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

Optimal. Leaf size=694 \[ \frac {3 c \sqrt {a+b x^3}}{20 x^4}+\frac {d \sqrt {a+b x^3}}{3 x^3}+\frac {3 e \sqrt {a+b x^3}}{2 x^2}-\frac {3 (b c+8 a f) \sqrt {a+b x^3}}{8 a x}+\frac {3 \sqrt [3]{b} (b c+8 a f) \sqrt {a+b x^3}}{8 a \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 \sqrt {a+b x^3} \left (3 c x+5 d x^2+15 e x^3-15 f x^4-5 g x^5\right )}{15 x^5}-\frac {(b d+2 a g) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{b} (b c+8 a f) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{16 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} \sqrt [3]{b} \left (4 a^{2/3} \sqrt [3]{b} e-\left (1-\sqrt {3}\right ) (b c+8 a f)\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{8 a^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[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]
time = 0.72, antiderivative size = 694, normalized size of antiderivative = 1.00, 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 = {1840, 1849, 1846, 272, 65, 214, 1892, 224, 1891} \begin {gather*} \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}} F\left (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right ) \left (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 (\text {ArcSin}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{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 {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 {(2 a g+b d) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}}+\frac {d \sqrt {a+b x^3}}{3 x^3}+\frac {3 e \sqrt {a+b x^3}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.45, size = 855, normalized size = 1.23 \begin {gather*} \frac {\sqrt {a+b x^3} \left (-6 a c-9 b c x^3-4 a x \left (2 d+x \left (3 e+6 f x-4 g x^2\right )\right )\right )}{24 a x^4}-\frac {8 \sqrt {a} b d \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a+b x^3} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+16 a^{3/2} g \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a+b x^3} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+36 a b^{2/3} e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )-9 \sqrt {2} \sqrt [3]{a} b^{4/3} c \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {i \left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}} \left (-\left (\left (-1+(-1)^{2/3}\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )-F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )-72 \sqrt {2} a^{4/3} \sqrt [3]{b} f \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{-1} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {i \left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}} \left (-\left (\left (-1+(-1)^{2/3}\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )-F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )}{24 a \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a+b x^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[a + b*x^3]*(-6*a*c - 9*b*c*x^3 - 4*a*x*(2*d + x*(3*e + 6*f*x - 4*g*x^2))))/(24*a*x^4) - (8*Sqrt[a]*b*d*S
qrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3]*ArcTanh[Sqrt[a + b*x^3]/Sqrt[
a]] + 16*a^(3/2)*g*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3]*ArcTanh[S
qrt[a + b*x^3]/Sqrt[a]] + 36*a*b^(2/3)*e*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1
)^(1/3))*a^(1/3))]*Sqrt[((-1)^(1/3)*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[Ar
cSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)] - 9*Sqrt[2]*a^(1/3)*b^(4/
3)*c*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(
1/3))]*Sqrt[(I*(1 + (b^(1/3)*x)/a^(1/3)))/(3*I + Sqrt[3])]*(-((-1 + (-1)^(2/3))*EllipticE[ArcSin[Sqrt[(-1)^(1/
6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))]) - EllipticF[ArcSin[Sqrt[(-1)^(1/6) - (I*b
^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))]) - 72*Sqrt[2]*a^(4/3)*b^(1/3)*f*((-1)^(1/3)*a^(1/3)
- b^(1/3)*x)*Sqrt[((-1)^(1/3)*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[(I*(1 + (b^(
1/3)*x)/a^(1/3)))/(3*I + Sqrt[3])]*(-((-1 + (-1)^(2/3))*EllipticE[ArcSin[Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/
3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^(1/3))]) - EllipticF[ArcSin[Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/
4)], (-1)^(1/3)/(-1 + (-1)^(1/3))]))/(24*a*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*S
qrt[a + b*x^3])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1285 vs. \(2 (532 ) = 1064\).
time = 0.52, size = 1286, normalized size = 1.85

method	result	size
elliptic	\(-\frac {c \sqrt {b \,x^{3}+a}}{4 x^{4}}-\frac {d \sqrt {b \,x^{3}+a}}{3 x^{3}}-\frac {e \sqrt {b \,x^{3}+a}}{2 x^{2}}-\frac {\left (8 a f +3 b c \right ) \sqrt {b \,x^{3}+a}}{8 a x}+\frac {2 g \sqrt {b \,x^{3}+a}}{3}-\frac {i e \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{2 \sqrt {b \,x^{3}+a}}-\frac {2 i \left (b f +\frac {b \left (8 a f +3 b c \right )}{16 a}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \EllipticE \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )+\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \EllipticF \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{b}\right )}{3 b \sqrt {b \,x^{3}+a}}-\frac {2 \left (a g +\frac {b d}{2}\right ) \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 \sqrt {a}}\)	\(845\)
risch	\(\text {Expression too large to display}\)	\(1243\)
default	\(\text {Expression too large to display}\)	\(1286\)

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.38, size = 330, normalized size = 0.48 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [A]
time = 3.84, size = 274, normalized size = 0.39 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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