3.6.22 \(\int \frac {(c+d x+e x^2+f x^3) (a+b x^4)^{3/2}}{x^8} \, dx\) [522]

Optimal. Leaf size=412 \[ -\frac {12 b e \sqrt {a+b x^4}}{5 x}+\frac {12 b^{3/2} e x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 b \left (5 c-21 e x^2\right ) \sqrt {a+b x^4}}{35 x^3}-\frac {b \left (2 d-3 f x^2\right ) \sqrt {a+b x^4}}{4 x^2}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3}{4} \sqrt {a} b f \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {12 \sqrt [4]{a} b^{5/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} c+21 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{a} \sqrt {a+b x^4}} \]

[Out]

-1/420*(60*c/x^7+70*d/x^6+84*e/x^5+105*f/x^4)*(b*x^4+a)^(3/2)+1/2*b^(3/2)*d*arctanh(x^2*b^(1/2)/(b*x^4+a)^(1/2
))-3/4*b*f*arctanh((b*x^4+a)^(1/2)/a^(1/2))*a^(1/2)-12/5*b*e*(b*x^4+a)^(1/2)/x-2/35*b*(-21*e*x^2+5*c)*(b*x^4+a
)^(1/2)/x^3-1/4*b*(-3*f*x^2+2*d)*(b*x^4+a)^(1/2)/x^2+12/5*b^(3/2)*e*x*(b*x^4+a)^(1/2)/(a^(1/2)+x^2*b^(1/2))-12
/5*a^(1/4)*b^(5/4)*e*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*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)/(b*
x^4+a)^(1/2)+2/35*b^(5/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*Elliptic
F(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(21*e*a^(1/2)+5*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^(1/4)/(b*x^4+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.27, antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 16, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {14, 1839, 1847, 1286, 1296, 1212, 226, 1210, 1266, 827, 858, 223, 212, 272, 65, 214} \begin {gather*} \frac {2 b^{5/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (21 \sqrt {a} e+5 \sqrt {b} c\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{a} \sqrt {a+b x^4}}-\frac {12 \sqrt [4]{a} b^{5/4} 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 )}{5 \sqrt {a+b x^4}}+\frac {1}{2} b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )+\frac {12 b^{3/2} e x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {1}{420} \left (a+b x^4\right )^{3/2} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right )-\frac {2 b \sqrt {a+b x^4} \left (5 c-21 e x^2\right )}{35 x^3}-\frac {b \sqrt {a+b x^4} \left (2 d-3 f x^2\right )}{4 x^2}-\frac {12 b e \sqrt {a+b x^4}}{5 x}-\frac {3}{4} \sqrt {a} b f \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^(3/2))/x^8,x]

[Out]

(-12*b*e*Sqrt[a + b*x^4])/(5*x) + (12*b^(3/2)*e*x*Sqrt[a + b*x^4])/(5*(Sqrt[a] + Sqrt[b]*x^2)) - (2*b*(5*c - 2
1*e*x^2)*Sqrt[a + b*x^4])/(35*x^3) - (b*(2*d - 3*f*x^2)*Sqrt[a + b*x^4])/(4*x^2) - (((60*c)/x^7 + (70*d)/x^6 +
(84*e)/x^5 + (105*f)/x^4)*(a + b*x^4)^(3/2))/420 + (b^(3/2)*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/2 - (3*
Sqrt[a]*b*f*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/4 - (12*a^(1/4)*b^(5/4)*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])/(5*Sqrt[a + b*x^4]) + (2*b^(5/4)*
(5*Sqrt[b]*c + 21*Sqrt[a]*e)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*A
rcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(35*a^(1/4)*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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 827

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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] &&  !IGtQ[m, 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 1286

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(a +
c*x^4)^p*((d*(m + 4*p + 3) + e*(m + 1)*x^2)/(f*(m + 1)*(m + 4*p + 3))), x] + Dist[4*(p/(f^2*(m + 1)*(m + 4*p
+ 3))), Int[(f*x)^(m + 2)*(a + c*x^4)^(p - 1)*(a*e*(m + 1) - c*d*(m + 4*p + 3)*x^2), x], x] /; FreeQ[{a, c, d,
e, f}, x] && GtQ[p, 0] && LtQ[m, -1] && m + 4*p + 3 != 0 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

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^8} \, dx &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \frac {\left (-\frac {c}{7}-\frac {d x}{6}-\frac {e x^2}{5}-\frac {f x^3}{4}\right ) \sqrt {a+b x^4}}{x^4} \, dx\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \left (\frac {\left (-\frac {c}{7}-\frac {e x^2}{5}\right ) \sqrt {a+b x^4}}{x^4}+\frac {\left (-\frac {d}{6}-\frac {f x^2}{4}\right ) \sqrt {a+b x^4}}{x^3}\right ) \, dx\\ &=-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \frac {\left (-\frac {c}{7}-\frac {e x^2}{5}\right ) \sqrt {a+b x^4}}{x^4} \, dx-(6 b) \int \frac {\left (-\frac {d}{6}-\frac {f x^2}{4}\right ) \sqrt {a+b x^4}}{x^3} \, dx\\ &=-\frac {2 b \left (5 c-21 e x^2\right ) \sqrt {a+b x^4}}{35 x^3}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}-(3 b) \text {Subst}\left (\int \frac {\left (-\frac {d}{6}-\frac {f x}{4}\right ) \sqrt {a+b x^2}}{x^2} \, dx,x,x^2\right )+(4 b) \int \frac {\frac {3 a e}{5}+\frac {1}{7} b c x^2}{x^2 \sqrt {a+b x^4}} \, dx\\ &=-\frac {12 b e \sqrt {a+b x^4}}{5 x}-\frac {2 b \left (5 c-21 e x^2\right ) \sqrt {a+b x^4}}{35 x^3}-\frac {b \left (2 d-3 f x^2\right ) \sqrt {a+b x^4}}{4 x^2}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} (3 b) \text {Subst}\left (\int \frac {\frac {a f}{2}+\frac {b d x}{3}}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {(4 b) \int \frac {-\frac {1}{7} a b c-\frac {3}{5} a b e x^2}{\sqrt {a+b x^4}} \, dx}{a}\\ &=-\frac {12 b e \sqrt {a+b x^4}}{5 x}-\frac {2 b \left (5 c-21 e x^2\right ) \sqrt {a+b x^4}}{35 x^3}-\frac {b \left (2 d-3 f x^2\right ) \sqrt {a+b x^4}}{4 x^2}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} \left (b^2 d\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {1}{5} \left (12 \sqrt {a} b^{3/2} e\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx+\frac {1}{35} \left (4 b^{3/2} \left (5 \sqrt {b} c+21 \sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx+\frac {1}{4} (3 a b f) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )\\ &=-\frac {12 b e \sqrt {a+b x^4}}{5 x}+\frac {12 b^{3/2} e x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 b \left (5 c-21 e x^2\right ) \sqrt {a+b x^4}}{35 x^3}-\frac {b \left (2 d-3 f x^2\right ) \sqrt {a+b x^4}}{4 x^2}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}-\frac {12 \sqrt [4]{a} b^{5/4} 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 )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} c+21 \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 )}{35 \sqrt [4]{a} \sqrt {a+b x^4}}+\frac {1}{2} \left (b^2 d\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )+\frac {1}{8} (3 a b f) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )\\ &=-\frac {12 b e \sqrt {a+b x^4}}{5 x}+\frac {12 b^{3/2} e x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 b \left (5 c-21 e x^2\right ) \sqrt {a+b x^4}}{35 x^3}-\frac {b \left (2 d-3 f x^2\right ) \sqrt {a+b x^4}}{4 x^2}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {12 \sqrt [4]{a} b^{5/4} 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 )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} c+21 \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 )}{35 \sqrt [4]{a} \sqrt {a+b x^4}}+\frac {1}{4} (3 a f) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )\\ &=-\frac {12 b e \sqrt {a+b x^4}}{5 x}+\frac {12 b^{3/2} e x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 b \left (5 c-21 e x^2\right ) \sqrt {a+b x^4}}{35 x^3}-\frac {b \left (2 d-3 f x^2\right ) \sqrt {a+b x^4}}{4 x^2}-\frac {1}{420} \left (\frac {60 c}{x^7}+\frac {70 d}{x^6}+\frac {84 e}{x^5}+\frac {105 f}{x^4}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} b^{3/2} d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3}{4} \sqrt {a} b f \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {12 \sqrt [4]{a} b^{5/4} 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 )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} c+21 \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 )}{35 \sqrt [4]{a} \sqrt {a+b x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.57, size = 330, normalized size = 0.80 \begin {gather*} \frac {-\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \left (\left (a+b x^4\right ) \left (2 b x^4 (90 c+7 x (20 d+3 x (14 e-5 f x)))+a (60 c+7 x (10 d+3 x (4 e+5 f x)))\right )-210 b^{3/2} d x^7 \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )+315 \sqrt {a} b f x^7 \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )\right )+1008 \sqrt {a} b^{3/2} e x^7 \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 )-48 b^{3/2} \left (5 i \sqrt {b} c+21 \sqrt {a} e\right ) x^7 \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 )}{420 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x^7 \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^8,x]

[Out]

(-(Sqrt[(I*Sqrt[b])/Sqrt[a]]*((a + b*x^4)*(2*b*x^4*(90*c + 7*x*(20*d + 3*x*(14*e - 5*f*x))) + a*(60*c + 7*x*(1
0*d + 3*x*(4*e + 5*f*x)))) - 210*b^(3/2)*d*x^7*Sqrt[a + b*x^4]*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]] + 315*Sq
rt[a]*b*f*x^7*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])) + 1008*Sqrt[a]*b^(3/2)*e*x^7*Sqrt[1 + (b*x^4)
/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] - 48*b^(3/2)*((5*I)*Sqrt[b]*c + 21*Sqrt[a]*e)*x^7*Sq
rt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(420*Sqrt[(I*Sqrt[b])/Sqrt[a]]*x^7*Sq
rt[a + b*x^4])

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Maple [C] Result contains complex when optimal does not.
time = 0.41, size = 352, normalized size = 0.85

method	result	size
elliptic	\(-\frac {a c \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {a d \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {a e \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {a f \sqrt {b \,x^{4}+a}}{4 x^{4}}-\frac {3 b c \sqrt {b \,x^{4}+a}}{7 x^{3}}-\frac {2 b d \sqrt {b \,x^{4}+a}}{3 x^{2}}-\frac {7 b e \sqrt {b \,x^{4}+a}}{5 x}+\frac {b f \sqrt {b \,x^{4}+a}}{2}+\frac {4 b^{2} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{\frac {3}{2}} d \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{2}+\frac {12 i b^{\frac {3}{2}} e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 \sqrt {a}\, b f \arctanh \left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{4}\)	\(346\)
default	\(e \left (-\frac {a \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {7 b \sqrt {b \,x^{4}+a}}{5 x}+\frac {12 i b^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+f \left (\frac {b \sqrt {b \,x^{4}+a}}{2}-\frac {3 \sqrt {a}\, b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4}-\frac {a \sqrt {b \,x^{4}+a}}{4 x^{4}}\right )+d \left (\frac {b^{\frac {3}{2}} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2}-\frac {a \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {2 b \sqrt {b \,x^{4}+a}}{3 x^{2}}\right )+c \left (-\frac {a \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {3 b \sqrt {b \,x^{4}+a}}{7 x^{3}}+\frac {4 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\)	\(352\)
risch	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (588 b e \,x^{6}+280 b d \,x^{5}+180 b c \,x^{4}+105 a f \,x^{3}+84 a e \,x^{2}+70 a d x +60 a c \right )}{420 x^{7}}+\frac {b f \sqrt {b \,x^{4}+a}}{2}+\frac {12 i b^{\frac {3}{2}} e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {12 i b^{\frac {3}{2}} e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{\frac {3}{2}} d \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2}+\frac {4 b^{2} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b \sqrt {a}\, f \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4}\)	\(358\)

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^8,x,method=_RETURNVERBOSE)

[Out]

e*(-1/5*a*(b*x^4+a)^(1/2)/x^5-7/5*b*(b*x^4+a)^(1/2)/x+12/5*I*b^(3/2)*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*(1/2*b*(b*x^4+a)^(1/2)-3/4*a^(1/2)*b*ln((2*a+2*a^(1/2)*(b*x^
4+a)^(1/2))/x^2)-1/4*a*(b*x^4+a)^(1/2)/x^4)+d*(1/2*b^(3/2)*ln(x^2*b^(1/2)+(b*x^4+a)^(1/2))-1/6*a/x^6*(b*x^4+a)
^(1/2)-2/3*b/x^2*(b*x^4+a)^(1/2))+c*(-1/7*a*(b*x^4+a)^(1/2)/x^7-3/7*b*(b*x^4+a)^(1/2)/x^3+4/7*b^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))

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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^8,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + x^2*e + d*x + c)/x^8, x)

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Fricas [F]
time = 0.27, size = 59, normalized size = 0.14 \begin {gather*} {\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^{8}}, x\right ) \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^8,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^8, x)

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Sympy [C] Result contains complex when optimal does not.
time = 5.71, size = 415, normalized size = 1.01 \begin {gather*} \frac {a^{\frac {3}{2}} 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 {a^{\frac {3}{2}} 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 {\sqrt {a} b c \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} - \frac {\sqrt {a} b d}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b e \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {3 \sqrt {a} b f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4} - \frac {a \sqrt {b} d \sqrt {\frac {a}{b x^{4}} + 1}}{6 x^{4}} - \frac {a \sqrt {b} f \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} + \frac {a \sqrt {b} f}{2 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{4}} + 1}}{6} + \frac {b^{\frac {3}{2}} d \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2} + \frac {b^{\frac {3}{2}} f x^{2}}{2 \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{2} d x^{2}}{2 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{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**8,x)

[Out]

a**(3/2)*c*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**(3/2)*
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)) + sqrt(a)*b*c*gamma(
-3/4)*hyper((-3/4, -1/2), (1/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**3*gamma(1/4)) - sqrt(a)*b*d/(2*x**2*sqrt(1 +
b*x**4/a)) + sqrt(a)*b*e*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), b*x**4*exp_polar(I*pi)/a)/(4*x*gamma(3/4)) -
3*sqrt(a)*b*f*asinh(sqrt(a)/(sqrt(b)*x**2))/4 - a*sqrt(b)*d*sqrt(a/(b*x**4) + 1)/(6*x**4) - a*sqrt(b)*f*sqrt(
a/(b*x**4) + 1)/(4*x**2) + a*sqrt(b)*f/(2*x**2*sqrt(a/(b*x**4) + 1)) - b**(3/2)*d*sqrt(a/(b*x**4) + 1)/6 + b**
(3/2)*d*asinh(sqrt(b)*x**2/sqrt(a))/2 + b**(3/2)*f*x**2/(2*sqrt(a/(b*x**4) + 1)) - b**2*d*x**2/(2*sqrt(a)*sqrt
(1 + b*x**4/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^8,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + x^2*e + d*x + c)/x^8, 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^8} \,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^8,x)

[Out]

int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^8, x)

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