3.6.20 \(\int \frac {(c+d x+e x^2+f x^3) (a+b x^4)^{3/2}}{x^6} \, dx\) [520]

Optimal. Leaf size=387 \[ \frac {12 b^{3/2} c x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 b \left (9 c-5 e x^2\right ) \sqrt {a+b x^4}}{15 x}+\frac {3}{4} b \left (d+f x^2\right ) \sqrt {a+b x^4}-\frac {1}{60} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right ) \left (a+b x^4\right )^{3/2}+\frac {3}{4} a \sqrt {b} f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3}{4} \sqrt {a} b d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {12 \sqrt [4]{a} b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 \sqrt [4]{a} b^{3/4} \left (9 \sqrt {b} c+5 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 \sqrt {a+b x^4}} \]

[Out]

-1/60*(12*c/x^5+15*d/x^4+20*e/x^3+30*f/x^2)*(b*x^4+a)^(3/2)-3/4*b*d*arctanh((b*x^4+a)^(1/2)/a^(1/2))*a^(1/2)+3
/4*a*f*arctanh(x^2*b^(1/2)/(b*x^4+a)^(1/2))*b^(1/2)-2/15*b*(-5*e*x^2+9*c)*(b*x^4+a)^(1/2)/x+3/4*b*(f*x^2+d)*(b
*x^4+a)^(1/2)+12/5*b^(3/2)*c*x*(b*x^4+a)^(1/2)/(a^(1/2)+x^2*b^(1/2))-12/5*a^(1/4)*b^(5/4)*c*(cos(2*arctan(b^(1
/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(b^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/
2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/(b*x^4+a)^(1/2)+2/15*a^(1/4)*b^(3/4)*(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))*(5*e*a^(1/2)+9*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)/(b*
x^4+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.23, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 15, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {14, 1839, 1847, 1286, 1212, 226, 1210, 1266, 829, 858, 223, 212, 272, 65, 214} \begin {gather*} \frac {2 \sqrt [4]{a} b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (5 \sqrt {a} e+9 \sqrt {b} c\right ) F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 \sqrt {a+b x^4}}-\frac {12 \sqrt [4]{a} b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {12 b^{3/2} c x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {1}{60} \left (a+b x^4\right )^{3/2} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right )-\frac {2 b \sqrt {a+b x^4} \left (9 c-5 e x^2\right )}{15 x}+\frac {3}{4} b \sqrt {a+b x^4} \left (d+f x^2\right )-\frac {3}{4} \sqrt {a} b d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )+\frac {3}{4} a \sqrt {b} f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\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^6,x]

[Out]

(12*b^(3/2)*c*x*Sqrt[a + b*x^4])/(5*(Sqrt[a] + Sqrt[b]*x^2)) - (2*b*(9*c - 5*e*x^2)*Sqrt[a + b*x^4])/(15*x) +
(3*b*(d + f*x^2)*Sqrt[a + b*x^4])/4 - (((12*c)/x^5 + (15*d)/x^4 + (20*e)/x^3 + (30*f)/x^2)*(a + b*x^4)^(3/2))/
60 + (3*a*Sqrt[b]*f*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/4 - (3*Sqrt[a]*b*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]
])/4 - (12*a^(1/4)*b^(5/4)*c*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*A
rcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(5*Sqrt[a + b*x^4]) + (2*a^(1/4)*b^(3/4)*(9*Sqrt[b]*c + 5*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])/(1
5*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 829

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m +
2*p + 2))), x] + Dist[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 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 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^6} \, dx &=-\frac {1}{60} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \frac {\left (-\frac {c}{5}-\frac {d x}{4}-\frac {e x^2}{3}-\frac {f x^3}{2}\right ) \sqrt {a+b x^4}}{x^2} \, dx\\ &=-\frac {1}{60} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \left (\frac {\left (-\frac {c}{5}-\frac {e x^2}{3}\right ) \sqrt {a+b x^4}}{x^2}+\frac {\left (-\frac {d}{4}-\frac {f x^2}{2}\right ) \sqrt {a+b x^4}}{x}\right ) \, dx\\ &=-\frac {1}{60} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \frac {\left (-\frac {c}{5}-\frac {e x^2}{3}\right ) \sqrt {a+b x^4}}{x^2} \, dx-(6 b) \int \frac {\left (-\frac {d}{4}-\frac {f x^2}{2}\right ) \sqrt {a+b x^4}}{x} \, dx\\ &=-\frac {2 b \left (9 c-5 e x^2\right ) \sqrt {a+b x^4}}{15 x}-\frac {1}{60} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right ) \left (a+b x^4\right )^{3/2}-(3 b) \text {Subst}\left (\int \frac {\left (-\frac {d}{4}-\frac {f x}{2}\right ) \sqrt {a+b x^2}}{x} \, dx,x,x^2\right )+(4 b) \int \frac {\frac {a e}{3}+\frac {3}{5} b c x^2}{\sqrt {a+b x^4}} \, dx\\ &=-\frac {2 b \left (9 c-5 e x^2\right ) \sqrt {a+b x^4}}{15 x}+\frac {3}{4} b \left (d+f x^2\right ) \sqrt {a+b x^4}-\frac {1}{60} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right ) \left (a+b x^4\right )^{3/2}-\frac {3}{2} \text {Subst}\left (\int \frac {-\frac {1}{2} a b d-\frac {1}{2} a b f x}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {1}{5} \left (12 \sqrt {a} b^{3/2} c\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx+\frac {1}{15} \left (4 \sqrt {a} b \left (9 \sqrt {b} c+5 \sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx\\ &=\frac {12 b^{3/2} c x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 b \left (9 c-5 e x^2\right ) \sqrt {a+b x^4}}{15 x}+\frac {3}{4} b \left (d+f x^2\right ) \sqrt {a+b x^4}-\frac {1}{60} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right ) \left (a+b x^4\right )^{3/2}-\frac {12 \sqrt [4]{a} b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 \sqrt [4]{a} b^{3/4} \left (9 \sqrt {b} c+5 \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 )}{15 \sqrt {a+b x^4}}+\frac {1}{4} (3 a b d) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )+\frac {1}{4} (3 a b f) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )\\ &=\frac {12 b^{3/2} c x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 b \left (9 c-5 e x^2\right ) \sqrt {a+b x^4}}{15 x}+\frac {3}{4} b \left (d+f x^2\right ) \sqrt {a+b x^4}-\frac {1}{60} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right ) \left (a+b x^4\right )^{3/2}-\frac {12 \sqrt [4]{a} b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 \sqrt [4]{a} b^{3/4} \left (9 \sqrt {b} c+5 \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 )}{15 \sqrt {a+b x^4}}+\frac {1}{8} (3 a b d) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )+\frac {1}{4} (3 a b f) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )\\ &=\frac {12 b^{3/2} c x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 b \left (9 c-5 e x^2\right ) \sqrt {a+b x^4}}{15 x}+\frac {3}{4} b \left (d+f x^2\right ) \sqrt {a+b x^4}-\frac {1}{60} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right ) \left (a+b x^4\right )^{3/2}+\frac {3}{4} a \sqrt {b} f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {12 \sqrt [4]{a} b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 \sqrt [4]{a} b^{3/4} \left (9 \sqrt {b} c+5 \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 )}{15 \sqrt {a+b x^4}}+\frac {1}{4} (3 a d) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )\\ &=\frac {12 b^{3/2} c x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {2 b \left (9 c-5 e x^2\right ) \sqrt {a+b x^4}}{15 x}+\frac {3}{4} b \left (d+f x^2\right ) \sqrt {a+b x^4}-\frac {1}{60} \left (\frac {12 c}{x^5}+\frac {15 d}{x^4}+\frac {20 e}{x^3}+\frac {30 f}{x^2}\right ) \left (a+b x^4\right )^{3/2}+\frac {3}{4} a \sqrt {b} f \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3}{4} \sqrt {a} b d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {12 \sqrt [4]{a} b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 \sqrt [4]{a} b^{3/4} \left (9 \sqrt {b} c+5 \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 )}{15 \sqrt {a+b x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.52, size = 331, normalized size = 0.86 \begin {gather*} \frac {-\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \left (\left (a+b x^4\right ) \left (12 a c+84 b c x^4+5 a x \left (3 d+4 e x+6 f x^2\right )-5 b x^5 (6 d+x (4 e+3 f x))\right )-45 a \sqrt {b} f x^5 \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )+45 \sqrt {a} b d x^5 \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )\right )+144 \sqrt {a} b^{3/2} c x^5 \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 )-16 i \sqrt {a} b \left (-9 i \sqrt {b} c+5 \sqrt {a} e\right ) x^5 \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 )}{60 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x^5 \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^6,x]

[Out]

(-(Sqrt[(I*Sqrt[b])/Sqrt[a]]*((a + b*x^4)*(12*a*c + 84*b*c*x^4 + 5*a*x*(3*d + 4*e*x + 6*f*x^2) - 5*b*x^5*(6*d
+ x*(4*e + 3*f*x))) - 45*a*Sqrt[b]*f*x^5*Sqrt[a + b*x^4]*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]] + 45*Sqrt[a]*b
*d*x^5*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])) + 144*Sqrt[a]*b^(3/2)*c*x^5*Sqrt[1 + (b*x^4)/a]*Elli
pticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] - (16*I)*Sqrt[a]*b*((-9*I)*Sqrt[b]*c + 5*Sqrt[a]*e)*x^5*Sqrt
[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(60*Sqrt[(I*Sqrt[b])/Sqrt[a]]*x^5*Sqrt[
a + b*x^4])

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Maple [C] Result contains complex when optimal does not.
time = 0.46, size = 350, normalized size = 0.90

method	result	size
elliptic	\(-\frac {a c \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {a d \sqrt {b \,x^{4}+a}}{4 x^{4}}-\frac {a e \sqrt {b \,x^{4}+a}}{3 x^{3}}-\frac {a f \sqrt {b \,x^{4}+a}}{2 x^{2}}-\frac {7 b c \sqrt {b \,x^{4}+a}}{5 x}+\frac {b f \,x^{2} \sqrt {b \,x^{4}+a}}{4}+\frac {b e x \sqrt {b \,x^{4}+a}}{3}+\frac {b d \sqrt {b \,x^{4}+a}}{2}+\frac {4 b e a \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {3 a \sqrt {b}\, f \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{4}+\frac {12 i b^{\frac {3}{2}} c \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 \sqrt {a}\, b d \arctanh \left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{4}\)	\(344\)
default	\(c \left (-\frac {a \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {7 b \sqrt {b \,x^{4}+a}}{5 x}+\frac {12 i b^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (\frac {b \sqrt {b \,x^{4}+a}}{2}-\frac {3 \sqrt {a}\, b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4}-\frac {a \sqrt {b \,x^{4}+a}}{4 x^{4}}\right )+f \left (\frac {b \,x^{2} \sqrt {b \,x^{4}+a}}{4}+\frac {3 a \sqrt {b}\, \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{4}-\frac {a \sqrt {b \,x^{4}+a}}{2 x^{2}}\right )+e \left (-\frac {a \sqrt {b \,x^{4}+a}}{3 x^{3}}+\frac {b x \sqrt {b \,x^{4}+a}}{3}+\frac {4 a b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\)	\(350\)
risch	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (84 b c \,x^{4}+30 a f \,x^{3}+20 a e \,x^{2}+15 a d x +12 a c \right )}{60 x^{5}}+\frac {b f \,x^{2} \sqrt {b \,x^{4}+a}}{4}+\frac {3 \sqrt {b}\, a f \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{4}+\frac {b e x \sqrt {b \,x^{4}+a}}{3}+\frac {4 b e a \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b d \sqrt {b \,x^{4}+a}}{2}+\frac {12 i b^{\frac {3}{2}} c \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {12 i b^{\frac {3}{2}} c \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b \sqrt {a}\, d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4}\)	\(374\)

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^6,x,method=_RETURNVERBOSE)

[Out]

c*(-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)))+d*(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)+f*(1/4*b*x^2*(b*x^4+a)^(1/2)+3/4*a*b^(1/2)*ln(x^2*b^(1/2)+(b*x^4+a
)^(1/2))-1/2*a/x^2*(b*x^4+a)^(1/2))+e*(-1/3*a*(b*x^4+a)^(1/2)/x^3+1/3*b*x*(b*x^4+a)^(1/2)+4/3*a*b/(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^6,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + x^2*e + d*x + c)/x^6, x)

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Fricas [F]
time = 0.26, size = 59, normalized size = 0.15 \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^{6}}, 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^6,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^6, x)

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Sympy [C] Result contains complex when optimal does not.
time = 5.85, size = 386, normalized size = 1.00 \begin {gather*} \frac {a^{\frac {3}{2}} c \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^{\frac {3}{2}} e \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 {a^{\frac {3}{2}} f}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b c \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 d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4} + \frac {\sqrt {a} b e x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {a} b f x^{2} \sqrt {1 + \frac {b x^{4}}{a}}}{4} - \frac {\sqrt {a} b f x^{2}}{2 \sqrt {1 + \frac {b x^{4}}{a}}} - \frac {a \sqrt {b} d \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} + \frac {a \sqrt {b} d}{2 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} + \frac {3 a \sqrt {b} f \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{4} + \frac {b^{\frac {3}{2}} d x^{2}}{2 \sqrt {\frac {a}{b x^{4}} + 1}} \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**6,x)

[Out]

a**(3/2)*c*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**(3/2)*
e*gamma(-3/4)*hyper((-3/4, -1/2), (1/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**3*gamma(1/4)) - a**(3/2)*f/(2*x**2*s
qrt(1 + b*x**4/a)) + sqrt(a)*b*c*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*d*asinh(sqrt(a)/(sqrt(b)*x**2))/4 + sqrt(a)*b*e*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b*
x**4*exp_polar(I*pi)/a)/(4*gamma(5/4)) + sqrt(a)*b*f*x**2*sqrt(1 + b*x**4/a)/4 - sqrt(a)*b*f*x**2/(2*sqrt(1 +
b*x**4/a)) - a*sqrt(b)*d*sqrt(a/(b*x**4) + 1)/(4*x**2) + a*sqrt(b)*d/(2*x**2*sqrt(a/(b*x**4) + 1)) + 3*a*sqrt(
b)*f*asinh(sqrt(b)*x**2/sqrt(a))/4 + b**(3/2)*d*x**2/(2*sqrt(a/(b*x**4) + 1))

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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^6,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + x^2*e + d*x + c)/x^6, 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^6} \,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^6,x)

[Out]

int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^6, x)

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