {2 a^2 (7 b d-2 a g) \sqrt {a+b x^3}}{105 b^2}+\frac {54 a^2 e x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {2 x \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}+\frac {\left (27 a^2 (19 b c-4 a f)\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 (27 a^{7/3} \left (3458 a^{2/3} \sqrt [3]{b} e+935 \left (1-\sqrt {3}\right ) (19 b c-4 a f)\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{1616615 b^{4/3}}\\ &=\frac {2 a^2 (7 b d-2 a g) \sqrt {a+b x^3}}{105 b^2}+\frac {54 a^2 e x \sqrt {a+b x^3}}{935 b}+\frac {54 a^2 f x^2 \sqrt {a+b x^3}}{1729 b}+\frac {2 a^2 g x^3 \sqrt {a+b x^3}}{105 b}+\frac {54 a^2 (19 b c-4 a f) \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 \left (a+b x^3\right )^{3/2} \left (33915 c x+29393 d x^2+25935 e x^3+23205 f x^4+20995 g x^5\right )}{440895}+\frac {2 a x \sqrt {a+b x^3} \left (479655 c x+323323 d x^2+233415 e x^3+176715 f x^4+138567 g x^5\right )}{4849845}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{7/3} (19 b c-4 a f) \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 {18\ 3^{3/4} \sqrt {2+\sqrt {3}} a^{7/3} \left (3458 a^{2/3} \sqrt [3]{b} e+935 \left (1-\sqrt {3}\right ) (19 b c-4 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{1616615 b^{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}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 9.74, size = 148, normalized size = 0.20 \begin {gather*} -\frac {\sqrt {a+b x^3} \left (4 \left (a+b x^3\right )^2 \sqrt {1+\frac {b x^3}{a}} (-2261 b d+646 a g-5 b x (399 e+17 x (21 f+19 g x)))+7980 a^2 b e x \, _2F_1\left (-\frac {3}{2},\frac {1}{3};\frac {4}{3};-\frac {b x^3}{a}\right )+1785 a b (-19 b c+4 a f) x^2 \, _2F_1\left (-\frac {3}{2},\frac {2}{3};\frac {5}{3};-\frac {b x^3}{a}\right )\right )}{67830 b^2 \sqrt {1+\frac {b x^3}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/67830*(Sqrt[a + b*x^3]*(4*(a + b*x^3)^2*Sqrt[1 + (b*x^3)/a]*(-2261*b*d + 646*a*g - 5*b*x*(399*e + 17*x*(21*
f + 19*g*x))) + 7980*a^2*b*e*x*Hypergeometric2F1[-3/2, 1/3, 4/3, -((b*x^3)/a)] + 1785*a*b*(-19*b*c + 4*a*f)*x^
2*Hypergeometric2F1[-3/2, 2/3, 5/3, -((b*x^3)/a)]))/(b^2*Sqrt[1 + (b*x^3)/a])

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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. 1382 vs. \(2 (565 ) = 1130\).
time = 0.39, size = 1383, normalized size = 1.91

method	result	size
risch	\(-\frac {2 \left (-230945 b^{3} g \,x^{9}-255255 b^{3} f \,x^{8}-285285 b^{3} e \,x^{7}-369512 a \,b^{2} g \,x^{6}-323323 b^{3} d \,x^{6}-431970 a \,b^{2} f \,x^{5}-373065 b^{3} c \,x^{5}-518700 a \,b^{2} e \,x^{4}-46189 a^{2} b g \,x^{3}-646646 a \,b^{2} d \,x^{3}-75735 a^{2} b f \,x^{2}-852720 a \,b^{2} c \,x^{2}-140049 a^{2} b e x +92378 a^{3} g -323323 d \,a^{2} b \right ) \sqrt {b \,x^{3}+a}}{4849845 b^{2}}-\frac {27 a^{2} \left (-\frac {2 i \left (3740 a f -17765 b c \right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{3 b \sqrt {b \,x^{3}+a}}-\frac {6916 i a e \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{3 b \sqrt {b \,x^{3}+a}}\right )}{1616615 b}\)	\(889\)
elliptic	\(\text {Expression too large to display}\)	\(1045\)
default	\(\text {Expression too large to display}\)	\(1383\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g*(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))+f*(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+72/17
29*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))))+e*(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
*d*(b*x^3+a)^(5/2)/b+c*(2/13*b*x^5*(b*x^3+a)^(1/2)+32/91*a*x^2*(b*x^3+a)^(1/2)-18/91*I*a^2*3^(1/2)/b*(-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))))

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.09, size = 207, normalized size = 0.29 \begin {gather*} -\frac {2 \, {\left (280098 \, a^{3} \sqrt {b} e {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 75735 \, {\left (19 \, a^{2} b c - 4 \, a^{3} f\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (230945 \, b^{3} g x^{9} + 255255 \, b^{3} f x^{8} + 285285 \, b^{3} e x^{7} + 518700 \, a b^{2} e x^{4} + 46189 \, {\left (7 \, b^{3} d + 8 \, a b^{2} g\right )} x^{6} + 19635 \, {\left (19 \, b^{3} c + 22 \, a b^{2} f\right )} x^{5} + 140049 \, a^{2} b e x + 323323 \, a^{2} b d - 92378 \, a^{3} g + 46189 \, {\left (14 \, a b^{2} d + a^{2} b g\right )} x^{3} + 2805 \, {\left (304 \, a b^{2} c + 27 \, a^{2} b f\right )} x^{2}\right )} \sqrt {b x^{3} + a}\right )}}{4849845 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/4849845*(280098*a^3*sqrt(b)*e*weierstrassPInverse(0, -4*a/b, x) + 75735*(19*a^2*b*c - 4*a^3*f)*sqrt(b)*weie
rstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) - (230945*b^3*g*x^9 + 255255*b^3*f*x^8 + 285285*b^3*
e*x^7 + 518700*a*b^2*e*x^4 + 46189*(7*b^3*d + 8*a*b^2*g)*x^6 + 19635*(19*b^3*c + 22*a*b^2*f)*x^5 + 140049*a^2*
b*e*x + 323323*a^2*b*d - 92378*a^3*g + 46189*(14*a*b^2*d + a^2*b*g)*x^3 + 2805*(304*a*b^2*c + 27*a^2*b*f)*x^2)
*sqrt(b*x^3 + a))/b^2

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Sympy [A]
time = 4.09, size = 525, normalized size = 0.73 \begin {gather*} \frac {a^{\frac {3}{2}} c x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {a^{\frac {3}{2}} e 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}} f x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {\sqrt {a} b c x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} + \frac {\sqrt {a} b e 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 f 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 )} + a d \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 g \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 d \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 g \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*(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**2*gamma(2/3)*hyper((-1/2, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(5/3)) + a**(3/2)*e*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)*f*x**5*gamma(5/3)
*hyper((-1/2, 5/3), (8/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(8/3)) + sqrt(a)*b*c*x**5*gamma(5/3)*hyper((-1/2
, 5/3), (8/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(8/3)) + sqrt(a)*b*e*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*f*x**8*gamma(8/3)*hyper((-1/2, 8/3), (11/3,), b*x**
3*exp_polar(I*pi)/a)/(3*gamma(11/3)) + a*d*Piecewise((sqrt(a)*x**3/3, Eq(b, 0)), (2*(a + b*x**3)**(3/2)/(9*b),
True)) + a*g*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*d*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*g*Piecewise((1
6*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*(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, x)

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

[Out]

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

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