3.470 \(\int \frac{(a+b x^3)^{3/2} (c+d x+e x^2+f x^3+g x^4)}{x^9} \, dx\)

Optimal. Leaf size=705 \[ -\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right ) \left (7 \sqrt [3]{b} (b c-16 a f)+20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (14 a g+b d)\right )}{2240 a \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (14 a g+b d) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{224 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 b^2 c \sqrt{a+b x^3}}{320 a x^2}+\frac{27 b^{4/3} \sqrt{a+b x^3} (14 a g+b d)}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{27 b^2 d \sqrt{a+b x^3}}{112 a x}-\frac{b^2 e \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{1}{560} b \sqrt{a+b x^3} \left (\frac{63 c}{x^5}+\frac{90 d}{x^4}+\frac{140 e}{x^3}+\frac{252 f}{x^2}+\frac{630 g}{x}\right )-\frac{1}{840} \left (a+b x^3\right )^{3/2} \left (\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}+\frac{210 g}{x^4}\right ) \]

[Out]

-(b*((63*c)/x^5 + (90*d)/x^4 + (140*e)/x^3 + (252*f)/x^2 + (630*g)/x)*Sqrt[a + b*x^3])/560 - (27*b^2*c*Sqrt[a
+ b*x^3])/(320*a*x^2) - (27*b^2*d*Sqrt[a + b*x^3])/(112*a*x) + (27*b^(4/3)*(b*d + 14*a*g)*Sqrt[a + b*x^3])/(11
2*a*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (((105*c)/x^8 + (120*d)/x^7 + (140*e)/x^6 + (168*f)/x^5 + (210*g)/x
^4)*(a + b*x^3)^(3/2))/840 - (b^2*e*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(4*Sqrt[a]) - (27*3^(1/4)*Sqrt[2 - Sqrt[
3]]*b^(4/3)*(b*d + 14*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]*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^(4/3)*(7*b^(1/3)*(b*c - 16*a*f) + 20*(1 - Sqrt[3])*a^(1/
3)*(b*d + 14*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 = 1.00792, antiderivative size = 705, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ -\frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right ) \left (7 \sqrt [3]{b} (b c-16 a f)+20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (14 a g+b d)\right )}{2240 a \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (14 a g+b d) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{224 a^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 b^2 c \sqrt{a+b x^3}}{320 a x^2}+\frac{27 b^{4/3} \sqrt{a+b x^3} (14 a g+b d)}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{27 b^2 d \sqrt{a+b x^3}}{112 a x}-\frac{b^2 e \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{1}{560} b \sqrt{a+b x^3} \left (\frac{63 c}{x^5}+\frac{90 d}{x^4}+\frac{140 e}{x^3}+\frac{252 f}{x^2}+\frac{630 g}{x}\right )-\frac{1}{840} \left (a+b x^3\right )^{3/2} \left (\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}+\frac{210 g}{x^4}\right ) \]

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^9,x]

[Out]

-(b*((63*c)/x^5 + (90*d)/x^4 + (140*e)/x^3 + (252*f)/x^2 + (630*g)/x)*Sqrt[a + b*x^3])/560 - (27*b^2*c*Sqrt[a
+ b*x^3])/(320*a*x^2) - (27*b^2*d*Sqrt[a + b*x^3])/(112*a*x) + (27*b^(4/3)*(b*d + 14*a*g)*Sqrt[a + b*x^3])/(11
2*a*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (((105*c)/x^8 + (120*d)/x^7 + (140*e)/x^6 + (168*f)/x^5 + (210*g)/x
^4)*(a + b*x^3)^(3/2))/840 - (b^2*e*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(4*Sqrt[a]) - (27*3^(1/4)*Sqrt[2 - Sqrt[
3]]*b^(4/3)*(b*d + 14*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]*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^(4/3)*(7*b^(1/3)*(b*c - 16*a*f) + 20*(1 - Sqrt[3])*a^(1/
3)*(b*d + 14*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^9} \, dx &=-\frac{1}{840} \left (\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}+\frac{210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}-\frac{1}{2} (9 b) \int \frac{\sqrt{a+b x^3} \left (-\frac{c}{8}-\frac{d x}{7}-\frac{e x^2}{6}-\frac{f x^3}{5}-\frac{g x^4}{4}\right )}{x^6} \, dx\\ &=-\frac{1}{560} b \left (\frac{63 c}{x^5}+\frac{90 d}{x^4}+\frac{140 e}{x^3}+\frac{252 f}{x^2}+\frac{630 g}{x}\right ) \sqrt{a+b x^3}-\frac{1}{840} \left (\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}+\frac{210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}+\frac{1}{4} \left (27 b^2\right ) \int \frac{\frac{c}{40}+\frac{d x}{28}+\frac{e x^2}{18}+\frac{f x^3}{10}+\frac{g x^4}{4}}{x^3 \sqrt{a+b x^3}} \, dx\\ &=-\frac{1}{560} b \left (\frac{63 c}{x^5}+\frac{90 d}{x^4}+\frac{140 e}{x^3}+\frac{252 f}{x^2}+\frac{630 g}{x}\right ) \sqrt{a+b x^3}-\frac{27 b^2 c \sqrt{a+b x^3}}{320 a x^2}-\frac{1}{840} \left (\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}+\frac{210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}-\frac{\left (27 b^2\right ) \int \frac{-\frac{a d}{7}-\frac{2 a e x}{9}+\frac{1}{40} (b c-16 a f) x^2-a g x^3}{x^2 \sqrt{a+b x^3}} \, dx}{16 a}\\ &=-\frac{1}{560} b \left (\frac{63 c}{x^5}+\frac{90 d}{x^4}+\frac{140 e}{x^3}+\frac{252 f}{x^2}+\frac{630 g}{x}\right ) \sqrt{a+b x^3}-\frac{27 b^2 c \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 d \sqrt{a+b x^3}}{112 a x}-\frac{1}{840} \left (\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}+\frac{210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}+\frac{\left (27 b^2\right ) \int \frac{\frac{4 a^2 e}{9}-\frac{1}{20} a (b c-16 a f) x+\frac{1}{7} a (b d+14 a g) x^2}{x \sqrt{a+b x^3}} \, dx}{32 a^2}\\ &=-\frac{1}{560} b \left (\frac{63 c}{x^5}+\frac{90 d}{x^4}+\frac{140 e}{x^3}+\frac{252 f}{x^2}+\frac{630 g}{x}\right ) \sqrt{a+b x^3}-\frac{27 b^2 c \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 d \sqrt{a+b x^3}}{112 a x}-\frac{1}{840} \left (\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}+\frac{210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}+\frac{\left (27 b^2\right ) \int \frac{-\frac{1}{20} a (b c-16 a f)+\frac{1}{7} a (b d+14 a g) x}{\sqrt{a+b x^3}} \, dx}{32 a^2}+\frac{1}{8} \left (3 b^2 e\right ) \int \frac{1}{x \sqrt{a+b x^3}} \, dx\\ &=-\frac{1}{560} b \left (\frac{63 c}{x^5}+\frac{90 d}{x^4}+\frac{140 e}{x^3}+\frac{252 f}{x^2}+\frac{630 g}{x}\right ) \sqrt{a+b x^3}-\frac{27 b^2 c \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 d \sqrt{a+b x^3}}{112 a x}-\frac{1}{840} \left (\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}+\frac{210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}+\frac{1}{8} \left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )+\frac{\left (27 b^{5/3} (b d+14 a g)\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 (27 b^2 \left (7 (b c-16 a f)+\frac{20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (b d+14 a g)}{\sqrt [3]{b}}\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{4480 a}\\ &=-\frac{1}{560} b \left (\frac{63 c}{x^5}+\frac{90 d}{x^4}+\frac{140 e}{x^3}+\frac{252 f}{x^2}+\frac{630 g}{x}\right ) \sqrt{a+b x^3}-\frac{27 b^2 c \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 d \sqrt{a+b x^3}}{112 a x}+\frac{27 b^{4/3} (b d+14 a g) \sqrt{a+b x^3}}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{1}{840} \left (\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}+\frac{210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} (b d+14 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{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 c-16 a f)+\frac{20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (b d+14 a g)}{\sqrt [3]{b}}\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{1}{4} (b e) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )\\ &=-\frac{1}{560} b \left (\frac{63 c}{x^5}+\frac{90 d}{x^4}+\frac{140 e}{x^3}+\frac{252 f}{x^2}+\frac{630 g}{x}\right ) \sqrt{a+b x^3}-\frac{27 b^2 c \sqrt{a+b x^3}}{320 a x^2}-\frac{27 b^2 d \sqrt{a+b x^3}}{112 a x}+\frac{27 b^{4/3} (b d+14 a g) \sqrt{a+b x^3}}{112 a \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{1}{840} \left (\frac{105 c}{x^8}+\frac{120 d}{x^7}+\frac{140 e}{x^6}+\frac{168 f}{x^5}+\frac{210 g}{x^4}\right ) \left (a+b x^3\right )^{3/2}-\frac{b^2 e \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} (b d+14 a g) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{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 c-16 a f)+\frac{20 \left (1-\sqrt{3}\right ) \sqrt [3]{a} (b d+14 a g)}{\sqrt [3]{b}}\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 = 0.460713, size = 202, normalized size = 0.29 \[ -\frac{\sqrt{a+b x^3} \left (2 x \left (7 x \left (5 \left (3 a^2 g x^2 \, _2F_1\left (-\frac{3}{2},-\frac{4}{3};-\frac{1}{3};-\frac{b x^3}{a}\right )+3 b^2 e x^6 \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )+a e \left (2 a+5 b x^3\right ) \sqrt{\frac{b x^3}{a}+1}\right )+12 a^2 f x \, _2F_1\left (-\frac{5}{3},-\frac{3}{2};-\frac{2}{3};-\frac{b x^3}{a}\right )\right )+60 a^2 d \, _2F_1\left (-\frac{7}{3},-\frac{3}{2};-\frac{4}{3};-\frac{b x^3}{a}\right )\right )+105 a^2 c \, _2F_1\left (-\frac{8}{3},-\frac{3}{2};-\frac{5}{3};-\frac{b x^3}{a}\right )\right )}{840 a x^8 \sqrt{\frac{b x^3}{a}+1}} \]

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^9,x]

[Out]

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

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Maple [B]  time = 0.01, size = 1663, normalized size = 2.4 \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^9,x)

[Out]

f*(-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)))+g*(-1/4*a*(b*x^3+a)^(1/2)/x^4-11/8*b*(b*x^3+a)^(1/2)/x-9/8*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)*((-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*(-1/8*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)))+e*(-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))+d*(-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))))

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Maxima [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^{9}}\,{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^9,x, algorithm="maxima")

[Out]

integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^9, x)

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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^{9}}, 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^9,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^9, x)

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Sympy [A]  time = 15.5129, size = 527, normalized size = 0.75 \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**9,x)

[Out]

a**(3/2)*c*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)*
d*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)*f*gamma(-
5/3)*hyper((-5/3, -1/2), (-2/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**5*gamma(-2/3)) + a**(3/2)*g*gamma(-4/3)*hype
r((-4/3, -1/2), (-1/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**4*gamma(-1/3)) + sqrt(a)*b*c*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(-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*f*gamma(-2/3)*hyper((-2/3, -1/2), (1/3,),
b*x**3*exp_polar(I*pi)/a)/(3*x**2*gamma(1/3)) + sqrt(a)*b*g*gamma(-1/3)*hyper((-1/2, -1/3), (2/3,), b*x**3*exp
_polar(I*pi)/a)/(3*x*gamma(2/3)) - a**2*e/(6*sqrt(b)*x**(15/2)*sqrt(a/(b*x**3) + 1)) - a*sqrt(b)*e/(4*x**(9/2)
*sqrt(a/(b*x**3) + 1)) - b**(3/2)*e*sqrt(a/(b*x**3) + 1)/(3*x**(3/2)) - b**(3/2)*e/(12*x**(3/2)*sqrt(a/(b*x**3
) + 1)) - b**2*e*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(4*sqrt(a))

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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^{9}}\,{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^9,x, algorithm="giac")

[Out]

integrate((g*x^4 + f*x^3 + e*x^2 + d*x + c)*(b*x^3 + a)^(3/2)/x^9, x)