3.6.27 \(\int \frac {(c+d x+e x^2+f x^3) (a+b x^4)^{3/2}}{x^{13}} \, dx\) [527]

Optimal. Leaf size=449 \[ -\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 e \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} f x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}+\frac {b^3 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{32 a^{3/2}}-\frac {4 b^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}-\frac {2 b^{9/4} \left (15 \sqrt {b} d-77 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 a^{5/4} \sqrt {a+b x^4}} \]

[Out]

-1/1980*(165*c/x^12+180*d/x^11+198*e/x^10+220*f/x^9)*(b*x^4+a)^(3/2)+1/32*b^3*c*arctanh((b*x^4+a)^(1/2)/a^(1/2
))/a^(3/2)-1/18480*b*(1155*c/x^8+1440*d/x^7+1848*e/x^6+2464*f/x^5)*(b*x^4+a)^(1/2)-1/32*b^2*c*(b*x^4+a)^(1/2)/
a/x^4-4/77*b^2*d*(b*x^4+a)^(1/2)/a/x^3-1/10*b^2*e*(b*x^4+a)^(1/2)/a/x^2-4/15*b^2*f*(b*x^4+a)^(1/2)/a/x+4/15*b^
(5/2)*f*x*(b*x^4+a)^(1/2)/a/(a^(1/2)+x^2*b^(1/2))-4/15*b^(9/4)*f*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/co
s(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^(3/4)/(b*x^4+a)^(1/2)-2/1155*b^(9/4)*(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*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(-77*f*a
^(1/2)+15*d*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^(5/4)/(b*x^4+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.33, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {14, 1839, 1847, 1266, 849, 821, 272, 65, 214, 1296, 1212, 226, 1210} \begin {gather*} -\frac {2 b^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (15 \sqrt {b} d-77 \sqrt {a} f\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 a^{5/4} \sqrt {a+b x^4}}-\frac {4 b^{9/4} f \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 )}{15 a^{3/4} \sqrt {a+b x^4}}+\frac {b^3 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{32 a^{3/2}}+\frac {4 b^{5/2} f x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 e \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right )}{1980}-\frac {b \sqrt {a+b x^4} \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right )}{18480} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^13,x]

[Out]

-1/18480*(b*((1155*c)/x^8 + (1440*d)/x^7 + (1848*e)/x^6 + (2464*f)/x^5)*Sqrt[a + b*x^4]) - (b^2*c*Sqrt[a + b*x
^4])/(32*a*x^4) - (4*b^2*d*Sqrt[a + b*x^4])/(77*a*x^3) - (b^2*e*Sqrt[a + b*x^4])/(10*a*x^2) - (4*b^2*f*Sqrt[a
+ b*x^4])/(15*a*x) + (4*b^(5/2)*f*x*Sqrt[a + b*x^4])/(15*a*(Sqrt[a] + Sqrt[b]*x^2)) - (((165*c)/x^12 + (180*d)
/x^11 + (198*e)/x^10 + (220*f)/x^9)*(a + b*x^4)^(3/2))/1980 + (b^3*c*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(32*a^(
3/2)) - (4*b^(9/4)*f*(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])/(15*a^(3/4)*Sqrt[a + b*x^4]) - (2*b^(9/4)*(15*Sqrt[b]*d - 77*Sqrt[a]*f)*(Sqrt[a] + S
qrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(1155*a
^(5/4)*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^{13}} \, dx &=-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}-(6 b) \int \frac {\left (-\frac {c}{12}-\frac {d x}{11}-\frac {e x^2}{10}-\frac {f x^3}{9}\right ) \sqrt {a+b x^4}}{x^9} \, dx\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}+\left (12 b^2\right ) \int \frac {\frac {c}{96}+\frac {d x}{77}+\frac {e x^2}{60}+\frac {f x^3}{45}}{x^5 \sqrt {a+b x^4}} \, dx\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}+\left (12 b^2\right ) \int \left (\frac {\frac {c}{96}+\frac {e x^2}{60}}{x^5 \sqrt {a+b x^4}}+\frac {\frac {d}{77}+\frac {f x^2}{45}}{x^4 \sqrt {a+b x^4}}\right ) \, dx\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}+\left (12 b^2\right ) \int \frac {\frac {c}{96}+\frac {e x^2}{60}}{x^5 \sqrt {a+b x^4}} \, dx+\left (12 b^2\right ) \int \frac {\frac {d}{77}+\frac {f x^2}{45}}{x^4 \sqrt {a+b x^4}} \, dx\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}+\left (6 b^2\right ) \text {Subst}\left (\int \frac {\frac {c}{96}+\frac {e x}{60}}{x^3 \sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {\left (4 b^2\right ) \int \frac {-\frac {a f}{15}+\frac {1}{77} b d x^2}{x^2 \sqrt {a+b x^4}} \, dx}{a}\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}+\frac {\left (4 b^2\right ) \int \frac {-\frac {1}{77} a b d+\frac {1}{15} a b f x^2}{\sqrt {a+b x^4}} \, dx}{a^2}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {-\frac {a e}{30}+\frac {b c x}{96}}{x^2 \sqrt {a+b x^2}} \, dx,x,x^2\right )}{a}\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 e \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}-\frac {\left (b^3 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )}{32 a}-\frac {\left (4 b^{5/2} f\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{15 \sqrt {a}}-\frac {\left (4 b^{5/2} \left (15 \sqrt {b} d-77 \sqrt {a} f\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{1155 a}\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 e \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} f x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}-\frac {4 b^{9/4} f \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 )}{15 a^{3/4} \sqrt {a+b x^4}}-\frac {2 b^{9/4} \left (15 \sqrt {b} d-77 \sqrt {a} f\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 )}{1155 a^{5/4} \sqrt {a+b x^4}}-\frac {\left (b^3 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )}{64 a}\\ &=-\frac {b \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 e \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} f x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}-\frac {4 b^{9/4} f \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 )}{15 a^{3/4} \sqrt {a+b x^4}}-\frac {2 b^{9/4} \left (15 \sqrt {b} d-77 \sqrt {a} f\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 )}{1155 a^{5/4} \sqrt {a+b x^4}}-\frac {\left (b^2 c\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 {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right ) \sqrt {a+b x^4}}{18480}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 e \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} f x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right ) \left (a+b x^4\right )^{3/2}}{1980}+\frac {b^3 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{32 a^{3/2}}-\frac {4 b^{9/4} f \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 )}{15 a^{3/4} \sqrt {a+b x^4}}-\frac {2 b^{9/4} \left (15 \sqrt {b} d-77 \sqrt {a} f\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 )}{1155 a^{5/4} \sqrt {a+b x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.47, size = 328, normalized size = 0.73 \begin {gather*} \frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \left (-\sqrt {a} \left (a+b x^4\right ) \left (56 a^2 \left (165 c+2 x \left (90 d+99 e x+110 f x^2\right )\right )+3 b^2 x^8 (1155 c+16 x (120 d+77 x (3 e+8 f x)))+2 a b x^4 (8085 c+16 x (585 d+77 x (9 e+11 f x)))\right )+3465 b^3 c x^{12} \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )\right )+29568 a b^{5/2} f x^{12} \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^{5/2} \left (-15 i \sqrt {b} d+77 \sqrt {a} f\right ) x^{12} \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 )}{110880 a^{3/2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x^{12} \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^13,x]

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(-(Sqrt[a]*(a + b*x^4)*(56*a^2*(165*c + 2*x*(90*d + 99*e*x + 110*f*x^2)) + 3*b^2*x^
8*(1155*c + 16*x*(120*d + 77*x*(3*e + 8*f*x))) + 2*a*b*x^4*(8085*c + 16*x*(585*d + 77*x*(9*e + 11*f*x))))) + 3
465*b^3*c*x^12*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]) + 29568*a*b^(5/2)*f*x^12*Sqrt[1 + (b*x^4)/a]*
EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] - 384*Sqrt[a]*b^(5/2)*((-15*I)*Sqrt[b]*d + 77*Sqrt[a]*f)
*x^12*Sqrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(110880*a^(3/2)*Sqrt[(I*Sqrt[
b])/Sqrt[a]]*x^12*Sqrt[a + b*x^4])

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Maple [C] Result contains complex when optimal does not.
time = 0.44, size = 400, normalized size = 0.89

method	result	size
risch	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (29568 b^{2} f \,x^{11}+11088 b^{2} e \,x^{10}+5760 b^{2} d \,x^{9}+3465 b^{2} c \,x^{8}+27104 a b f \,x^{7}+22176 a b e \,x^{6}+18720 a b d \,x^{5}+16170 a b c \,x^{4}+12320 a^{2} f \,x^{3}+11088 a^{2} e \,x^{2}+10080 a^{2} d x +9240 a^{2} c \right )}{110880 x^{12} a}+\frac {4 i b^{\frac {5}{2}} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {4 i b^{\frac {5}{2}} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {4 b^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{3} c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}\)	\(384\)
default	\(d \left (-\frac {a \sqrt {b \,x^{4}+a}}{11 x^{11}}-\frac {13 b \sqrt {b \,x^{4}+a}}{77 x^{7}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{77 a \,x^{3}}-\frac {4 b^{3} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+f \left (-\frac {a \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {11 b \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{15 a x}+\frac {4 i b^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+c \left (-\frac {7 b \sqrt {b \,x^{4}+a}}{48 x^{8}}-\frac {b^{2} \sqrt {b \,x^{4}+a}}{32 a \,x^{4}}+\frac {b^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {a \sqrt {b \,x^{4}+a}}{12 x^{12}}\right )-\frac {e \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right ) \sqrt {b \,x^{4}+a}}{10 a \,x^{10}}\)	\(400\)
elliptic	\(-\frac {a c \sqrt {b \,x^{4}+a}}{12 x^{12}}-\frac {a d \sqrt {b \,x^{4}+a}}{11 x^{11}}-\frac {a e \sqrt {b \,x^{4}+a}}{10 x^{10}}-\frac {a f \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {7 b c \sqrt {b \,x^{4}+a}}{48 x^{8}}-\frac {13 b d \sqrt {b \,x^{4}+a}}{77 x^{7}}-\frac {b e \sqrt {b \,x^{4}+a}}{5 x^{6}}-\frac {11 b f \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {b^{2} c \sqrt {b \,x^{4}+a}}{32 a \,x^{4}}-\frac {4 b^{2} d \sqrt {b \,x^{4}+a}}{77 a \,x^{3}}-\frac {b^{2} e \sqrt {b \,x^{4}+a}}{10 a \,x^{2}}-\frac {4 b^{2} f \sqrt {b \,x^{4}+a}}{15 a x}-\frac {4 b^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {4 i b^{\frac {5}{2}} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{3} c \arctanh \left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{32 a^{\frac {3}{2}}}\)	\(411\)

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^13,x,method=_RETURNVERBOSE)

[Out]

d*(-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))+f*(-1/9*a*(b*x^4+a)^(1/2)/x^9-11/45*b*(b*x^4+a)^(1/2)/x^5-4/15*b^2/a*(b*x^4+a)^(
1/2)/x+4/15*I*b^(5/2)/a^(1/2)/(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)))
+c*(-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*ln((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))-1/10*e*(b^2*x^8+2*a*b*x^4+a^2)/a/x^10*(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^13,x, algorithm="maxima")

[Out]

-1/192*(3*b^3*log((sqrt(b*x^4 + a) - sqrt(a))/(sqrt(b*x^4 + a) + sqrt(a)))/a^(3/2) + 2*(3*(b*x^4 + a)^(5/2)*b^
3 + 8*(b*x^4 + a)^(3/2)*a*b^3 - 3*sqrt(b*x^4 + a)*a^2*b^3)/((b*x^4 + a)^3*a - 3*(b*x^4 + a)^2*a^2 + 3*(b*x^4 +
a)*a^3 - a^4))*c + integrate((b*f*x^6 + b*x^5*e + b*d*x^4 + a*f*x^2 + a*x*e + a*d)*sqrt(b*x^4 + a)/x^12, x)

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Fricas [A]
time = 0.12, size = 250, normalized size = 0.56 \begin {gather*} -\frac {59136 \, a^{\frac {3}{2}} b^{2} f x^{12} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 3465 \, \sqrt {a} b^{3} c x^{12} \log \left (-\frac {b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 768 \, {\left (15 \, a b^{2} d + 77 \, a b^{2} f\right )} \sqrt {a} x^{12} \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 \, a b^{2} f x^{11} + 11088 \, a b^{2} e x^{10} + 5760 \, a b^{2} d x^{9} + 3465 \, a b^{2} c x^{8} + 27104 \, a^{2} b f x^{7} + 22176 \, a^{2} b e x^{6} + 18720 \, a^{2} b d x^{5} + 16170 \, a^{2} b c x^{4} + 12320 \, a^{3} f x^{3} + 11088 \, a^{3} e x^{2} + 10080 \, a^{3} d x + 9240 \, a^{3} c\right )} \sqrt {b x^{4} + a}}{221760 \, a^{2} x^{12}} \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^13,x, algorithm="fricas")

[Out]

-1/221760*(59136*a^(3/2)*b^2*f*x^12*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^(1/4)), -1) - 3465*sqrt(a)*b^3*c*x
^12*log(-(b*x^4 + 2*sqrt(b*x^4 + a)*sqrt(a) + 2*a)/x^4) - 768*(15*a*b^2*d + 77*a*b^2*f)*sqrt(a)*x^12*(-b/a)^(3
/4)*elliptic_f(arcsin(x*(-b/a)^(1/4)), -1) + 2*(29568*a*b^2*f*x^11 + 11088*a*b^2*e*x^10 + 5760*a*b^2*d*x^9 + 3
465*a*b^2*c*x^8 + 27104*a^2*b*f*x^7 + 22176*a^2*b*e*x^6 + 18720*a^2*b*d*x^5 + 16170*a^2*b*c*x^4 + 12320*a^3*f*
x^3 + 11088*a^3*e*x^2 + 10080*a^3*d*x + 9240*a^3*c)*sqrt(b*x^4 + a))/(a^2*x^12)

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Sympy [C] Result contains complex when optimal does not.
time = 12.06, size = 403, normalized size = 0.90 \begin {gather*} \frac {a^{\frac {3}{2}} d \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 {a^{\frac {3}{2}} f \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 d \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 {\sqrt {a} b f \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} - \frac {a^{2} c}{12 \sqrt {b} x^{14} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {11 a \sqrt {b} c}{48 x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} e \sqrt {\frac {a}{b x^{4}} + 1}}{10 x^{8}} - \frac {17 b^{\frac {3}{2}} c}{96 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} e \sqrt {\frac {a}{b x^{4}} + 1}}{5 x^{4}} - \frac {b^{\frac {5}{2}} c}{32 a x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {5}{2}} e \sqrt {\frac {a}{b x^{4}} + 1}}{10 a} + \frac {b^{3} c \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**13,x)

[Out]

a**(3/2)*d*gamma(-11/4)*hyper((-11/4, -1/2), (-7/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**11*gamma(-7/4)) + a**(3/
2)*f*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*d*gam
ma(-7/4)*hyper((-7/4, -1/2), (-3/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**7*gamma(-3/4)) + sqrt(a)*b*f*gamma(-5/4)
*hyper((-5/4, -1/2), (-1/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**5*gamma(-1/4)) - a**2*c/(12*sqrt(b)*x**14*sqrt(a
/(b*x**4) + 1)) - 11*a*sqrt(b)*c/(48*x**10*sqrt(a/(b*x**4) + 1)) - a*sqrt(b)*e*sqrt(a/(b*x**4) + 1)/(10*x**8)
- 17*b**(3/2)*c/(96*x**6*sqrt(a/(b*x**4) + 1)) - b**(3/2)*e*sqrt(a/(b*x**4) + 1)/(5*x**4) - b**(5/2)*c/(32*a*x
**2*sqrt(a/(b*x**4) + 1)) - b**(5/2)*e*sqrt(a/(b*x**4) + 1)/(10*a) + b**3*c*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^13,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + x^2*e + d*x + c)/x^13, 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^{13}} \,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^13,x)

[Out]

int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^13, x)

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