3.471 \(\int \frac{(a+b x^3)^{3/2} (c+d x+e x^2+f x^3+g x^4)}{x^{10}} \, dx\)

Optimal. Leaf size=714 \[ -\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-112 a g+7 b d\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{2240 a \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{b^2 (b c-6 a f) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{3/2}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{224 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}+\frac{27 b^{7/3} e \sqrt{a+b x^3}}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}-\frac{b \sqrt{a+b x^3} \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right )}{1680}-\frac{\left (a+b x^3\right )^{3/2} \left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right )}{2520} \]

[Out]

-(b*((140*c)/x^6 + (189*d)/x^5 + (270*e)/x^4 + (420*f)/x^3 + (756*g)/x^2)*Sqrt[a + b*x^3])/1680 - (b^2*c*Sqrt[
a + b*x^3])/(24*a*x^3) - (27*b^2*d*Sqrt[a + b*x^3])/(320*a*x^2) - (27*b^2*e*Sqrt[a + b*x^3])/(112*a*x) + (27*b
^(7/3)*e*Sqrt[a + b*x^3])/(112*a*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (((280*c)/x^9 + (315*d)/x^8 + (360*e)/
x^7 + (420*f)/x^6 + (504*g)/x^5)*(a + b*x^3)^(3/2))/2520 + (b^2*(b*c - 6*a*f)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]
)/(24*a^(3/2)) - (27*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(7/3)*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]])/(224*a^(2/3)*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]) - (9*3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(5/3)*(7*b*d + 20*(1 - Sqr
t[3])*a^(1/3)*b^(2/3)*e - 112*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*Sqrt[3]])/(2240*a*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])

________________________________________________________________________________________

Rubi [A]  time = 1.12299, antiderivative size = 714, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {14, 1825, 1835, 1832, 266, 63, 208, 1878, 218, 1877} \[ \frac{b^2 (b c-6 a f) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{3/2}}-\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-112 a g+7 b d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{2240 a \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{7/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{224 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}+\frac{27 b^{7/3} e \sqrt{a+b x^3}}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}-\frac{b \sqrt{a+b x^3} \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right )}{1680}-\frac{\left (a+b x^3\right )^{3/2} \left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right )}{2520} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^10,x]

[Out]

-(b*((140*c)/x^6 + (189*d)/x^5 + (270*e)/x^4 + (420*f)/x^3 + (756*g)/x^2)*Sqrt[a + b*x^3])/1680 - (b^2*c*Sqrt[
a + b*x^3])/(24*a*x^3) - (27*b^2*d*Sqrt[a + b*x^3])/(320*a*x^2) - (27*b^2*e*Sqrt[a + b*x^3])/(112*a*x) + (27*b
^(7/3)*e*Sqrt[a + b*x^3])/(112*a*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (((280*c)/x^9 + (315*d)/x^8 + (360*e)/
x^7 + (420*f)/x^6 + (504*g)/x^5)*(a + b*x^3)^(3/2))/2520 + (b^2*(b*c - 6*a*f)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]]
)/(24*a^(3/2)) - (27*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(7/3)*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]])/(224*a^(2/3)*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]) - (9*3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(5/3)*(7*b*d + 20*(1 - Sqr
t[3])*a^(1/3)*b^(2/3)*e - 112*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*Sqrt[3]])/(2240*a*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 1825

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{u = IntHide[x^m*Pq, x]}, Simp[u*(a +
b*x^n)^p, x] - Dist[b*n*p, Int[x^(m + n)*(a + b*x^n)^(p - 1)*ExpandToSum[u/x^(m + 1), x], x], x]] /; FreeQ[{a
, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && GtQ[p, 0] && LtQ[m + Expon[Pq, x] + 1, 0]

Rule 1835

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{Pq0 = Coeff[Pq, x, 0]}, Simp[(Pq
0*(c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(2*a*c*(m + 1)), Int[(c*x)^(m + 1)*ExpandToSum
[(2*a*(m + 1)*(Pq - Pq0))/x - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b*x^n)^p, x], x] /; NeQ[Pq0, 0]]
/; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]

Rule 1832

Int[(Pq_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[Coeff[Pq, x, 0], Int[1/(x*Sqrt[a + b*x^n]), x
], x] + Int[ExpandToSum[(Pq - Coeff[Pq, x, 0])/x, x]/Sqrt[a + b*x^n], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] &
& IGtQ[n, 0] && NeQ[Coeff[Pq, x, 0], 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 1878

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a,
3]]}, Dist[(c*r - (1 - Sqrt[3])*d*s)/r, Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/r, Int[((1 - Sqrt[3])*s + r*x)
/Sqrt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 1877

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 - Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 - Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 + Sqrt[3])*s + r*x)), x
] - Simp[(3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^{10}} \, dx &=-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac{1}{2} (9 b) \int \frac{\sqrt{a+b x^3} \left (-\frac{c}{9}-\frac{d x}{8}-\frac{e x^2}{7}-\frac{f x^3}{6}-\frac{g x^4}{5}\right )}{x^7} \, dx\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac{1}{4} \left (27 b^2\right ) \int \frac{\frac{c}{54}+\frac{d x}{40}+\frac{e x^2}{28}+\frac{f x^3}{18}+\frac{g x^4}{10}}{x^4 \sqrt{a+b x^3}} \, dx\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac{\left (9 b^2\right ) \int \frac{-\frac{3 a d}{20}-\frac{3 a e x}{14}+\frac{1}{18} (b c-6 a f) x^2-\frac{3}{5} a g x^3}{x^3 \sqrt{a+b x^3}} \, dx}{8 a}\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac{\left (9 b^2\right ) \int \frac{\frac{6 a^2 e}{7}-\frac{2}{9} a (b c-6 a f) x-\frac{3}{20} a (b d-16 a g) x^2}{x^2 \sqrt{a+b x^3}} \, dx}{32 a^2}\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac{\left (9 b^2\right ) \int \frac{\frac{4}{9} a^2 (b c-6 a f)+\frac{3}{10} a^2 (b d-16 a g) x-\frac{6}{7} a^2 b e x^2}{x \sqrt{a+b x^3}} \, dx}{64 a^3}\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac{\left (9 b^2\right ) \int \frac{\frac{3}{10} a^2 (b d-16 a g)-\frac{6}{7} a^2 b e x}{\sqrt{a+b x^3}} \, dx}{64 a^3}-\frac{\left (b^2 (b c-6 a f)\right ) \int \frac{1}{x \sqrt{a+b x^3}} \, dx}{16 a}\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac{\left (27 b^{8/3} e\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{224 a}-\frac{\left (b^2 (b c-6 a f)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{48 a}-\frac{\left (27 b^2 \left (7 b d+20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-112 a g\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{4480 a}\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}+\frac{27 b^{7/3} e \sqrt{a+b x^3}}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{7/3} 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 )}{224 a^{2/3} \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{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left (7 b d+20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-112 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 )}{2240 a \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{(b (b c-6 a f)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{24 a}\\ &=-\frac{b \left (\frac{140 c}{x^6}+\frac{189 d}{x^5}+\frac{270 e}{x^4}+\frac{420 f}{x^3}+\frac{756 g}{x^2}\right ) \sqrt{a+b x^3}}{1680}-\frac{b^2 c \sqrt{a+b x^3}}{24 a x^3}-\frac{27 b^2 d \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 e \sqrt{a+b x^3}}{112 a x}+\frac{27 b^{7/3} e \sqrt{a+b x^3}}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{\left (\frac{280 c}{x^9}+\frac{315 d}{x^8}+\frac{360 e}{x^7}+\frac{420 f}{x^6}+\frac{504 g}{x^5}\right ) \left (a+b x^3\right )^{3/2}}{2520}+\frac{b^2 (b c-6 a f) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{3/2}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{7/3} 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 )}{224 a^{2/3} \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{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{5/3} \left (7 b d+20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e-112 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 )}{2240 a \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}\\ \end{align*}

Mathematica [C]  time = 0.644999, size = 226, normalized size = 0.32 \[ -\frac{\sqrt{a+b x^3} \left (2 x \left (7 x \left (5 a^3 f \left (3 b^2 x^6 \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )+a \left (2 a+5 b x^3\right ) \sqrt{\frac{b x^3}{a}+1}\right )+12 a^5 g x \, _2F_1\left (-\frac{5}{3},-\frac{3}{2};-\frac{2}{3};-\frac{b x^3}{a}\right )-8 b^3 c x^6 \left (a+b x^3\right )^2 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{b x^3}{a}+1\right )\right )+60 a^5 e \, _2F_1\left (-\frac{7}{3},-\frac{3}{2};-\frac{4}{3};-\frac{b x^3}{a}\right )\right )+105 a^5 d \, _2F_1\left (-\frac{8}{3},-\frac{3}{2};-\frac{5}{3};-\frac{b x^3}{a}\right )\right )}{840 a^4 x^8 \sqrt{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^10,x]

[Out]

-(Sqrt[a + b*x^3]*(105*a^5*d*Hypergeometric2F1[-8/3, -3/2, -5/3, -((b*x^3)/a)] + 2*x*(60*a^5*e*Hypergeometric2
F1[-7/3, -3/2, -4/3, -((b*x^3)/a)] + 7*x*(5*a^3*f*(a*(2*a + 5*b*x^3)*Sqrt[1 + (b*x^3)/a] + 3*b^2*x^6*ArcTanh[S
qrt[1 + (b*x^3)/a]]) + 12*a^5*g*x*Hypergeometric2F1[-5/3, -3/2, -2/3, -((b*x^3)/a)] - 8*b^3*c*x^6*(a + b*x^3)^
2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[5/2, 4, 7/2, 1 + (b*x^3)/a]))))/(840*a^4*x^8*Sqrt[1 + (b*x^3)/a])

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Maple [B]  time = 0.025, size = 1273, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^10,x)

[Out]

g*(-1/5*a*(b*x^3+a)^(1/2)/x^5-13/20*b*(b*x^3+a)^(1/2)/x^2-9/20*I*b*3^(1/2)*(-b^2*a)^(1/3)*(I*(x+1/2/b*(-b^2*a)
^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2)*((x-1/b*(-b^2*a)^(1/3))/(-3/2/b*(-b^2*a
)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(
1/2)*b/(-b^2*a)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*
(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/
2)/b*(-b^2*a)^(1/3)))^(1/2)))+d*(-1/8*a*(b*x^3+a)^(1/2)/x^8-19/80*b*(b*x^3+a)^(1/2)/x^5-27/320/a*b^2*(b*x^3+a)
^(1/2)/x^2+9/320*I/a*b^2*3^(1/2)*(-b^2*a)^(1/3)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(
1/2)*b/(-b^2*a)^(1/3))^(1/2)*((x-1/b*(-b^2*a)^(1/3))/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(
1/2)*(-I*(x+1/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2)/(b*x^3+a)^(1/
2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^
(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2)))+f*(-1/6*a*(b
*x^3+a)^(1/2)/x^6-5/12*b*(b*x^3+a)^(1/2)/x^3-1/4*b^2*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(1/2))+e*(-1/7*a*(b*x^
3+a)^(1/2)/x^7-17/56*b*(b*x^3+a)^(1/2)/x^4-27/112/a*b^2*(b*x^3+a)^(1/2)/x-9/112*I/a*b^2*3^(1/2)*(-b^2*a)^(1/3)
*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2)*((x-1/b*(-b^2*a)^(
1/3))/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/
b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^
2*a)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a
)^(1/3))^(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2))+1/b*
(-b^2*a)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^
2*a)^(1/3))^(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2))))
+c*(-1/9*a*(b*x^3+a)^(1/2)/x^9-7/36*b*(b*x^3+a)^(1/2)/x^6-1/24/a*b^2*(b*x^3+a)^(1/2)/x^3+1/24/a^(3/2)*b^3*arct
anh((b*x^3+a)^(1/2)/a^(1/2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^10,x, algorithm="fricas")

[Out]

integral((b*g*x^7 + b*f*x^6 + b*e*x^5 + (b*d + a*g)*x^4 + a*e*x^2 + (b*c + a*f)*x^3 + a*d*x + a*c)*sqrt(b*x^3
+ a)/x^10, x)

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Sympy [A]  time = 23.7195, size = 573, normalized size = 0.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(3/2)*(g*x**4+f*x**3+e*x**2+d*x+c)/x**10,x)

[Out]

a**(3/2)*d*gamma(-8/3)*hyper((-8/3, -1/2), (-5/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**8*gamma(-5/3)) + a**(3/2)*
e*gamma(-7/3)*hyper((-7/3, -1/2), (-4/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**7*gamma(-4/3)) + a**(3/2)*g*gamma(-
5/3)*hyper((-5/3, -1/2), (-2/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**5*gamma(-2/3)) + sqrt(a)*b*d*gamma(-5/3)*hyp
er((-5/3, -1/2), (-2/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**5*gamma(-2/3)) + sqrt(a)*b*e*gamma(-4/3)*hyper((-4/3
, -1/2), (-1/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**4*gamma(-1/3)) + sqrt(a)*b*g*gamma(-2/3)*hyper((-2/3, -1/2),
(1/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**2*gamma(1/3)) - a**2*c/(9*sqrt(b)*x**(21/2)*sqrt(a/(b*x**3) + 1)) - a
**2*f/(6*sqrt(b)*x**(15/2)*sqrt(a/(b*x**3) + 1)) - 11*a*sqrt(b)*c/(36*x**(15/2)*sqrt(a/(b*x**3) + 1)) - a*sqrt
(b)*f/(4*x**(9/2)*sqrt(a/(b*x**3) + 1)) - 17*b**(3/2)*c/(72*x**(9/2)*sqrt(a/(b*x**3) + 1)) - b**(3/2)*f*sqrt(a
/(b*x**3) + 1)/(3*x**(3/2)) - b**(3/2)*f/(12*x**(3/2)*sqrt(a/(b*x**3) + 1)) - b**(5/2)*c/(24*a*x**(3/2)*sqrt(a
/(b*x**3) + 1)) - b**2*f*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(4*sqrt(a)) + b**3*c*asinh(sqrt(a)/(sqrt(b)*x**(3/2
)))/(24*a**(3/2))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{x^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^10,x, algorithm="giac")

[Out]

integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^10, x)