3.6.39 \(\int \frac {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx\) [539]

Optimal. Leaf size=377 \[ -\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2}+\frac {3 b c \sqrt {a+b x^4}}{5 a^2 x}-\frac {3 b^{3/2} c x \sqrt {a+b x^4}}{5 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}+\frac {3 b^{5/4} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{7/4} \sqrt {a+b x^4}}-\frac {b^{3/4} \left (9 \sqrt {b} c+5 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 a^{7/4} \sqrt {a+b x^4}} \]

[Out]

1/4*b*d*arctanh((b*x^4+a)^(1/2)/a^(1/2))/a^(3/2)-1/5*c*(b*x^4+a)^(1/2)/a/x^5-1/4*d*(b*x^4+a)^(1/2)/a/x^4-1/3*e
*(b*x^4+a)^(1/2)/a/x^3-1/2*f*(b*x^4+a)^(1/2)/a/x^2+3/5*b*c*(b*x^4+a)^(1/2)/a^2/x-3/5*b^(3/2)*c*x*(b*x^4+a)^(1/
2)/a^2/(a^(1/2)+x^2*b^(1/2))+3/5*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)/a^(7/4)/(b*x^4+a)^(1/2)-1/30*b^(3/4)*(cos(2*arctan(b^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arcta
n(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)/a^(7/4)/(b*x^4+a)^(1/2)

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Rubi [A]
time = 0.22, antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {1847, 1296, 1212, 226, 1210, 1266, 849, 821, 272, 65, 214} \begin {gather*} -\frac {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 )}{30 a^{7/4} \sqrt {a+b x^4}}+\frac {3 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 a^{7/4} \sqrt {a+b x^4}}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {3 b^{3/2} c x \sqrt {a+b x^4}}{5 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {3 b c \sqrt {a+b x^4}}{5 a^2 x}-\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3)/(x^6*Sqrt[a + b*x^4]),x]

[Out]

-1/5*(c*Sqrt[a + b*x^4])/(a*x^5) - (d*Sqrt[a + b*x^4])/(4*a*x^4) - (e*Sqrt[a + b*x^4])/(3*a*x^3) - (f*Sqrt[a +
b*x^4])/(2*a*x^2) + (3*b*c*Sqrt[a + b*x^4])/(5*a^2*x) - (3*b^(3/2)*c*x*Sqrt[a + b*x^4])/(5*a^2*(Sqrt[a] + Sqr
t[b]*x^2)) + (b*d*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(4*a^(3/2)) + (3*b^(5/4)*c*(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*a^(7/4)*Sqrt[a + b*x^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]*E
llipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(30*a^(7/4)*Sqrt[a + b*x^4])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 849

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*
(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1212

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(e + d*q)/q, Int
[1/Sqrt[a + c*x^4], x], x] - Dist[e/q, Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a
, c, d, e}, x] && PosQ[c/a]

Rule 1266

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rule 1296

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a
+ c*x^4)^(p + 1)/(a*f*(m + 1))), x] + Dist[1/(a*f^2*(m + 1)), Int[(f*x)^(m + 2)*(a + c*x^4)^p*(a*e*(m + 1) -
c*d*(m + 4*p + 5)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p]
|| IntegerQ[m])

Rule 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 {c+d x+e x^2+f x^3}{x^6 \sqrt {a+b x^4}} \, dx &=\int \left (\frac {c+e x^2}{x^6 \sqrt {a+b x^4}}+\frac {d+f x^2}{x^5 \sqrt {a+b x^4}}\right ) \, dx\\ &=\int \frac {c+e x^2}{x^6 \sqrt {a+b x^4}} \, dx+\int \frac {d+f x^2}{x^5 \sqrt {a+b x^4}} \, dx\\ &=-\frac {c \sqrt {a+b x^4}}{5 a x^5}+\frac {1}{2} \text {Subst}\left (\int \frac {d+f x}{x^3 \sqrt {a+b x^2}} \, dx,x,x^2\right )-\frac {\int \frac {-5 a e+3 b c x^2}{x^4 \sqrt {a+b x^4}} \, dx}{5 a}\\ &=-\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}+\frac {\int \frac {-9 a b c-5 a b e x^2}{x^2 \sqrt {a+b x^4}} \, dx}{15 a^2}-\frac {\text {Subst}\left (\int \frac {-2 a f+b d x}{x^2 \sqrt {a+b x^2}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2}+\frac {3 b c \sqrt {a+b x^4}}{5 a^2 x}-\frac {\int \frac {5 a^2 b e+9 a b^2 c x^2}{\sqrt {a+b x^4}} \, dx}{15 a^3}-\frac {(b d) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )}{4 a}\\ &=-\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2}+\frac {3 b c \sqrt {a+b x^4}}{5 a^2 x}+\frac {\left (3 b^{3/2} c\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{5 a^{3/2}}-\frac {(b d) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )}{8 a}-\frac {\left (b \left (9 \sqrt {b} c+5 \sqrt {a} e\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{15 a^{3/2}}\\ &=-\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2}+\frac {3 b c \sqrt {a+b x^4}}{5 a^2 x}-\frac {3 b^{3/2} c x \sqrt {a+b x^4}}{5 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {3 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 a^{7/4} \sqrt {a+b x^4}}-\frac {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 )}{30 a^{7/4} \sqrt {a+b x^4}}-\frac {d \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )}{4 a}\\ &=-\frac {c \sqrt {a+b x^4}}{5 a x^5}-\frac {d \sqrt {a+b x^4}}{4 a x^4}-\frac {e \sqrt {a+b x^4}}{3 a x^3}-\frac {f \sqrt {a+b x^4}}{2 a x^2}+\frac {3 b c \sqrt {a+b x^4}}{5 a^2 x}-\frac {3 b^{3/2} c x \sqrt {a+b x^4}}{5 a^2 \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}+\frac {3 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 a^{7/4} \sqrt {a+b x^4}}-\frac {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 )}{30 a^{7/4} \sqrt {a+b x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.34, size = 268, normalized size = 0.71 \begin {gather*} \frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \left (-\left (\left (a+b x^4\right ) \left (12 a c-36 b c x^4+5 a x \left (3 d+4 e x+6 f x^2\right )\right )\right )+15 \sqrt {a} b d x^5 \sqrt {a+b x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )\right )-36 \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 )+4 \sqrt {a} b \left (9 \sqrt {b} c+5 i \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 a^2 \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)/(x^6*Sqrt[a + b*x^4]),x]

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(-((a + b*x^4)*(12*a*c - 36*b*c*x^4 + 5*a*x*(3*d + 4*e*x + 6*f*x^2))) + 15*Sqrt[a]*
b*d*x^5*Sqrt[a + b*x^4]*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]) - 36*Sqrt[a]*b^(3/2)*c*x^5*Sqrt[1 + (b*x^4)/a]*Ellip
ticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] + 4*Sqrt[a]*b*(9*Sqrt[b]*c + (5*I)*Sqrt[a]*e)*x^5*Sqrt[1 + (b
*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(60*a^2*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 = 0.40, size = 297, normalized size = 0.79

method	result	size
elliptic	\(-\frac {c \sqrt {b \,x^{4}+a}}{5 a \,x^{5}}-\frac {d \sqrt {b \,x^{4}+a}}{4 a \,x^{4}}-\frac {e \sqrt {b \,x^{4}+a}}{3 a \,x^{3}}-\frac {f \sqrt {b \,x^{4}+a}}{2 a \,x^{2}}+\frac {3 b c \sqrt {b \,x^{4}+a}}{5 a^{2} x}-\frac {b e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 i b^{\frac {3}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b d \arctanh \left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{4 a^{\frac {3}{2}}}\)	\(286\)
default	\(c \left (-\frac {\sqrt {b \,x^{4}+a}}{5 a \,x^{5}}+\frac {3 b \sqrt {b \,x^{4}+a}}{5 a^{2} x}-\frac {3 i b^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+d \left (-\frac {\sqrt {b \,x^{4}+a}}{4 a \,x^{4}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\right )-\frac {f \sqrt {b \,x^{4}+a}}{2 a \,x^{2}}+e \left (-\frac {\sqrt {b \,x^{4}+a}}{3 a \,x^{3}}-\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\)	\(297\)
risch	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (-36 b c \,x^{4}+30 a f \,x^{3}+20 a e \,x^{2}+15 a d x +12 a c \right )}{60 a^{2} x^{5}}-\frac {3 i b^{\frac {3}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {3 i b^{\frac {3}{2}} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {b e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\)	\(311\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^3+e*x^2+d*x+c)/x^6/(b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

c*(-1/5*(b*x^4+a)^(1/2)/a/x^5+3/5*b*(b*x^4+a)^(1/2)/a^2/x-3/5*I*b^(3/2)/a^(3/2)/(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*(-1/4*(b*x^4+a)^(1/2)/a/x^4+1/4*b/a^(3/2)*ln((2*a+2*a^(1/
2)*(b*x^4+a)^(1/2))/x^2))-1/2*f*(b*x^4+a)^(1/2)/a/x^2+e*(-1/3*(b*x^4+a)^(1/2)/a/x^3-1/3*b/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))

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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)/x^6/(b*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.13, size = 159, normalized size = 0.42 \begin {gather*} \frac {72 \, \sqrt {a} b c x^{5} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 15 \, \sqrt {a} b d x^{5} \log \left (-\frac {b x^{4} + 2 \, \sqrt {b x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 8 \, {\left (9 \, b c - 5 \, a e\right )} \sqrt {a} x^{5} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + 2 \, {\left (36 \, b c x^{4} - 30 \, a f x^{3} - 20 \, a e x^{2} - 15 \, a d x - 12 \, a c\right )} \sqrt {b x^{4} + a}}{120 \, a^{2} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(72*sqrt(a)*b*c*x^5*(-b/a)^(3/4)*elliptic_e(arcsin(x*(-b/a)^(1/4)), -1) + 15*sqrt(a)*b*d*x^5*log(-(b*x^4
+ 2*sqrt(b*x^4 + a)*sqrt(a) + 2*a)/x^4) - 8*(9*b*c - 5*a*e)*sqrt(a)*x^5*(-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/
a)^(1/4)), -1) + 2*(36*b*c*x^4 - 30*a*f*x^3 - 20*a*e*x^2 - 15*a*d*x - 12*a*c)*sqrt(b*x^4 + a))/(a^2*x^5)

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Sympy [C] Result contains complex when optimal does not.
time = 2.73, size = 163, normalized size = 0.43 \begin {gather*} - \frac {\sqrt {b} d \sqrt {\frac {a}{b x^{4}} + 1}}{4 a x^{2}} - \frac {\sqrt {b} f \sqrt {\frac {a}{b x^{4}} + 1}}{2 a} + \frac {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 \sqrt {a} x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {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 \sqrt {a} x^{3} \Gamma \left (\frac {1}{4}\right )} + \frac {b d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4 a^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**3+e*x**2+d*x+c)/x**6/(b*x**4+a)**(1/2),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {f\,x^3+e\,x^2+d\,x+c}{x^6\,\sqrt {b\,x^4+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x + e*x^2 + f*x^3)/(x^6*(a + b*x^4)^(1/2)),x)

[Out]

int((c + d*x + e*x^2 + f*x^3)/(x^6*(a + b*x^4)^(1/2)), x)

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