3.5.68 \(\int \frac {(a+b x^3)^{3/2} (c+d x+e x^2+f x^3+g x^4)}{x^7} \, dx\) [468]

Optimal. Leaf size=692 \[ \frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x}+\frac {27 b^{4/3} e \sqrt {a+b x^3}}{8 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac {b (b c+4 a f) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} b^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{16 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \left (2 b d-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{40 \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[Out]

-1/60*(10*c/x^6+12*d/x^5+15*e/x^4+20*f/x^3+30*g/x^2)*(b*x^3+a)^(3/2)-1/4*b*(4*a*f+b*c)*arctanh((b*x^3+a)^(1/2)
/a^(1/2))/a^(1/2)+1/4*b*c*(b*x^3+a)^(1/2)/x^3+27/20*b*d*(b*x^3+a)^(1/2)/x^2-27/8*b*e*(b*x^3+a)^(1/2)/x-1/20*b*
(-18*g*x^5-20*f*x^4-45*e*x^3+36*d*x^2+10*c*x)*(b*x^3+a)^(1/2)/x^4+27/8*b^(4/3)*e*(b*x^3+a)^(1/2)/(b^(1/3)*x+a^
(1/3)*(1+3^(1/2)))-27/16*3^(1/4)*a^(1/3)*b^(4/3)*e*(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)/(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)+9/40*3^(3/4)*b^(2/3)*(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)*(2*b*d+4*a*g-5*a^(1/3)*b^(2/3)*e*(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)/(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 = 0.69, antiderivative size = 692, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {14, 1839, 1840, 1849, 1846, 272, 65, 214, 1892, 224, 1891} \begin {gather*} \frac {9\ 3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \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 (\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 (-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g+2 b d\right )}{40 \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}} \sqrt [3]{a} b^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\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 \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^{4/3} e \sqrt {a+b x^3}}{8 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac {1}{60} \left (a+b x^3\right )^{3/2} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right )-\frac {b (4 a f+b c) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}+\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x} \end {gather*}

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^7,x]

[Out]

(b*c*Sqrt[a + b*x^3])/(4*x^3) + (27*b*d*Sqrt[a + b*x^3])/(20*x^2) - (27*b*e*Sqrt[a + b*x^3])/(8*x) + (27*b^(4/
3)*e*Sqrt[a + b*x^3])/(8*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (((10*c)/x^6 + (12*d)/x^5 + (15*e)/x^4 + (20*f
)/x^3 + (30*g)/x^2)*(a + b*x^3)^(3/2))/60 - (b*Sqrt[a + b*x^3]*(10*c*x + 36*d*x^2 - 45*e*x^3 - 20*f*x^4 - 18*g
*x^5))/(20*x^4) - (b*(b*c + 4*a*f)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(4*Sqrt[a]) - (27*3^(1/4)*Sqrt[2 - Sqrt[3
]]*a^(1/3)*b^(4/3)*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]])/(16*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^(2/3)*(2*b*d - 5*(1 - Sqrt[3])*a^(1/3)*b^(2/3)*e + 4*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[Arc
Sin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(40*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 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 1839

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 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 {\left (a+b x^3\right )^{3/2} \left (c+d x+e x^2+f x^3+g x^4\right )}{x^7} \, dx &=-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {1}{2} (9 b) \int \frac {\sqrt {a+b x^3} \left (-\frac {c}{6}-\frac {d x}{5}-\frac {e x^2}{4}-\frac {f x^3}{3}-\frac {g x^4}{2}\right )}{x^4} \, dx\\ &=-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac {1}{4} (27 a b) \int \frac {\frac {c}{9}+\frac {2 d x}{5}-\frac {e x^2}{2}-\frac {2 f x^3}{9}-\frac {g x^4}{5}}{x^4 \sqrt {a+b x^3}} \, dx\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}+\frac {1}{8} (9 b) \int \frac {-\frac {12 a d}{5}+3 a e x+\frac {1}{3} (b c+4 a f) x^2+\frac {6}{5} a g x^3}{x^3 \sqrt {a+b x^3}} \, dx\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac {(9 b) \int \frac {-12 a^2 e-\frac {4}{3} a (b c+4 a f) x-\frac {12}{5} a (b d+2 a g) x^2}{x^2 \sqrt {a+b x^3}} \, dx}{32 a}\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}+\frac {(9 b) \int \frac {\frac {8}{3} a^2 (b c+4 a f)+\frac {24}{5} a^2 (b d+2 a g) x+12 a^2 b e x^2}{x \sqrt {a+b x^3}} \, dx}{64 a^2}\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}+\frac {(9 b) \int \frac {\frac {24}{5} a^2 (b d+2 a g)+12 a^2 b e x}{\sqrt {a+b x^3}} \, dx}{64 a^2}+\frac {1}{8} (3 b (b c+4 a f)) \int \frac {1}{x \sqrt {a+b x^3}} \, dx\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}+\frac {1}{16} \left (27 b^{5/3} e\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx+\frac {1}{8} (b (b c+4 a f)) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )+\frac {1}{80} \left (27 b \left (2 b d-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+4 a g\right )\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x}+\frac {27 b^{4/3} e \sqrt {a+b x^3}}{8 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} b^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{16 \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^{2/3} \left (2 b d-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+4 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 )}{40 \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 c+4 a f) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )\\ &=\frac {b c \sqrt {a+b x^3}}{4 x^3}+\frac {27 b d \sqrt {a+b x^3}}{20 x^2}-\frac {27 b e \sqrt {a+b x^3}}{8 x}+\frac {27 b^{4/3} e \sqrt {a+b x^3}}{8 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {1}{60} \left (\frac {10 c}{x^6}+\frac {12 d}{x^5}+\frac {15 e}{x^4}+\frac {20 f}{x^3}+\frac {30 g}{x^2}\right ) \left (a+b x^3\right )^{3/2}-\frac {b \sqrt {a+b x^3} \left (10 c x+36 d x^2-45 e x^3-20 f x^4-18 g x^5\right )}{20 x^4}-\frac {b (b c+4 a f) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {27 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} b^{4/3} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{16 \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^{2/3} \left (2 b d-5 \left (1-\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} e+4 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 )}{40 \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 = 11.81, size = 805, normalized size = 1.16 \begin {gather*} -\frac {\sqrt {a+b x^3} \left (b x^3 \left (50 c+x \left (78 d+x \left (165 e-80 f x-48 g x^2\right )\right )\right )+a \left (20 c+2 x \left (12 d+5 x \left (3 e+4 f x+6 g x^2\right )\right )\right )\right )}{120 x^6}+\frac {3}{80} b \left (-\frac {20 b c \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}}-\frac {80}{3} \sqrt {a} f \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )-\frac {36 b^{2/3} d \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} \sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\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 )}{\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}}-\frac {72 a g \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} \sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\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 )}{\sqrt [3]{b} \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}}-\frac {90 \sqrt {2} \sqrt [3]{a} \sqrt [3]{b} e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{-1} \sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\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 (-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 )+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 )}{\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}}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^7,x]

[Out]

-1/120*(Sqrt[a + b*x^3]*(b*x^3*(50*c + x*(78*d + x*(165*e - 80*f*x - 48*g*x^2))) + a*(20*c + 2*x*(12*d + 5*x*(
3*e + 4*f*x + 6*g*x^2)))))/x^6 + (3*b*((-20*b*c*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(3*Sqrt[a]) - (80*Sqrt[a]*f*
ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/3 - (36*b^(2/3)*d*((-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)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Ell
ipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/(Sqrt[(a^(1/3)
+ (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3]) - (72*a*g*((-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)^(2/3)*b^(1/3)*x)/((1
+ (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (
-1)^(1/3)])/(b^(1/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3]) - (90*
Sqrt[2]*a^(1/3)*b^(1/3)*e*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[((-1)^(1/3)*a^(1/3) - (-1)^(2/3)*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[Arc
Sin[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))]))/(Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/
3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3])))/80

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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. 1195 vs. \(2 (532 ) = 1064\).
time = 0.54, size = 1196, normalized size = 1.73

method	result	size
elliptic	\(-\frac {a c \sqrt {b \,x^{3}+a}}{6 x^{6}}-\frac {a d \sqrt {b \,x^{3}+a}}{5 x^{5}}-\frac {a e \sqrt {b \,x^{3}+a}}{4 x^{4}}-\frac {\left (a f +\frac {5 b c}{4}\right ) \sqrt {b \,x^{3}+a}}{3 x^{3}}-\frac {\left (a g +\frac {13 b d}{10}\right ) \sqrt {b \,x^{3}+a}}{2 x^{2}}-\frac {11 b e \sqrt {b \,x^{3}+a}}{8 x}+\frac {2 b g x \sqrt {b \,x^{3}+a}}{5}+\frac {2 b f \sqrt {b \,x^{3}+a}}{3}-\frac {2 i \left (\frac {8 a b g}{5}+b^{2} d -\frac {b \left (10 a g +13 b d \right )}{40}\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}}}}\, \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 )}{3 b \sqrt {b \,x^{3}+a}}-\frac {9 i b 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}}}}\, \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 )}{8 \sqrt {b \,x^{3}+a}}-\frac {b \left (4 a f +b c \right ) \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{4 \sqrt {a}}\)	\(903\)
default	\(\text {Expression too large to display}\)	\(1196\)
risch	\(\text {Expression too large to display}\)	\(1411\)

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^7,x,method=_RETURNVERBOSE)

[Out]

d*(-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)*(-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)))+e*(-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)*(-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))))+c*(-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
))+g*(-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)*(-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)))+f*(-1/3*a*(b*x^3+a)^(1/2)/x^3+2/3*b*(b*x^3+a)^(1/2)-b*arctanh((b*x^3+a)^(1/2)/a^(
1/2))*a^(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((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^7,x, algorithm="maxima")

[Out]

1/24*(3*b^2*log((sqrt(b*x^3 + a) - sqrt(a))/(sqrt(b*x^3 + a) + sqrt(a)))/sqrt(a) - 2*(5*(b*x^3 + a)^(3/2)*b^2
- 3*sqrt(b*x^3 + a)*a*b^2)/((b*x^3 + a)^2 - 2*(b*x^3 + a)*a + a^2))*c + integrate((b*g*x^6 + b*f*x^5 + b*x^4*e
+ a*f*x^2 + (b*d + a*g)*x^3 + a*x*e + a*d)*sqrt(b*x^3 + a)/x^6, x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.37, size = 430, normalized size = 0.62 \begin {gather*} \left [-\frac {810 \, a b^{\frac {3}{2}} e x^{6} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - 15 \, {\left (b^{2} c + 4 \, a b f\right )} \sqrt {a} x^{6} \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 ) - 324 \, {\left (a b d + 2 \, a^{2} g\right )} \sqrt {b} x^{6} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 2 \, {\left (48 \, a b g x^{7} + 80 \, a b f x^{6} - 165 \, a b e x^{5} - 30 \, a^{2} e x^{2} - 6 \, {\left (13 \, a b d + 10 \, a^{2} g\right )} x^{4} - 24 \, a^{2} d x - 10 \, {\left (5 \, a b c + 4 \, a^{2} f\right )} x^{3} - 20 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{240 \, a x^{6}}, -\frac {405 \, a b^{\frac {3}{2}} e x^{6} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - 15 \, {\left (b^{2} c + 4 \, a b f\right )} \sqrt {-a} x^{6} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-a}}{b x^{3} + 2 \, a}\right ) - 162 \, {\left (a b d + 2 \, a^{2} g\right )} \sqrt {b} x^{6} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - {\left (48 \, a b g x^{7} + 80 \, a b f x^{6} - 165 \, a b e x^{5} - 30 \, a^{2} e x^{2} - 6 \, {\left (13 \, a b d + 10 \, a^{2} g\right )} x^{4} - 24 \, a^{2} d x - 10 \, {\left (5 \, a b c + 4 \, a^{2} f\right )} x^{3} - 20 \, a^{2} c\right )} \sqrt {b x^{3} + a}}{120 \, a x^{6}}\right ] \end {gather*}

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^7,x, algorithm="fricas")

[Out]

[-1/240*(810*a*b^(3/2)*e*x^6*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) - 15*(b^2*c + 4*a*b
*f)*sqrt(a)*x^6*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) - 324*(a*b*d
+ 2*a^2*g)*sqrt(b)*x^6*weierstrassPInverse(0, -4*a/b, x) - 2*(48*a*b*g*x^7 + 80*a*b*f*x^6 - 165*a*b*e*x^5 - 3
0*a^2*e*x^2 - 6*(13*a*b*d + 10*a^2*g)*x^4 - 24*a^2*d*x - 10*(5*a*b*c + 4*a^2*f)*x^3 - 20*a^2*c)*sqrt(b*x^3 + a
))/(a*x^6), -1/120*(405*a*b^(3/2)*e*x^6*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) - 15*(b^
2*c + 4*a*b*f)*sqrt(-a)*x^6*arctan(2*sqrt(b*x^3 + a)*sqrt(-a)/(b*x^3 + 2*a)) - 162*(a*b*d + 2*a^2*g)*sqrt(b)*x
^6*weierstrassPInverse(0, -4*a/b, x) - (48*a*b*g*x^7 + 80*a*b*f*x^6 - 165*a*b*e*x^5 - 30*a^2*e*x^2 - 6*(13*a*b
*d + 10*a^2*g)*x^4 - 24*a^2*d*x - 10*(5*a*b*c + 4*a^2*f)*x^3 - 20*a^2*c)*sqrt(b*x^3 + a))/(a*x^6)]

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Sympy [A]
time = 8.75, size = 524, normalized size = 0.76 \begin {gather*} \frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {a^{\frac {3}{2}} e \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 {a^{\frac {3}{2}} g \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} b d \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} b e \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 )} - \sqrt {a} b f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )} + \frac {\sqrt {a} b g x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} - \frac {a^{2} c}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} c}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {a \sqrt {b} f \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} + \frac {2 a \sqrt {b} f}{3 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b^{\frac {3}{2}} c}{12 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 b^{\frac {3}{2}} f x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{2} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{4 \sqrt {a}} \end {gather*}

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**7,x)

[Out]

a**(3/2)*d*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)*
e*gamma(-4/3)*hyper((-4/3, -1/2), (-1/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**4*gamma(-1/3)) + a**(3/2)*g*gamma(-
2/3)*hyper((-2/3, -1/2), (1/3,), b*x**3*exp_polar(I*pi)/a)/(3*x**2*gamma(1/3)) + sqrt(a)*b*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)) + sqrt(a)*b*e*gamma(-1/3)*hyper((-1/2, -1
/3), (2/3,), b*x**3*exp_polar(I*pi)/a)/(3*x*gamma(2/3)) - sqrt(a)*b*f*asinh(sqrt(a)/(sqrt(b)*x**(3/2))) + sqrt
(a)*b*g*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(4/3)) - a**2*c/(6*sqrt(b)*x
**(15/2)*sqrt(a/(b*x**3) + 1)) - a*sqrt(b)*c/(4*x**(9/2)*sqrt(a/(b*x**3) + 1)) - a*sqrt(b)*f*sqrt(a/(b*x**3) +
1)/(3*x**(3/2)) + 2*a*sqrt(b)*f/(3*x**(3/2)*sqrt(a/(b*x**3) + 1)) - b**(3/2)*c*sqrt(a/(b*x**3) + 1)/(3*x**(3/
2)) - b**(3/2)*c/(12*x**(3/2)*sqrt(a/(b*x**3) + 1)) + 2*b**(3/2)*f*x**(3/2)/(3*sqrt(a/(b*x**3) + 1)) - b**2*c*
asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(4*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((b*x^3+a)^(3/2)*(g*x^4+f*x^3+e*x^2+d*x+c)/x^7,x, algorithm="giac")

[Out]

integrate((g*x^4 + f*x^3 + x^2*e + d*x + c)*(b*x^3 + a)^(3/2)/x^7, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^{3/2}\,\left (g\,x^4+f\,x^3+e\,x^2+d\,x+c\right )}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^7,x)

[Out]

int(((a + b*x^3)^(3/2)*(c + d*x + e*x^2 + f*x^3 + g*x^4))/x^7, x)

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