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

Optimal. Leaf size=692 \[ \frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e+28 a g+77 b d\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{770 \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \sqrt [3]{b} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{14 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}-\frac{2 a \sqrt{a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{1155 x^4}-\frac{1}{3} \sqrt{a} (2 a f+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{a c \sqrt{a+b x^3}}{x^3}+\frac{27 a d \sqrt{a+b x^3}}{10 x^2}-\frac{27 a e \sqrt{a+b x^3}}{7 x}+\frac{27 a \sqrt [3]{b} e \sqrt{a+b x^3}}{7 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )} \]

[Out]

(a*c*Sqrt[a + b*x^3])/x^3 + (27*a*d*Sqrt[a + b*x^3])/(10*x^2) - (27*a*e*Sqrt[a + b*x^3])/(7*x) + (27*a*b^(1/3)
*e*Sqrt[a + b*x^3])/(7*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (2*a*Sqrt[a + b*x^3]*(1155*c*x + 2079*d*x^2 - 14
85*e*x^3 - 385*f*x^4 - 189*g*x^5))/(1155*x^4) + (2*(a + b*x^3)^(3/2)*(1155*c*x + 693*d*x^2 + 495*e*x^3 + 385*f
*x^4 + 315*g*x^5))/(3465*x^4) - (Sqrt[a]*(3*b*c + 2*a*f)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/3 - (27*3^(1/4)*Sqr
t[2 - Sqrt[3]]*a^(4/3)*b^(1/3)*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]])/(14*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]]*a*(77*b*d - 110*(1 - Sqrt[3])*a^(1/3)*b^(2/3)*e + 28*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]*Ell
ipticF[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]])/(770*
b^(1/3)*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])

________________________________________________________________________________________

Rubi [A]  time = 0.956585, antiderivative size = 692, normalized size of antiderivative = 1., 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, 1835, 1832, 266, 63, 208, 1878, 218, 1877} \[ \frac{9\ 3^{3/4} \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e+28 a g+77 b d\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{770 \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \sqrt [3]{b} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{14 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}-\frac{2 a \sqrt{a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{1155 x^4}-\frac{1}{3} \sqrt{a} (2 a f+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )+\frac{a c \sqrt{a+b x^3}}{x^3}+\frac{27 a d \sqrt{a+b x^3}}{10 x^2}-\frac{27 a e \sqrt{a+b x^3}}{7 x}+\frac{27 a \sqrt [3]{b} e \sqrt{a+b x^3}}{7 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )} \]

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^4,x]

[Out]

(a*c*Sqrt[a + b*x^3])/x^3 + (27*a*d*Sqrt[a + b*x^3])/(10*x^2) - (27*a*e*Sqrt[a + b*x^3])/(7*x) + (27*a*b^(1/3)
*e*Sqrt[a + b*x^3])/(7*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (2*a*Sqrt[a + b*x^3]*(1155*c*x + 2079*d*x^2 - 14
85*e*x^3 - 385*f*x^4 - 189*g*x^5))/(1155*x^4) + (2*(a + b*x^3)^(3/2)*(1155*c*x + 693*d*x^2 + 495*e*x^3 + 385*f
*x^4 + 315*g*x^5))/(3465*x^4) - (Sqrt[a]*(3*b*c + 2*a*f)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/3 - (27*3^(1/4)*Sqr
t[2 - Sqrt[3]]*a^(4/3)*b^(1/3)*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]])/(14*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]]*a*(77*b*d - 110*(1 - Sqrt[3])*a^(1/3)*b^(2/3)*e + 28*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]*Ell
ipticF[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]])/(770*
b^(1/3)*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 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 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^4} \, dx &=\frac{2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}+\frac{1}{2} (9 a) \int \frac{\sqrt{a+b x^3} \left (\frac{2 c}{3}+\frac{2 d x}{5}+\frac{2 e x^2}{7}+\frac{2 f x^3}{9}+\frac{2 g x^4}{11}\right )}{x^4} \, dx\\ &=-\frac{2 a \sqrt{a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{1155 x^4}+\frac{2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}+\frac{1}{4} \left (27 a^2\right ) \int \frac{-\frac{4 c}{9}-\frac{4 d x}{5}+\frac{4 e x^2}{7}+\frac{4 f x^3}{27}+\frac{4 g x^4}{55}}{x^4 \sqrt{a+b x^3}} \, dx\\ &=\frac{a c \sqrt{a+b x^3}}{x^3}-\frac{2 a \sqrt{a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{1155 x^4}+\frac{2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}-\frac{1}{8} (9 a) \int \frac{\frac{24 a d}{5}-\frac{24 a e x}{7}-\frac{4}{9} (3 b c+2 a f) x^2-\frac{24}{55} a g x^3}{x^3 \sqrt{a+b x^3}} \, dx\\ &=\frac{a c \sqrt{a+b x^3}}{x^3}+\frac{27 a d \sqrt{a+b x^3}}{10 x^2}-\frac{2 a \sqrt{a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{1155 x^4}+\frac{2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}+\frac{9}{32} \int \frac{\frac{96 a^2 e}{7}+\frac{16}{9} a (3 b c+2 a f) x+\frac{24}{55} a (11 b d+4 a g) x^2}{x^2 \sqrt{a+b x^3}} \, dx\\ &=\frac{a c \sqrt{a+b x^3}}{x^3}+\frac{27 a d \sqrt{a+b x^3}}{10 x^2}-\frac{27 a e \sqrt{a+b x^3}}{7 x}-\frac{2 a \sqrt{a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{1155 x^4}+\frac{2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}-\frac{9 \int \frac{-\frac{32}{9} a^2 (3 b c+2 a f)-\frac{48}{55} a^2 (11 b d+4 a g) x-\frac{96}{7} a^2 b e x^2}{x \sqrt{a+b x^3}} \, dx}{64 a}\\ &=\frac{a c \sqrt{a+b x^3}}{x^3}+\frac{27 a d \sqrt{a+b x^3}}{10 x^2}-\frac{27 a e \sqrt{a+b x^3}}{7 x}-\frac{2 a \sqrt{a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{1155 x^4}+\frac{2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}-\frac{9 \int \frac{-\frac{48}{55} a^2 (11 b d+4 a g)-\frac{96}{7} a^2 b e x}{\sqrt{a+b x^3}} \, dx}{64 a}+\frac{1}{2} (a (3 b c+2 a f)) \int \frac{1}{x \sqrt{a+b x^3}} \, dx\\ &=\frac{a c \sqrt{a+b x^3}}{x^3}+\frac{27 a d \sqrt{a+b x^3}}{10 x^2}-\frac{27 a e \sqrt{a+b x^3}}{7 x}-\frac{2 a \sqrt{a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{1155 x^4}+\frac{2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}+\frac{1}{14} \left (27 a b^{2/3} e\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx+\frac{1}{6} (a (3 b c+2 a f)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )+\frac{\left (27 a \left (77 b d-110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e+28 a g\right )\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{1540}\\ &=\frac{a c \sqrt{a+b x^3}}{x^3}+\frac{27 a d \sqrt{a+b x^3}}{10 x^2}-\frac{27 a e \sqrt{a+b x^3}}{7 x}+\frac{27 a \sqrt [3]{b} e \sqrt{a+b x^3}}{7 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{2 a \sqrt{a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{1155 x^4}+\frac{2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \sqrt [3]{b} 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 )}{14 \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}} a \left (77 b d-110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e+28 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 )}{770 \sqrt [3]{b} \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{(a (3 b c+2 a f)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{3 b}\\ &=\frac{a c \sqrt{a+b x^3}}{x^3}+\frac{27 a d \sqrt{a+b x^3}}{10 x^2}-\frac{27 a e \sqrt{a+b x^3}}{7 x}+\frac{27 a \sqrt [3]{b} e \sqrt{a+b x^3}}{7 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{2 a \sqrt{a+b x^3} \left (1155 c x+2079 d x^2-1485 e x^3-385 f x^4-189 g x^5\right )}{1155 x^4}+\frac{2 \left (a+b x^3\right )^{3/2} \left (1155 c x+693 d x^2+495 e x^3+385 f x^4+315 g x^5\right )}{3465 x^4}-\frac{1}{3} \sqrt{a} (3 b c+2 a f) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )-\frac{27 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} \sqrt [3]{b} 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 )}{14 \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}} a \left (77 b d-110 \left (1-\sqrt{3}\right ) \sqrt [3]{a} b^{2/3} e+28 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 )}{770 \sqrt [3]{b} \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 = 0.611858, size = 243, normalized size = 0.35 \[ \frac{4 x^2 \sqrt{\frac{b x^3}{a}+1} \left (5 a^2 f \left (\sqrt{a+b x^3} \left (4 a+b x^3\right )-3 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )\right )+3 b c \left (a+b x^3\right )^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{b x^3}{a}+1\right )\right )-45 a^3 d \sqrt{a+b x^3} \, _2F_1\left (-\frac{3}{2},-\frac{2}{3};\frac{1}{3};-\frac{b x^3}{a}\right )-90 a^3 e x \sqrt{a+b x^3} \, _2F_1\left (-\frac{3}{2},-\frac{1}{3};\frac{2}{3};-\frac{b x^3}{a}\right )+90 a^3 g x^3 \sqrt{a+b x^3} \, _2F_1\left (-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{90 a^2 x^2 \sqrt{\frac{b x^3}{a}+1}} \]

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^4,x]

[Out]

(-45*a^3*d*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, -2/3, 1/3, -((b*x^3)/a)] - 90*a^3*e*x*Sqrt[a + b*x^3]*Hyper
geometric2F1[-3/2, -1/3, 2/3, -((b*x^3)/a)] + 90*a^3*g*x^3*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, 1/3, 4/3, -
((b*x^3)/a)] + 4*x^2*Sqrt[1 + (b*x^3)/a]*(5*a^2*f*(Sqrt[a + b*x^3]*(4*a + b*x^3) - 3*a^(3/2)*ArcTanh[Sqrt[a +
b*x^3]/Sqrt[a]]) + 3*b*c*(a + b*x^3)^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + (b*x^3)/a]))/(90*a^2*x^2*Sqrt[1
+ (b*x^3)/a])

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Maple [B]  time = 0.01, size = 1193, normalized size = 1.7 \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^4,x)

[Out]

g*(2/11*b*x^4*(b*x^3+a)^(1/2)+28/55*a*x*(b*x^3+a)^(1/2)-18/55*I*a^2*3^(1/2)/b*(-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*(2/9*b*x^3*(b*x^3+a)^(1/2)+8/9*a*(b*x^3+a)^(1/2)-2/3*a^(3/2)*arctanh((b*x^3
+a)^(1/2)/a^(1/2)))+c*(-1/3*a*(b*x^3+a)^(1/2)/x^3+2/3*b*(b*x^3+a)^(1/2)-a^(1/2)*b*arctanh((b*x^3+a)^(1/2)/a^(1
/2)))+d*(-1/2*a*(b*x^3+a)^(1/2)/x^2+2/5*b*x*(b*x^3+a)^(1/2)-9/10*I*a*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*(-a*(b*x^3+a)^(1/2)/x+2/7*b*x^2*(b*x^3+a)^(1/2)-9/7*I*a*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^{4}}\,{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^4,x, algorithm="maxima")

[Out]

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

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Sympy [A]  time = 11.9664, size = 484, normalized size = 0.7 \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**4,x)

[Out]

a**(3/2)*d*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**(3/2)*e*
gamma(-1/3)*hyper((-1/2, -1/3), (2/3,), b*x**3*exp_polar(I*pi)/a)/(3*x*gamma(2/3)) - 2*a**(3/2)*f*asinh(sqrt(a
)/(sqrt(b)*x**(3/2)))/3 + a**(3/2)*g*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamm
a(4/3)) - sqrt(a)*b*c*asinh(sqrt(a)/(sqrt(b)*x**(3/2))) + sqrt(a)*b*d*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,),
b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) + sqrt(a)*b*e*x**2*gamma(2/3)*hyper((-1/2, 2/3), (5/3,), b*x**3*exp_p
olar(I*pi)/a)/(3*gamma(5/3)) + sqrt(a)*b*g*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a
)/(3*gamma(7/3)) + 2*a**2*f/(3*sqrt(b)*x**(3/2)*sqrt(a/(b*x**3) + 1)) - a*sqrt(b)*c*sqrt(a/(b*x**3) + 1)/(3*x*
*(3/2)) + 2*a*sqrt(b)*c/(3*x**(3/2)*sqrt(a/(b*x**3) + 1)) + 2*a*sqrt(b)*f*x**(3/2)/(3*sqrt(a/(b*x**3) + 1)) +
2*b**(3/2)*c*x**(3/2)/(3*sqrt(a/(b*x**3) + 1)) + b*f*Piecewise((sqrt(a)*x**3/3, Eq(b, 0)), (2*(a + b*x**3)**(3
/2)/(9*b), True))

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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^{4}}\,{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^4,x, algorithm="giac")

[Out]

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