3.6.28 \(\int \frac {(c+d x+e x^2+f x^3) (a+b x^4)^{3/2}}{x^{14}} \, dx\) [528]

Optimal. Leaf size=474 \[ -\frac {b \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right ) \sqrt {a+b x^4}}{240240}-\frac {4 b^2 c \sqrt {a+b x^4}}{195 a x^5}-\frac {b^2 d \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 e \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 f \sqrt {a+b x^4}}{10 a x^2}+\frac {4 b^3 c \sqrt {a+b x^4}}{65 a^2 x}-\frac {4 b^{7/2} c x \sqrt {a+b x^4}}{65 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right ) \left (a+b x^4\right )^{3/2}}{8580}+\frac {b^3 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{32 a^{3/2}}+\frac {4 b^{13/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 a^{7/4} \sqrt {a+b x^4}}-\frac {2 b^{11/4} \left (77 \sqrt {b} c+65 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5005 a^{7/4} \sqrt {a+b x^4}} \]

[Out]

-1/8580*(660*c/x^13+715*d/x^12+780*e/x^11+858*f/x^10)*(b*x^4+a)^(3/2)+1/32*b^3*d*arctanh((b*x^4+a)^(1/2)/a^(1/
2))/a^(3/2)-1/240240*b*(12320*c/x^9+15015*d/x^8+18720*e/x^7+24024*f/x^6)*(b*x^4+a)^(1/2)-4/195*b^2*c*(b*x^4+a)
^(1/2)/a/x^5-1/32*b^2*d*(b*x^4+a)^(1/2)/a/x^4-4/77*b^2*e*(b*x^4+a)^(1/2)/a/x^3-1/10*b^2*f*(b*x^4+a)^(1/2)/a/x^
2+4/65*b^3*c*(b*x^4+a)^(1/2)/a^2/x-4/65*b^(7/2)*c*x*(b*x^4+a)^(1/2)/a^2/(a^(1/2)+x^2*b^(1/2))+4/65*b^(13/4)*c*
(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a
^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(7/4)/(b*x^4+a)^(1/2)-
2/5005*b^(11/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticF(sin(2*ar
ctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(65*e*a^(1/2)+77*c*b^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^
2*b^(1/2))^2)^(1/2)/a^(7/4)/(b*x^4+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.38, antiderivative size = 474, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {14, 1839, 1847, 1296, 1212, 226, 1210, 1266, 849, 821, 272, 65, 214} \begin {gather*} -\frac {2 b^{11/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (65 \sqrt {a} e+77 \sqrt {b} c\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5005 a^{7/4} \sqrt {a+b x^4}}+\frac {4 b^{13/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 a^{7/4} \sqrt {a+b x^4}}+\frac {b^3 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{32 a^{3/2}}-\frac {4 b^{7/2} c x \sqrt {a+b x^4}}{65 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {4 b^3 c \sqrt {a+b x^4}}{65 a^2 x}-\frac {4 b^2 c \sqrt {a+b x^4}}{195 a x^5}-\frac {b^2 d \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 e \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 f \sqrt {a+b x^4}}{10 a x^2}-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right )}{8580}-\frac {b \sqrt {a+b x^4} \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right )}{240240} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^14,x]

[Out]

-1/240240*(b*((12320*c)/x^9 + (15015*d)/x^8 + (18720*e)/x^7 + (24024*f)/x^6)*Sqrt[a + b*x^4]) - (4*b^2*c*Sqrt[
a + b*x^4])/(195*a*x^5) - (b^2*d*Sqrt[a + b*x^4])/(32*a*x^4) - (4*b^2*e*Sqrt[a + b*x^4])/(77*a*x^3) - (b^2*f*S
qrt[a + b*x^4])/(10*a*x^2) + (4*b^3*c*Sqrt[a + b*x^4])/(65*a^2*x) - (4*b^(7/2)*c*x*Sqrt[a + b*x^4])/(65*a^2*(S
qrt[a] + Sqrt[b]*x^2)) - (((660*c)/x^13 + (715*d)/x^12 + (780*e)/x^11 + (858*f)/x^10)*(a + b*x^4)^(3/2))/8580
+ (b^3*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(32*a^(3/2)) + (4*b^(13/4)*c*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^
4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(65*a^(7/4)*Sqrt[a + b*x^4]) - (2
*b^(11/4)*(77*Sqrt[b]*c + 65*Sqrt[a]*e)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*El
lipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(5005*a^(7/4)*Sqrt[a + b*x^4])

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 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/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 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 849

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*
(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1212

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(e + d*q)/q, Int
[1/Sqrt[a + c*x^4], x], x] - Dist[e/q, Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a
, c, d, e}, x] && PosQ[c/a]

Rule 1266

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rule 1296

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a
+ c*x^4)^(p + 1)/(a*f*(m + 1))), x] + Dist[1/(a*f^2*(m + 1)), Int[(f*x)^(m + 2)*(a + c*x^4)^p*(a*e*(m + 1) -
c*d*(m + 4*p + 5)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p]
|| IntegerQ[m])

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 1847

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[
Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {
j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

Rubi steps

\begin {align*} \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x^{14}} \, dx &=-\frac {\left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right ) \left (a+b x^4\right )^{3/2}}{8580}-(6 b) \int \frac {\left (-\frac {c}{13}-\frac {d x}{12}-\frac {e x^2}{11}-\frac {f x^3}{10}\right ) \sqrt {a+b x^4}}{x^{10}} \, dx\\ &=-\frac {b \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right ) \sqrt {a+b x^4}}{240240}-\frac {\left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right ) \left (a+b x^4\right )^{3/2}}{8580}+\left (12 b^2\right ) \int \frac {\frac {c}{117}+\frac {d x}{96}+\frac {e x^2}{77}+\frac {f x^3}{60}}{x^6 \sqrt {a+b x^4}} \, dx\\ &=-\frac {b \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right ) \sqrt {a+b x^4}}{240240}-\frac {\left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right ) \left (a+b x^4\right )^{3/2}}{8580}+\left (12 b^2\right ) \int \left (\frac {\frac {c}{117}+\frac {e x^2}{77}}{x^6 \sqrt {a+b x^4}}+\frac {\frac {d}{96}+\frac {f x^2}{60}}{x^5 \sqrt {a+b x^4}}\right ) \, dx\\ &=-\frac {b \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right ) \sqrt {a+b x^4}}{240240}-\frac {\left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right ) \left (a+b x^4\right )^{3/2}}{8580}+\left (12 b^2\right ) \int \frac {\frac {c}{117}+\frac {e x^2}{77}}{x^6 \sqrt {a+b x^4}} \, dx+\left (12 b^2\right ) \int \frac {\frac {d}{96}+\frac {f x^2}{60}}{x^5 \sqrt {a+b x^4}} \, dx\\ &=-\frac {b \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right ) \sqrt {a+b x^4}}{240240}-\frac {4 b^2 c \sqrt {a+b x^4}}{195 a x^5}-\frac {\left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right ) \left (a+b x^4\right )^{3/2}}{8580}+\left (6 b^2\right ) \text {Subst}\left (\int \frac {\frac {d}{96}+\frac {f x}{60}}{x^3 \sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {\left (12 b^2\right ) \int \frac {-\frac {5 a e}{77}+\frac {1}{39} b c x^2}{x^4 \sqrt {a+b x^4}} \, dx}{5 a}\\ &=-\frac {b \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right ) \sqrt {a+b x^4}}{240240}-\frac {4 b^2 c \sqrt {a+b x^4}}{195 a x^5}-\frac {b^2 d \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 e \sqrt {a+b x^4}}{77 a x^3}-\frac {\left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right ) \left (a+b x^4\right )^{3/2}}{8580}+\frac {\left (4 b^2\right ) \int \frac {-\frac {1}{13} a b c-\frac {5}{77} a b e x^2}{x^2 \sqrt {a+b x^4}} \, dx}{5 a^2}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {-\frac {a f}{30}+\frac {b d x}{96}}{x^2 \sqrt {a+b x^2}} \, dx,x,x^2\right )}{a}\\ &=-\frac {b \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right ) \sqrt {a+b x^4}}{240240}-\frac {4 b^2 c \sqrt {a+b x^4}}{195 a x^5}-\frac {b^2 d \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 e \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 f \sqrt {a+b x^4}}{10 a x^2}+\frac {4 b^3 c \sqrt {a+b x^4}}{65 a^2 x}-\frac {\left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right ) \left (a+b x^4\right )^{3/2}}{8580}-\frac {\left (4 b^2\right ) \int \frac {\frac {5}{77} a^2 b e+\frac {1}{13} a b^2 c x^2}{\sqrt {a+b x^4}} \, dx}{5 a^3}-\frac {\left (b^3 d\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )}{32 a}\\ &=-\frac {b \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right ) \sqrt {a+b x^4}}{240240}-\frac {4 b^2 c \sqrt {a+b x^4}}{195 a x^5}-\frac {b^2 d \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 e \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 f \sqrt {a+b x^4}}{10 a x^2}+\frac {4 b^3 c \sqrt {a+b x^4}}{65 a^2 x}-\frac {\left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right ) \left (a+b x^4\right )^{3/2}}{8580}+\frac {\left (4 b^{7/2} c\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{65 a^{3/2}}-\frac {\left (b^3 d\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )}{64 a}-\frac {\left (4 b^3 \left (77 \sqrt {b} c+65 \sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{5005 a^{3/2}}\\ &=-\frac {b \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right ) \sqrt {a+b x^4}}{240240}-\frac {4 b^2 c \sqrt {a+b x^4}}{195 a x^5}-\frac {b^2 d \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 e \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 f \sqrt {a+b x^4}}{10 a x^2}+\frac {4 b^3 c \sqrt {a+b x^4}}{65 a^2 x}-\frac {4 b^{7/2} c x \sqrt {a+b x^4}}{65 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right ) \left (a+b x^4\right )^{3/2}}{8580}+\frac {4 b^{13/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 a^{7/4} \sqrt {a+b x^4}}-\frac {2 b^{11/4} \left (77 \sqrt {b} c+65 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5005 a^{7/4} \sqrt {a+b x^4}}-\frac {\left (b^2 d\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )}{32 a}\\ &=-\frac {b \left (\frac {12320 c}{x^9}+\frac {15015 d}{x^8}+\frac {18720 e}{x^7}+\frac {24024 f}{x^6}\right ) \sqrt {a+b x^4}}{240240}-\frac {4 b^2 c \sqrt {a+b x^4}}{195 a x^5}-\frac {b^2 d \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 e \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 f \sqrt {a+b x^4}}{10 a x^2}+\frac {4 b^3 c \sqrt {a+b x^4}}{65 a^2 x}-\frac {4 b^{7/2} c x \sqrt {a+b x^4}}{65 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {660 c}{x^{13}}+\frac {715 d}{x^{12}}+\frac {780 e}{x^{11}}+\frac {858 f}{x^{10}}\right ) \left (a+b x^4\right )^{3/2}}{8580}+\frac {b^3 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{32 a^{3/2}}+\frac {4 b^{13/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{65 a^{7/4} \sqrt {a+b x^4}}-\frac {2 b^{11/4} \left (77 \sqrt {b} c+65 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5005 a^{7/4} \sqrt {a+b x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.43, size = 339, normalized size = 0.72 \begin {gather*} \frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \left (-\left (\left (a+b x^4\right ) \left (-29568 b^3 c x^{12}+56 a^3 \left (660 c+13 x \left (55 d+60 e x+66 f x^2\right )\right )+a b^2 x^8 (9856 c+39 x (385 d+16 x (40 e+77 f x)))+2 a^2 b x^4 (30800 c+13 x (2695 d+48 x (65 e+77 f x)))\right )\right )+15015 \sqrt {a} b^3 d x^{13} \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )\right )-29568 \sqrt {a} b^{7/2} c x^{13} \sqrt {1+\frac {b x^4}{a}} E\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )+384 \sqrt {a} b^3 \left (77 \sqrt {b} c+65 i \sqrt {a} e\right ) x^{13} \sqrt {1+\frac {b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )}{480480 a^2 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x^{13} \sqrt {a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^14,x]

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(-((a + b*x^4)*(-29568*b^3*c*x^12 + 56*a^3*(660*c + 13*x*(55*d + 60*e*x + 66*f*x^2)
) + a*b^2*x^8*(9856*c + 39*x*(385*d + 16*x*(40*e + 77*f*x))) + 2*a^2*b*x^4*(30800*c + 13*x*(2695*d + 48*x*(65*
e + 77*f*x))))) + 15015*Sqrt[a]*b^3*d*x^13*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]) - 29568*Sqrt[a]*b
^(7/2)*c*x^13*Sqrt[1 + (b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] + 384*Sqrt[a]*b^3*(77*
Sqrt[b]*c + (65*I)*Sqrt[a]*e)*x^13*Sqrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/
(480480*a^2*Sqrt[(I*Sqrt[b])/Sqrt[a]]*x^13*Sqrt[a + b*x^4])

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Maple [C] Result contains complex when optimal does not.
time = 0.46, size = 420, normalized size = 0.89

method	result	size
risch	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (-29568 b^{3} c \,x^{12}+48048 a \,b^{2} f \,x^{11}+24960 a \,b^{2} e \,x^{10}+15015 a \,b^{2} d \,x^{9}+9856 a \,b^{2} c \,x^{8}+96096 a^{2} b f \,x^{7}+81120 a^{2} b e \,x^{6}+70070 a^{2} b d \,x^{5}+61600 a^{2} b c \,x^{4}+48048 a^{3} f \,x^{3}+43680 a^{3} e \,x^{2}+40040 a^{3} d x +36960 c \,a^{3}\right )}{480480 x^{13} a^{2}}-\frac {4 i b^{\frac {7}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{65 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {4 i b^{\frac {7}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{65 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {4 b^{3} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{3} d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}\)	\(405\)
default	\(e \left (-\frac {a \sqrt {b \,x^{4}+a}}{11 x^{11}}-\frac {13 b \sqrt {b \,x^{4}+a}}{77 x^{7}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{77 a \,x^{3}}-\frac {4 b^{3} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (-\frac {7 b \sqrt {b \,x^{4}+a}}{48 x^{8}}-\frac {b^{2} \sqrt {b \,x^{4}+a}}{32 a \,x^{4}}+\frac {b^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {a \sqrt {b \,x^{4}+a}}{12 x^{12}}\right )+c \left (-\frac {a \sqrt {b \,x^{4}+a}}{13 x^{13}}-\frac {5 b \sqrt {b \,x^{4}+a}}{39 x^{9}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{195 a \,x^{5}}+\frac {4 b^{3} \sqrt {b \,x^{4}+a}}{65 a^{2} x}-\frac {4 i b^{\frac {7}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{65 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )-\frac {f \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right ) \sqrt {b \,x^{4}+a}}{10 a \,x^{10}}\)	\(420\)
elliptic	\(-\frac {a c \sqrt {b \,x^{4}+a}}{13 x^{13}}-\frac {a d \sqrt {b \,x^{4}+a}}{12 x^{12}}-\frac {a e \sqrt {b \,x^{4}+a}}{11 x^{11}}-\frac {a f \sqrt {b \,x^{4}+a}}{10 x^{10}}-\frac {5 b c \sqrt {b \,x^{4}+a}}{39 x^{9}}-\frac {7 b d \sqrt {b \,x^{4}+a}}{48 x^{8}}-\frac {13 b e \sqrt {b \,x^{4}+a}}{77 x^{7}}-\frac {b f \sqrt {b \,x^{4}+a}}{5 x^{6}}-\frac {4 b^{2} c \sqrt {b \,x^{4}+a}}{195 a \,x^{5}}-\frac {b^{2} d \sqrt {b \,x^{4}+a}}{32 a \,x^{4}}-\frac {4 b^{2} e \sqrt {b \,x^{4}+a}}{77 a \,x^{3}}-\frac {b^{2} f \sqrt {b \,x^{4}+a}}{10 a \,x^{2}}+\frac {4 b^{3} c \sqrt {b \,x^{4}+a}}{65 a^{2} x}-\frac {4 b^{3} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {4 i b^{\frac {7}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{65 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{3} d \arctanh \left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{32 a^{\frac {3}{2}}}\)	\(432\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^14,x,method=_RETURNVERBOSE)

[Out]

e*(-1/11*a*(b*x^4+a)^(1/2)/x^11-13/77*b*(b*x^4+a)^(1/2)/x^7-4/77*b^2/a*(b*x^4+a)^(1/2)/x^3-4/77*b^3/a/(I/a^(1/
2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*
(I/a^(1/2)*b^(1/2))^(1/2),I))+d*(-7/48*b/x^8*(b*x^4+a)^(1/2)-1/32/a*b^2/x^4*(b*x^4+a)^(1/2)+1/32/a^(3/2)*b^3*l
n((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x^2)-1/12*a/x^12*(b*x^4+a)^(1/2))+c*(-1/13*a*(b*x^4+a)^(1/2)/x^13-5/39*b*(b*
x^4+a)^(1/2)/x^9-4/195*b^2/a*(b*x^4+a)^(1/2)/x^5+4/65*b^3/a^2*(b*x^4+a)^(1/2)/x-4/65*I*b^(7/2)/a^(3/2)/(I/a^(1
/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*(EllipticF(
x*(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I)))-1/10*f*(b^2*x^8+2*a*b*x^4+a^2)/a/x^1
0*(b*x^4+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((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^14,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + x^2*e + d*x + c)/x^14, x)

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Fricas [A]
time = 0.13, size = 258, normalized size = 0.54 \begin {gather*} \frac {59136 \, \sqrt {a} b^{3} c x^{13} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 15015 \, \sqrt {a} b^{3} d x^{13} \log \left (-\frac {b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 768 \, {\left (77 \, b^{3} c - 65 \, a b^{2} e\right )} \sqrt {a} x^{13} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (29568 \, b^{3} c x^{12} - 48048 \, a b^{2} f x^{11} - 24960 \, a b^{2} e x^{10} - 15015 \, a b^{2} d x^{9} - 9856 \, a b^{2} c x^{8} - 96096 \, a^{2} b f x^{7} - 81120 \, a^{2} b e x^{6} - 70070 \, a^{2} b d x^{5} - 61600 \, a^{2} b c x^{4} - 48048 \, a^{3} f x^{3} - 43680 \, a^{3} e x^{2} - 40040 \, a^{3} d x - 36960 \, a^{3} c\right )} \sqrt {b x^{4} + a}}{960960 \, a^{2} x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^14,x, algorithm="fricas")

[Out]

1/960960*(59136*sqrt(a)*b^3*c*x^13*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^(1/4)), -1) + 15015*sqrt(a)*b^3*d*x
^13*log(-(b*x^4 + 2*sqrt(b*x^4 + a)*sqrt(a) + 2*a)/x^4) - 768*(77*b^3*c - 65*a*b^2*e)*sqrt(a)*x^13*(-b/a)^(3/4
)*elliptic_f(arcsin(x*(-b/a)^(1/4)), -1) + 2*(29568*b^3*c*x^12 - 48048*a*b^2*f*x^11 - 24960*a*b^2*e*x^10 - 150
15*a*b^2*d*x^9 - 9856*a*b^2*c*x^8 - 96096*a^2*b*f*x^7 - 81120*a^2*b*e*x^6 - 70070*a^2*b*d*x^5 - 61600*a^2*b*c*
x^4 - 48048*a^3*f*x^3 - 43680*a^3*e*x^2 - 40040*a^3*d*x - 36960*a^3*c)*sqrt(b*x^4 + a))/(a^2*x^13)

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Sympy [C] Result contains complex when optimal does not.
time = 12.57, size = 403, normalized size = 0.85 \begin {gather*} \frac {a^{\frac {3}{2}} c \Gamma \left (- \frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {13}{4}, - \frac {1}{2} \\ - \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{13} \Gamma \left (- \frac {9}{4}\right )} + \frac {a^{\frac {3}{2}} e \Gamma \left (- \frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{4}, - \frac {1}{2} \\ - \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{11} \Gamma \left (- \frac {7}{4}\right )} + \frac {\sqrt {a} b c \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac {5}{4}\right )} + \frac {\sqrt {a} b e \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} - \frac {a^{2} d}{12 \sqrt {b} x^{14} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {11 a \sqrt {b} d}{48 x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} f \sqrt {\frac {a}{b x^{4}} + 1}}{10 x^{8}} - \frac {17 b^{\frac {3}{2}} d}{96 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} f \sqrt {\frac {a}{b x^{4}} + 1}}{5 x^{4}} - \frac {b^{\frac {5}{2}} d}{32 a x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {5}{2}} f \sqrt {\frac {a}{b x^{4}} + 1}}{10 a} + \frac {b^{3} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{32 a^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**(3/2)/x**14,x)

[Out]

a**(3/2)*c*gamma(-13/4)*hyper((-13/4, -1/2), (-9/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**13*gamma(-9/4)) + a**(3/
2)*e*gamma(-11/4)*hyper((-11/4, -1/2), (-7/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**11*gamma(-7/4)) + sqrt(a)*b*c*
gamma(-9/4)*hyper((-9/4, -1/2), (-5/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**9*gamma(-5/4)) + sqrt(a)*b*e*gamma(-7
/4)*hyper((-7/4, -1/2), (-3/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**7*gamma(-3/4)) - a**2*d/(12*sqrt(b)*x**14*sqr
t(a/(b*x**4) + 1)) - 11*a*sqrt(b)*d/(48*x**10*sqrt(a/(b*x**4) + 1)) - a*sqrt(b)*f*sqrt(a/(b*x**4) + 1)/(10*x**
8) - 17*b**(3/2)*d/(96*x**6*sqrt(a/(b*x**4) + 1)) - b**(3/2)*f*sqrt(a/(b*x**4) + 1)/(5*x**4) - b**(5/2)*d/(32*
a*x**2*sqrt(a/(b*x**4) + 1)) - b**(5/2)*f*sqrt(a/(b*x**4) + 1)/(10*a) + b**3*d*asinh(sqrt(a)/(sqrt(b)*x**2))/(
32*a**(3/2))

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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((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^14,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + x^2*e + d*x + c)/x^14, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^{14}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^14,x)

[Out]

int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^14, x)

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