3.6.26 \(\int \frac {(c+d x+e x^2+f x^3) (a+b x^4)^{3/2}}{x^{12}} \, dx\) [526]

Optimal. Leaf size=424 \[ -\frac {b \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right ) \sqrt {a+b x^4}}{18480}-\frac {4 b^2 c \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 d \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 e \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} e x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right ) \left (a+b x^4\right )^{3/2}}{3960}-\frac {3 b^2 f \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {4 b^{9/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 a^{3/4} \sqrt {a+b x^4}}-\frac {2 b^{9/4} \left (15 \sqrt {b} c-77 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{1155 a^{5/4} \sqrt {a+b x^4}} \]

[Out]

-1/3960*(360*c/x^11+396*d/x^10+440*e/x^9+495*f/x^8)*(b*x^4+a)^(3/2)-3/16*b^2*f*arctanh((b*x^4+a)^(1/2)/a^(1/2)
)/a^(1/2)-1/18480*b*(1440*c/x^7+1848*d/x^6+2464*e/x^5+3465*f/x^4)*(b*x^4+a)^(1/2)-4/77*b^2*c*(b*x^4+a)^(1/2)/a
/x^3-1/10*b^2*d*(b*x^4+a)^(1/2)/a/x^2-4/15*b^2*e*(b*x^4+a)^(1/2)/a/x+4/15*b^(5/2)*e*x*(b*x^4+a)^(1/2)/a/(a^(1/
2)+x^2*b^(1/2))-4/15*b^(9/4)*e*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*Ell
ipticE(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*e*a^(1/2)+15*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^(5/4)/(b*x^4+a)^(1/2)

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Rubi [A]
time = 0.32, antiderivative size = 424, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {14, 1839, 1847, 1296, 1212, 226, 1210, 1266, 821, 272, 65, 214} \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} c-77 \sqrt {a} e\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} e \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 {4 b^{5/2} e x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {4 b^2 c \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 d \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 e \sqrt {a+b x^4}}{15 a x}-\frac {3 b^2 f \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {b \sqrt {a+b x^4} \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right )}{18480}-\frac {\left (a+b x^4\right )^{3/2} \left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right )}{3960} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^12,x]

[Out]

-1/18480*(b*((1440*c)/x^7 + (1848*d)/x^6 + (2464*e)/x^5 + (3465*f)/x^4)*Sqrt[a + b*x^4]) - (4*b^2*c*Sqrt[a + b
*x^4])/(77*a*x^3) - (b^2*d*Sqrt[a + b*x^4])/(10*a*x^2) - (4*b^2*e*Sqrt[a + b*x^4])/(15*a*x) + (4*b^(5/2)*e*x*S
qrt[a + b*x^4])/(15*a*(Sqrt[a] + Sqrt[b]*x^2)) - (((360*c)/x^11 + (396*d)/x^10 + (440*e)/x^9 + (495*f)/x^8)*(a
+ b*x^4)^(3/2))/3960 - (3*b^2*f*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(16*Sqrt[a]) - (4*b^(9/4)*e*(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]*c - 77*Sqrt[a]*e)*(Sqrt[a] + Sqrt[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 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^{12}} \, dx &=-\frac {\left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right ) \left (a+b x^4\right )^{3/2}}{3960}-(6 b) \int \frac {\left (-\frac {c}{11}-\frac {d x}{10}-\frac {e x^2}{9}-\frac {f x^3}{8}\right ) \sqrt {a+b x^4}}{x^8} \, dx\\ &=-\frac {b \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right ) \sqrt {a+b x^4}}{18480}-\frac {\left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right ) \left (a+b x^4\right )^{3/2}}{3960}+\left (12 b^2\right ) \int \frac {\frac {c}{77}+\frac {d x}{60}+\frac {e x^2}{45}+\frac {f x^3}{32}}{x^4 \sqrt {a+b x^4}} \, dx\\ &=-\frac {b \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right ) \sqrt {a+b x^4}}{18480}-\frac {\left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right ) \left (a+b x^4\right )^{3/2}}{3960}+\left (12 b^2\right ) \int \left (\frac {\frac {c}{77}+\frac {e x^2}{45}}{x^4 \sqrt {a+b x^4}}+\frac {\frac {d}{60}+\frac {f x^2}{32}}{x^3 \sqrt {a+b x^4}}\right ) \, dx\\ &=-\frac {b \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right ) \sqrt {a+b x^4}}{18480}-\frac {\left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right ) \left (a+b x^4\right )^{3/2}}{3960}+\left (12 b^2\right ) \int \frac {\frac {c}{77}+\frac {e x^2}{45}}{x^4 \sqrt {a+b x^4}} \, dx+\left (12 b^2\right ) \int \frac {\frac {d}{60}+\frac {f x^2}{32}}{x^3 \sqrt {a+b x^4}} \, dx\\ &=-\frac {b \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right ) \sqrt {a+b x^4}}{18480}-\frac {4 b^2 c \sqrt {a+b x^4}}{77 a x^3}-\frac {\left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right ) \left (a+b x^4\right )^{3/2}}{3960}+\left (6 b^2\right ) \text {Subst}\left (\int \frac {\frac {d}{60}+\frac {f x}{32}}{x^2 \sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {\left (4 b^2\right ) \int \frac {-\frac {a e}{15}+\frac {1}{77} b c x^2}{x^2 \sqrt {a+b x^4}} \, dx}{a}\\ &=-\frac {b \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right ) \sqrt {a+b x^4}}{18480}-\frac {4 b^2 c \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 d \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 e \sqrt {a+b x^4}}{15 a x}-\frac {\left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right ) \left (a+b x^4\right )^{3/2}}{3960}+\frac {\left (4 b^2\right ) \int \frac {-\frac {1}{77} a b c+\frac {1}{15} a b e x^2}{\sqrt {a+b x^4}} \, dx}{a^2}+\frac {1}{16} \left (3 b^2 f\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )\\ &=-\frac {b \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right ) \sqrt {a+b x^4}}{18480}-\frac {4 b^2 c \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 d \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 e \sqrt {a+b x^4}}{15 a x}-\frac {\left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right ) \left (a+b x^4\right )^{3/2}}{3960}-\frac {\left (4 b^{5/2} e\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} c-77 \sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{1155 a}+\frac {1}{32} \left (3 b^2 f\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )\\ &=-\frac {b \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right ) \sqrt {a+b x^4}}{18480}-\frac {4 b^2 c \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 d \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 e \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} e x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right ) \left (a+b x^4\right )^{3/2}}{3960}-\frac {4 b^{9/4} e \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} c-77 \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 )}{1155 a^{5/4} \sqrt {a+b x^4}}+\frac {1}{16} (3 b f) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )\\ &=-\frac {b \left (\frac {1440 c}{x^7}+\frac {1848 d}{x^6}+\frac {2464 e}{x^5}+\frac {3465 f}{x^4}\right ) \sqrt {a+b x^4}}{18480}-\frac {4 b^2 c \sqrt {a+b x^4}}{77 a x^3}-\frac {b^2 d \sqrt {a+b x^4}}{10 a x^2}-\frac {4 b^2 e \sqrt {a+b x^4}}{15 a x}+\frac {4 b^{5/2} e x \sqrt {a+b x^4}}{15 a \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {\left (\frac {360 c}{x^{11}}+\frac {396 d}{x^{10}}+\frac {440 e}{x^9}+\frac {495 f}{x^8}\right ) \left (a+b x^4\right )^{3/2}}{3960}-\frac {3 b^2 f \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {4 b^{9/4} e \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} c-77 \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 )}{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 = 10.43, size = 317, normalized size = 0.75 \begin {gather*} \frac {-\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \left (\left (a+b x^4\right ) \left (24 b^2 x^8 (120 c+77 x (3 d+8 e x))+a b x^4 \left (9360 c+77 x \left (144 d+176 e x+225 f x^2\right )\right )+14 a^2 (360 c+11 x (36 d+5 x (8 e+9 f x)))\right )+10395 \sqrt {a} b^2 f x^{11} \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )\right )+14784 \sqrt {a} b^{5/2} e x^{11} \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 )-192 b^{5/2} \left (-15 i \sqrt {b} c+77 \sqrt {a} e\right ) x^{11} \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 )}{55440 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x^{11} \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^12,x]

[Out]

(-(Sqrt[(I*Sqrt[b])/Sqrt[a]]*((a + b*x^4)*(24*b^2*x^8*(120*c + 77*x*(3*d + 8*e*x)) + a*b*x^4*(9360*c + 77*x*(1
44*d + 176*e*x + 225*f*x^2)) + 14*a^2*(360*c + 11*x*(36*d + 5*x*(8*e + 9*f*x)))) + 10395*Sqrt[a]*b^2*f*x^11*Sq
rt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])) + 14784*Sqrt[a]*b^(5/2)*e*x^11*Sqrt[1 + (b*x^4)/a]*EllipticE[
I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] - 192*b^(5/2)*((-15*I)*Sqrt[b]*c + 77*Sqrt[a]*e)*x^11*Sqrt[1 + (b*
x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(55440*a*Sqrt[(I*Sqrt[b])/Sqrt[a]]*x^11*Sqrt[a
+ b*x^4])

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Maple [C] Result contains complex when optimal does not.
time = 0.44, size = 380, normalized size = 0.90

method	result	size
risch	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (14784 b^{2} e \,x^{10}+5544 b^{2} d \,x^{9}+2880 b^{2} c \,x^{8}+17325 a b f \,x^{7}+13552 a b e \,x^{6}+11088 a b d \,x^{5}+9360 a b c \,x^{4}+6930 a^{2} f \,x^{3}+6160 a^{2} e \,x^{2}+5544 a^{2} d x +5040 a^{2} c \right )}{55440 x^{11} a}+\frac {4 i b^{\frac {5}{2}} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {4 i b^{\frac {5}{2}} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {4 b^{3} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b^{2} f \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}\)	\(375\)
default	\(c \left (-\frac {a \sqrt {b \,x^{4}+a}}{11 x^{11}}-\frac {13 b \sqrt {b \,x^{4}+a}}{77 x^{7}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{77 a \,x^{3}}-\frac {4 b^{3} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+e \left (-\frac {a \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {11 b \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {4 b^{2} \sqrt {b \,x^{4}+a}}{15 a x}+\frac {4 i b^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+f \left (-\frac {a \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {5 b \sqrt {b \,x^{4}+a}}{16 x^{4}}-\frac {3 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}\right )-\frac {d \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right ) \sqrt {b \,x^{4}+a}}{10 a \,x^{10}}\)	\(380\)
elliptic	\(-\frac {a c \sqrt {b \,x^{4}+a}}{11 x^{11}}-\frac {a d \sqrt {b \,x^{4}+a}}{10 x^{10}}-\frac {a e \sqrt {b \,x^{4}+a}}{9 x^{9}}-\frac {a f \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {13 b c \sqrt {b \,x^{4}+a}}{77 x^{7}}-\frac {b d \sqrt {b \,x^{4}+a}}{5 x^{6}}-\frac {11 b e \sqrt {b \,x^{4}+a}}{45 x^{5}}-\frac {5 b f \sqrt {b \,x^{4}+a}}{16 x^{4}}-\frac {4 b^{2} c \sqrt {b \,x^{4}+a}}{77 a \,x^{3}}-\frac {b^{2} d \sqrt {b \,x^{4}+a}}{10 a \,x^{2}}-\frac {4 b^{2} e \sqrt {b \,x^{4}+a}}{15 a x}-\frac {4 b^{3} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{77 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {4 i b^{\frac {5}{2}} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b^{2} f \arctanh \left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{16 \sqrt {a}}\)	\(390\)

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^12,x,method=_RETURNVERBOSE)

[Out]

c*(-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))+e*(-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)))
+f*(-1/8*a/x^8*(b*x^4+a)^(1/2)-5/16*b/x^4*(b*x^4+a)^(1/2)-3/16*b^2/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/
x^2))-1/10*d*(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^12,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + x^2*e + d*x + c)/x^12, x)

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Fricas [A]
time = 0.12, size = 227, normalized size = 0.54 \begin {gather*} -\frac {29568 \, \sqrt {a} b^{2} e x^{11} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - 10395 \, \sqrt {a} b^{2} f x^{11} \log \left (-\frac {b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 384 \, {\left (15 \, b^{2} c + 77 \, b^{2} e\right )} \sqrt {a} x^{11} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (14784 \, b^{2} e x^{10} + 5544 \, b^{2} d x^{9} + 2880 \, b^{2} c x^{8} + 17325 \, a b f x^{7} + 13552 \, a b e x^{6} + 11088 \, a b d x^{5} + 9360 \, a b c x^{4} + 6930 \, a^{2} f x^{3} + 6160 \, a^{2} e x^{2} + 5544 \, a^{2} d x + 5040 \, a^{2} c\right )} \sqrt {b x^{4} + a}}{110880 \, a x^{11}} \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^12,x, algorithm="fricas")

[Out]

-1/110880*(29568*sqrt(a)*b^2*e*x^11*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^(1/4)), -1) - 10395*sqrt(a)*b^2*f*
x^11*log(-(b*x^4 - 2*sqrt(b*x^4 + a)*sqrt(a) + 2*a)/x^4) - 384*(15*b^2*c + 77*b^2*e)*sqrt(a)*x^11*(-b/a)^(3/4)
*elliptic_f(arcsin(x*(-b/a)^(1/4)), -1) + 2*(14784*b^2*e*x^10 + 5544*b^2*d*x^9 + 2880*b^2*c*x^8 + 17325*a*b*f*
x^7 + 13552*a*b*e*x^6 + 11088*a*b*d*x^5 + 9360*a*b*c*x^4 + 6930*a^2*f*x^3 + 6160*a^2*e*x^2 + 5544*a^2*d*x + 50
40*a^2*c)*sqrt(b*x^4 + a))/(a*x^11)

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Sympy [C] Result contains complex when optimal does not.
time = 7.65, size = 401, normalized size = 0.95 \begin {gather*} \frac {a^{\frac {3}{2}} c \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}} e \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 c \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 e \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} f}{8 \sqrt {b} x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} d \sqrt {\frac {a}{b x^{4}} + 1}}{10 x^{8}} - \frac {3 a \sqrt {b} f}{16 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{4}} + 1}}{5 x^{4}} - \frac {b^{\frac {3}{2}} f \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} - \frac {b^{\frac {3}{2}} f}{16 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {5}{2}} d \sqrt {\frac {a}{b x^{4}} + 1}}{10 a} - \frac {3 b^{2} f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{16 \sqrt {a}} \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**12,x)

[Out]

a**(3/2)*c*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)*e*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*c*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*e*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*f/(8*sqrt(b)*x**10*sqrt(a/
(b*x**4) + 1)) - a*sqrt(b)*d*sqrt(a/(b*x**4) + 1)/(10*x**8) - 3*a*sqrt(b)*f/(16*x**6*sqrt(a/(b*x**4) + 1)) - b
**(3/2)*d*sqrt(a/(b*x**4) + 1)/(5*x**4) - b**(3/2)*f*sqrt(a/(b*x**4) + 1)/(4*x**2) - b**(3/2)*f/(16*x**2*sqrt(
a/(b*x**4) + 1)) - b**(5/2)*d*sqrt(a/(b*x**4) + 1)/(10*a) - 3*b**2*f*asinh(sqrt(a)/(sqrt(b)*x**2))/(16*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((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^(3/2)/x^12,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + x^2*e + d*x + c)/x^12, 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^{12}} \,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^12,x)

[Out]

int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^12, x)

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