\(\int \frac {\sqrt {a+b x^3} (c+d x+e x^2+f x^3+g x^4)}{x^5} \, dx\) [453]

Optimal result

Integrand size = 35, antiderivative size = 694 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\frac {3 c \sqrt {a+b x^3}}{20 x^4}+\frac {d \sqrt {a+b x^3}}{3 x^3}+\frac {3 e \sqrt {a+b x^3}}{2 x^2}-\frac {3 (b c+8 a f) \sqrt {a+b x^3}}{8 a x}+\frac {3 \sqrt [3]{b} (b c+8 a f) \sqrt {a+b x^3}}{8 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 \sqrt {a+b x^3} \left (3 c x+5 d x^2+15 e x^3-15 f x^4-5 g x^5\right )}{15 x^5}-\frac {(b d+2 a g) \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{b} (b c+8 a f) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{16 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} \sqrt [3]{b} \left (4 a^{2/3} \sqrt [3]{b} e-\left (1-\sqrt {3}\right ) (b c+8 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{8 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[Out]

-1/3*(2*a*g+b*d)*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(1/2)+3/20*c*(b*x^3+a)^(1/2)/x^4+1/3*d*(b*x^3+a)^(1/2)/x^3
+3/2*e*(b*x^3+a)^(1/2)/x^2-3/8*(8*a*f+b*c)*(b*x^3+a)^(1/2)/a/x-2/15*(-5*g*x^5-15*f*x^4+15*e*x^3+5*d*x^2+3*c*x)
*(b*x^3+a)^(1/2)/x^5+3/8*b^(1/3)*(8*a*f+b*c)*(b*x^3+a)^(1/2)/a/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))-3/16*3^(1/4)*b^
(1/3)*(8*a*f+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)/a^(2/3)/(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)+1
/8*3^(3/4)*b^(1/3)*(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)*(4*a^(2/3)*b^(1/3)*e-(8*a*f+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)/a^(2/3)/(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] (verified)

Time = 0.71 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {1840, 1849, 1846, 272, 65, 214, 1892, 224, 1891} \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\frac {3^{3/4} \sqrt {2+\sqrt {3}} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right ) \left (4 a^{2/3} \sqrt [3]{b} e-\left (1-\sqrt {3}\right ) (8 a f+b c)\right )}{8 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (8 a f+b c) E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{16 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right ) (2 a g+b d)}{3 \sqrt {a}}-\frac {2 \sqrt {a+b x^3} \left (3 c x+5 d x^2+15 e x^3-15 f x^4-5 g x^5\right )}{15 x^5}-\frac {3 \sqrt {a+b x^3} (8 a f+b c)}{8 a x}+\frac {3 \sqrt [3]{b} \sqrt {a+b x^3} (8 a f+b c)}{8 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {3 c \sqrt {a+b x^3}}{20 x^4}+\frac {d \sqrt {a+b x^3}}{3 x^3}+\frac {3 e \sqrt {a+b x^3}}{2 x^2} \]

[In]

Int[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^5,x]

[Out]

(3*c*Sqrt[a + b*x^3])/(20*x^4) + (d*Sqrt[a + b*x^3])/(3*x^3) + (3*e*Sqrt[a + b*x^3])/(2*x^2) - (3*(b*c + 8*a*f
)*Sqrt[a + b*x^3])/(8*a*x) + (3*b^(1/3)*(b*c + 8*a*f)*Sqrt[a + b*x^3])/(8*a*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x
)) - (2*Sqrt[a + b*x^3]*(3*c*x + 5*d*x^2 + 15*e*x^3 - 15*f*x^4 - 5*g*x^5))/(15*x^5) - ((b*d + 2*a*g)*ArcTanh[S
qrt[a + b*x^3]/Sqrt[a]])/(3*Sqrt[a]) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*b^(1/3)*(b*c + 8*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]])/(16*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]) + (3^(3/4)*Sqrt[2 + Sqrt[3]
]*b^(1/3)*(4*a^(2/3)*b^(1/3)*e - (1 - Sqrt[3])*(b*c + 8*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]])/(8*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])

Rule 65

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 214

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

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 272

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 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 1846

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 1849

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{Pq0 = Coeff[Pq, x, 0]}, Simp[Pq0
*(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 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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a+b x^3} \left (3 c x+5 d x^2+15 e x^3-15 f x^4-5 g x^5\right )}{15 x^5}+\frac {1}{2} (3 a) \int \frac {-\frac {2 c}{5}-\frac {2 d x}{3}-2 e x^2+2 f x^3+\frac {2 g x^4}{3}}{x^5 \sqrt {a+b x^3}} \, dx \\ & = \frac {3 c \sqrt {a+b x^3}}{20 x^4}-\frac {2 \sqrt {a+b x^3} \left (3 c x+5 d x^2+15 e x^3-15 f x^4-5 g x^5\right )}{15 x^5}-\frac {3}{16} \int \frac {\frac {16 a d}{3}+16 a e x-2 (b c+8 a f) x^2-\frac {16}{3} a g x^3}{x^4 \sqrt {a+b x^3}} \, dx \\ & = \frac {3 c \sqrt {a+b x^3}}{20 x^4}+\frac {d \sqrt {a+b x^3}}{3 x^3}-\frac {2 \sqrt {a+b x^3} \left (3 c x+5 d x^2+15 e x^3-15 f x^4-5 g x^5\right )}{15 x^5}+\frac {\int \frac {-96 a^2 e+12 a (b c+8 a f) x+16 a (b d+2 a g) x^2}{x^3 \sqrt {a+b x^3}} \, dx}{32 a} \\ & = \frac {3 c \sqrt {a+b x^3}}{20 x^4}+\frac {d \sqrt {a+b x^3}}{3 x^3}+\frac {3 e \sqrt {a+b x^3}}{2 x^2}-\frac {2 \sqrt {a+b x^3} \left (3 c x+5 d x^2+15 e x^3-15 f x^4-5 g x^5\right )}{15 x^5}-\frac {\int \frac {-48 a^2 (b c+8 a f)-64 a^2 (b d+2 a g) x-96 a^2 b e x^2}{x^2 \sqrt {a+b x^3}} \, dx}{128 a^2} \\ & = \frac {3 c \sqrt {a+b x^3}}{20 x^4}+\frac {d \sqrt {a+b x^3}}{3 x^3}+\frac {3 e \sqrt {a+b x^3}}{2 x^2}-\frac {3 (b c+8 a f) \sqrt {a+b x^3}}{8 a x}-\frac {2 \sqrt {a+b x^3} \left (3 c x+5 d x^2+15 e x^3-15 f x^4-5 g x^5\right )}{15 x^5}+\frac {\int \frac {128 a^3 (b d+2 a g)+192 a^3 b e x+48 a^2 b (b c+8 a f) x^2}{x \sqrt {a+b x^3}} \, dx}{256 a^3} \\ & = \frac {3 c \sqrt {a+b x^3}}{20 x^4}+\frac {d \sqrt {a+b x^3}}{3 x^3}+\frac {3 e \sqrt {a+b x^3}}{2 x^2}-\frac {3 (b c+8 a f) \sqrt {a+b x^3}}{8 a x}-\frac {2 \sqrt {a+b x^3} \left (3 c x+5 d x^2+15 e x^3-15 f x^4-5 g x^5\right )}{15 x^5}+\frac {\int \frac {192 a^3 b e+48 a^2 b (b c+8 a f) x}{\sqrt {a+b x^3}} \, dx}{256 a^3}+\frac {1}{2} (b d+2 a g) \int \frac {1}{x \sqrt {a+b x^3}} \, dx \\ & = \frac {3 c \sqrt {a+b x^3}}{20 x^4}+\frac {d \sqrt {a+b x^3}}{3 x^3}+\frac {3 e \sqrt {a+b x^3}}{2 x^2}-\frac {3 (b c+8 a f) \sqrt {a+b x^3}}{8 a x}-\frac {2 \sqrt {a+b x^3} \left (3 c x+5 d x^2+15 e x^3-15 f x^4-5 g x^5\right )}{15 x^5}+\frac {\left (3 b^{2/3} (b c+8 a f)\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{16 a}+\frac {1}{16} \left (3 b^{2/3} \left (4 \sqrt [3]{b} e-\frac {\left (1-\sqrt {3}\right ) (b c+8 a f)}{a^{2/3}}\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx+\frac {1}{6} (b d+2 a g) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right ) \\ & = \frac {3 c \sqrt {a+b x^3}}{20 x^4}+\frac {d \sqrt {a+b x^3}}{3 x^3}+\frac {3 e \sqrt {a+b x^3}}{2 x^2}-\frac {3 (b c+8 a f) \sqrt {a+b x^3}}{8 a x}+\frac {3 \sqrt [3]{b} (b c+8 a f) \sqrt {a+b x^3}}{8 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 \sqrt {a+b x^3} \left (3 c x+5 d x^2+15 e x^3-15 f x^4-5 g x^5\right )}{15 x^5}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{b} (b c+8 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 )}{16 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 {3^{3/4} \sqrt {2+\sqrt {3}} \sqrt [3]{b} \left (4 \sqrt [3]{b} e-\frac {\left (1-\sqrt {3}\right ) (b c+8 a f)}{a^{2/3}}\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 )}{8 \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 d+2 a g) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 b} \\ & = \frac {3 c \sqrt {a+b x^3}}{20 x^4}+\frac {d \sqrt {a+b x^3}}{3 x^3}+\frac {3 e \sqrt {a+b x^3}}{2 x^2}-\frac {3 (b c+8 a f) \sqrt {a+b x^3}}{8 a x}+\frac {3 \sqrt [3]{b} (b c+8 a f) \sqrt {a+b x^3}}{8 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 \sqrt {a+b x^3} \left (3 c x+5 d x^2+15 e x^3-15 f x^4-5 g x^5\right )}{15 x^5}-\frac {(b d+2 a g) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{b} (b c+8 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 )}{16 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 {3^{3/4} \sqrt {2+\sqrt {3}} \sqrt [3]{b} \left (4 \sqrt [3]{b} e-\frac {\left (1-\sqrt {3}\right ) (b c+8 a f)}{a^{2/3}}\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 )}{8 \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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.45 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=-\frac {4 a d x+4 b d x^4-8 a g x^4-8 b g x^7+8 \sqrt {a} g x^4 \sqrt {a+b x^3} \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+4 b d x^4 \sqrt {1+\frac {b x^3}{a}} \text {arctanh}\left (\sqrt {1+\frac {b x^3}{a}}\right )+3 a c \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},-\frac {1}{2},-\frac {1}{3},-\frac {b x^3}{a}\right )+6 a e x^2 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{2},\frac {1}{3},-\frac {b x^3}{a}\right )+12 a f x^3 \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{3},\frac {2}{3},-\frac {b x^3}{a}\right )}{12 x^4 \sqrt {a+b x^3}} \]

[In]

Integrate[(Sqrt[a + b*x^3]*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^5,x]

[Out]

-1/12*(4*a*d*x + 4*b*d*x^4 - 8*a*g*x^4 - 8*b*g*x^7 + 8*Sqrt[a]*g*x^4*Sqrt[a + b*x^3]*ArcTanh[Sqrt[a + b*x^3]/S
qrt[a]] + 4*b*d*x^4*Sqrt[1 + (b*x^3)/a]*ArcTanh[Sqrt[1 + (b*x^3)/a]] + 3*a*c*Sqrt[1 + (b*x^3)/a]*Hypergeometri
c2F1[-4/3, -1/2, -1/3, -((b*x^3)/a)] + 6*a*e*x^2*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-2/3, -1/2, 1/3, -((b*x
^3)/a)] + 12*a*f*x^3*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-1/2, -1/3, 2/3, -((b*x^3)/a)])/(x^4*Sqrt[a + b*x^3
])

Maple [A] (verified)

Time = 1.78 (sec) , antiderivative size = 845, normalized size of antiderivative = 1.22

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(845\)
risch	\(\text {Expression too large to display}\)	\(1243\)
default	\(\text {Expression too large to display}\)	\(1286\)

[In]

int((g*x^4+f*x^3+e*x^2+d*x+c)*(b*x^3+a)^(1/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*c*(b*x^3+a)^(1/2)/x^4-1/3*d*(b*x^3+a)^(1/2)/x^3-1/2*e*(b*x^3+a)^(1/2)/x^2-1/8*(8*a*f+3*b*c)/a*(b*x^3+a)^(
1/2)/x+2/3*g*(b*x^3+a)^(1/2)-1/2*I*e*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
/3*I*(b*f+1/16*b*(8*a*f+3*b*c)/a)*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)))-2/3*(a*g+1/2*b*d)*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a
^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.64 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\left [\frac {36 \, a \sqrt {b} e x^{4} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 2 \, {\left (b d + 2 \, a g\right )} \sqrt {a} x^{4} \log \left (-\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) - 9 \, {\left (b c + 8 \, a f\right )} \sqrt {b} x^{4} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (16 \, a g x^{4} - 12 \, a e x^{2} - 3 \, {\left (3 \, b c + 8 \, a f\right )} x^{3} - 8 \, a d x - 6 \, a c\right )} \sqrt {b x^{3} + a}}{24 \, a x^{4}}, \frac {36 \, a \sqrt {b} e x^{4} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 4 \, {\left (b d + 2 \, a g\right )} \sqrt {-a} x^{4} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) - 9 \, {\left (b c + 8 \, a f\right )} \sqrt {b} x^{4} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (16 \, a g x^{4} - 12 \, a e x^{2} - 3 \, {\left (3 \, b c + 8 \, a f\right )} x^{3} - 8 \, a d x - 6 \, a c\right )} \sqrt {b x^{3} + a}}{24 \, a x^{4}}\right ] \]

[In]

integrate((g*x^4+f*x^3+e*x^2+d*x+c)*(b*x^3+a)^(1/2)/x^5,x, algorithm="fricas")

[Out]

[1/24*(36*a*sqrt(b)*e*x^4*weierstrassPInverse(0, -4*a/b, x) + 2*(b*d + 2*a*g)*sqrt(a)*x^4*log(-(b^2*x^6 + 8*a*
b*x^3 - 4*(b*x^3 + 2*a)*sqrt(b*x^3 + a)*sqrt(a) + 8*a^2)/x^6) - 9*(b*c + 8*a*f)*sqrt(b)*x^4*weierstrassZeta(0,
-4*a/b, weierstrassPInverse(0, -4*a/b, x)) + (16*a*g*x^4 - 12*a*e*x^2 - 3*(3*b*c + 8*a*f)*x^3 - 8*a*d*x - 6*a
*c)*sqrt(b*x^3 + a))/(a*x^4), 1/24*(36*a*sqrt(b)*e*x^4*weierstrassPInverse(0, -4*a/b, x) + 4*(b*d + 2*a*g)*sqr
t(-a)*x^4*arctan(2*sqrt(b*x^3 + a)*sqrt(-a)/(b*x^3 + 2*a)) - 9*(b*c + 8*a*f)*sqrt(b)*x^4*weierstrassZeta(0, -4
*a/b, weierstrassPInverse(0, -4*a/b, x)) + (16*a*g*x^4 - 12*a*e*x^2 - 3*(3*b*c + 8*a*f)*x^3 - 8*a*d*x - 6*a*c)
*sqrt(b*x^3 + a))/(a*x^4)]

Sympy [A] (verification not implemented)

Time = 3.61 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.39 \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\frac {\sqrt {a} c \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{2} \\ - \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt {a} e \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} + \frac {\sqrt {a} f \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} - \frac {2 \sqrt {a} g \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3} + \frac {2 a g}{3 \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {\sqrt {b} d \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} + \frac {2 \sqrt {b} g x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3 \sqrt {a}} \]

[In]

integrate((g*x**4+f*x**3+e*x**2+d*x+c)*(b*x**3+a)**(1/2)/x**5,x)

[Out]

sqrt(a)*c*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)*e*
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)*f*gamma(-1/3)*
hyper((-1/2, -1/3), (2/3,), b*x**3*exp_polar(I*pi)/a)/(3*x*gamma(2/3)) - 2*sqrt(a)*g*asinh(sqrt(a)/(sqrt(b)*x*
*(3/2)))/3 + 2*a*g/(3*sqrt(b)*x**(3/2)*sqrt(a/(b*x**3) + 1)) - sqrt(b)*d*sqrt(a/(b*x**3) + 1)/(3*x**(3/2)) + 2
*sqrt(b)*g*x**(3/2)/(3*sqrt(a/(b*x**3) + 1)) - b*d*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(3*sqrt(a))

Maxima [F]

\[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{5}} \,d x } \]

[In]

integrate((g*x^4+f*x^3+e*x^2+d*x+c)*(b*x^3+a)^(1/2)/x^5,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\int { \frac {{\left (g x^{4} + f x^{3} + e x^{2} + d x + c\right )} \sqrt {b x^{3} + a}}{x^{5}} \,d x } \]

[In]

integrate((g*x^4+f*x^3+e*x^2+d*x+c)*(b*x^3+a)^(1/2)/x^5,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^3} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^5} \, dx=\int \frac {\sqrt {b\,x^3+a}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^5} \,d x \]

[In]

int(((a + b*x^3)^(1/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^5,x)

[Out]

int(((a + b*x^3)^(1/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^5, x)