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

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

[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 = 0.49, 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, 1825, 1833, 1252, 835, 807, 266, 63, 208, 1282, 1198, 220, 1196} \[ \frac {b^3 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{32 a^{3/2}}-\frac {2 b^{9/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (15 \sqrt {b} d-77 \sqrt {a} f\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 a^{5/4} \sqrt {a+b x^4}}-\frac {4 b^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}-\frac {b^2 c \sqrt {a+b x^4}}{32 a x^4}-\frac {4 b^2 d \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 e \sqrt {a+b x^4}}{10 a x^2}+\frac {4 b^{5/2} f x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {4 b^2 f \sqrt {a+b x^4}}{15 a x}-\frac {b \sqrt {a+b x^4} \left (\frac {1155 c}{x^8}+\frac {1440 d}{x^7}+\frac {1848 e}{x^6}+\frac {2464 f}{x^5}\right )}{18480}-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {165 c}{x^{12}}+\frac {180 d}{x^{11}}+\frac {198 e}{x^{10}}+\frac {220 f}{x^9}\right )}{1980} \]

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]

-(b*((1155*c)/x^8 + (1440*d)/x^7 + (1848*e)/x^6 + (2464*f)/x^5)*Sqrt[a + b*x^4])/18480 - (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] + Sqr
t[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 63

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 208

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

Rule 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 266

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 807

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 835

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 1196

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)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1198

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 1252

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 1282

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 1825

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 1833

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)*Sum[Coeff[Pq, x, j + (k*n)/2]*x^((k*n)/2), {k, 0, (2*(q - j))/n + 1}]*(a + b*x^n)^p)/c^j, {
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}

________________________________________________________________________________________

Mathematica [C]  time = 0.21, size = 149, normalized size = 0.33 \[ -\frac {\sqrt {a+b x^4} \left (90 a^5 d \, _2F_1\left (-\frac {11}{4},-\frac {3}{2};-\frac {7}{4};-\frac {b x^4}{a}\right )+11 x \left (10 a^5 f x \, _2F_1\left (-\frac {9}{4},-\frac {3}{2};-\frac {5}{4};-\frac {b x^4}{a}\right )+9 \left (a+b x^4\right )^2 \sqrt {\frac {b x^4}{a}+1} \left (a^3 e-b^3 c x^{10} \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {b x^4}{a}+1\right )\right )\right )\right )}{990 a^4 x^{11} \sqrt {\frac {b x^4}{a}+1}} \]

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]

-1/990*(Sqrt[a + b*x^4]*(90*a^5*d*Hypergeometric2F1[-11/4, -3/2, -7/4, -((b*x^4)/a)] + 11*x*(10*a^5*f*x*Hyperg
eometric2F1[-9/4, -3/2, -5/4, -((b*x^4)/a)] + 9*(a + b*x^4)^2*Sqrt[1 + (b*x^4)/a]*(a^3*e - b^3*c*x^10*Hypergeo
metric2F1[5/2, 4, 7/2, 1 + (b*x^4)/a]))))/(a^4*x^11*Sqrt[1 + (b*x^4)/a])

________________________________________________________________________________________

fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b f x^{7} + b e x^{6} + b d x^{5} + b c x^{4} + a f x^{3} + a e x^{2} + a d x + a c\right )} \sqrt {b x^{4} + a}}{x^{13}}, x\right ) \]

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]

integral((b*f*x^7 + b*e*x^6 + b*d*x^5 + b*c*x^4 + a*f*x^3 + a*e*x^2 + a*d*x + a*c)*sqrt(b*x^4 + a)/x^13, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )}}{x^{13}}\,{d x} \]

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 + e*x^2 + d*x + c)/x^13, x)

________________________________________________________________________________________

maple [C]  time = 0.20, size = 462, normalized size = 1.03 \[ -\frac {4 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b^{\frac {5}{2}} f \EllipticE \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {a}}+\frac {4 i \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b^{\frac {5}{2}} f \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {a}}-\frac {4 \sqrt {-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, b^{3} d \EllipticF \left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, x , i\right )}{77 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, a}+\frac {b^{3} c \ln \left (\frac {2 a +2 \sqrt {b \,x^{4}+a}\, \sqrt {a}}{x^{2}}\right )}{32 a^{\frac {3}{2}}}-\frac {4 \sqrt {b \,x^{4}+a}\, b^{2} f}{15 a x}-\frac {4 \sqrt {b \,x^{4}+a}\, b^{2} d}{77 a \,x^{3}}-\frac {\sqrt {b \,x^{4}+a}\, b^{2} c}{32 a \,x^{4}}-\frac {11 \sqrt {b \,x^{4}+a}\, b f}{45 x^{5}}-\frac {13 \sqrt {b \,x^{4}+a}\, b d}{77 x^{7}}-\frac {7 \sqrt {b \,x^{4}+a}\, b c}{48 x^{8}}-\frac {\sqrt {b \,x^{4}+a}\, a f}{9 x^{9}}-\frac {\sqrt {b \,x^{4}+a}\, a d}{11 x^{11}}-\frac {\sqrt {b \,x^{4}+a}\, a c}{12 x^{12}}-\frac {\sqrt {b \,x^{4}+a}\, \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right ) e}{10 a \,x^{10}} \]

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)

[Out]

-7/48*c*b/x^8*(b*x^4+a)^(1/2)-1/32*b^2*c*(b*x^4+a)^(1/2)/a/x^4+1/32*c/a^(3/2)*b^3*ln((2*a+2*(b*x^4+a)^(1/2)*a^
(1/2))/x^2)-1/12*c*a/x^12*(b*x^4+a)^(1/2)-1/9*f*a*(b*x^4+a)^(1/2)/x^9-11/45*f*b*(b*x^4+a)^(1/2)/x^5-4/15*b^2*f
*(b*x^4+a)^(1/2)/a/x+4/15*I*f/a^(1/2)*b^(5/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(-I/a^(1/2)*b^(1/2)*x^2+1)^(1/2)*(I/a^
(1/2)*b^(1/2)*x^2+1)^(1/2)/(b*x^4+a)^(1/2)*EllipticF((I/a^(1/2)*b^(1/2))^(1/2)*x,I)-4/15*I*f/a^(1/2)*b^(5/2)/(
I/a^(1/2)*b^(1/2))^(1/2)*(-I/a^(1/2)*b^(1/2)*x^2+1)^(1/2)*(I/a^(1/2)*b^(1/2)*x^2+1)^(1/2)/(b*x^4+a)^(1/2)*Elli
pticE((I/a^(1/2)*b^(1/2))^(1/2)*x,I)-1/11*d*a*(b*x^4+a)^(1/2)/x^11-13/77*d*b*(b*x^4+a)^(1/2)/x^7-4/77*b^2*d*(b
*x^4+a)^(1/2)/a/x^3-4/77*d/a*b^3/(I/a^(1/2)*b^(1/2))^(1/2)*(-I/a^(1/2)*b^(1/2)*x^2+1)^(1/2)*(I/a^(1/2)*b^(1/2)
*x^2+1)^(1/2)/(b*x^4+a)^(1/2)*EllipticF((I/a^(1/2)*b^(1/2))^(1/2)*x,I)-1/10*e*(b*x^4+a)^(1/2)/x^10/a*(b^2*x^8+
2*a*b*x^4+a^2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{192} \, {\left (\frac {3 \, b^{3} \log \left (\frac {\sqrt {b x^{4} + a} - \sqrt {a}}{\sqrt {b x^{4} + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (b x^{4} + a\right )}^{\frac {5}{2}} b^{3} + 8 \, {\left (b x^{4} + a\right )}^{\frac {3}{2}} a b^{3} - 3 \, \sqrt {b x^{4} + a} a^{2} b^{3}\right )}}{{\left (b x^{4} + a\right )}^{3} a - 3 \, {\left (b x^{4} + a\right )}^{2} a^{2} + 3 \, {\left (b x^{4} + a\right )} a^{3} - a^{4}}\right )} c + \int \frac {{\left (b f x^{6} + b e x^{5} + b d x^{4} + a f x^{2} + a e x + a d\right )} \sqrt {b x^{4} + a}}{x^{12}}\,{d x} \]

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*e*x^5 + b*d*x^4 + a*f*x^2 + a*e*x + a*d)*sqrt(b*x^4 + a)/x^12, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x^{13}} \,d x \]

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)

________________________________________________________________________________________

sympy [C]  time = 31.88, size = 403, normalized size = 0.90 \[ \frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{4}, - \frac {1}{2} \\ - \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{11} \Gamma \left (- \frac {7}{4}\right )} + \frac {a^{\frac {3}{2}} f \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{9} \Gamma \left (- \frac {5}{4}\right )} + \frac {\sqrt {a} b d \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {\sqrt {a} b f \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} - \frac {a^{2} c}{12 \sqrt {b} x^{14} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {11 a \sqrt {b} c}{48 x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} e \sqrt {\frac {a}{b x^{4}} + 1}}{10 x^{8}} - \frac {17 b^{\frac {3}{2}} c}{96 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} e \sqrt {\frac {a}{b x^{4}} + 1}}{5 x^{4}} - \frac {b^{\frac {5}{2}} c}{32 a x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {5}{2}} e \sqrt {\frac {a}{b x^{4}} + 1}}{10 a} + \frac {b^{3} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{32 a^{\frac {3}{2}}} \]

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))

________________________________________________________________________________________