3.6.23 \(\int \frac {(c+d x+e x^2+f x^3) (a+b x^4)^{3/2}}{x^9} \, dx\) [523]

Optimal. Leaf size=377 \[ -\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}+\frac {12 b^{3/2} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} b^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3 b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {12 \sqrt [4]{a} b^{5/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} d+21 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{a} \sqrt {a+b x^4}} \]

[Out]

-1/840*(105*c/x^8+120*d/x^7+140*e/x^6+168*f/x^5)*(b*x^4+a)^(3/2)+1/2*b^(3/2)*e*arctanh(x^2*b^(1/2)/(b*x^4+a)^(
1/2))-3/16*b^2*c*arctanh((b*x^4+a)^(1/2)/a^(1/2))/a^(1/2)-1/560*b*(105*c/x^4+160*d/x^3+280*e/x^2+672*f/x)*(b*x
^4+a)^(1/2)+12/5*b^(3/2)*f*x*(b*x^4+a)^(1/2)/(a^(1/2)+x^2*b^(1/2))-12/5*a^(1/4)*b^(5/4)*f*(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)))*EllipticF(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^
(1/2))*(21*f*a^(1/2)+5*d*b^(1/2))*(a^(1/2)+x^2*b^(1/2))*((b*x^4+a)/(a^(1/2)+x^2*b^(1/2))^2)^(1/2)/a^(1/4)/(b*x
^4+a)^(1/2)

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Rubi [A]
time = 0.25, antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {14, 1839, 1846, 272, 65, 214, 1899, 281, 223, 212, 1212, 226, 1210} \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} f+5 \sqrt {b} d\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} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \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} e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )+\frac {12 b^{3/2} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {3 b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {1}{560} b \sqrt {a+b x^4} \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right )-\frac {1}{840} \left (a+b x^4\right )^{3/2} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\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^9,x]

[Out]

-1/560*(b*((105*c)/x^4 + (160*d)/x^3 + (280*e)/x^2 + (672*f)/x)*Sqrt[a + b*x^4]) + (12*b^(3/2)*f*x*Sqrt[a + b*
x^4])/(5*(Sqrt[a] + Sqrt[b]*x^2)) - (((105*c)/x^8 + (120*d)/x^7 + (140*e)/x^6 + (168*f)/x^5)*(a + b*x^4)^(3/2)
)/840 + (b^(3/2)*e*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/2 - (3*b^2*c*ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]])/(16*
Sqrt[a]) - (12*a^(1/4)*b^(5/4)*f*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE
[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(5*Sqrt[a + b*x^4]) + (2*b^(5/4)*(5*Sqrt[b]*d + 21*Sqrt[a]*f)*(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])/(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 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 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 1846

Int[(Pq_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[Coeff[Pq, x, 0], Int[1/(x*Sqrt[a + b*x^n]), x
], x] + Int[ExpandToSum[(Pq - Coeff[Pq, x, 0])/x, x]/Sqrt[a + b*x^n], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] &
& IGtQ[n, 0] && NeQ[Coeff[Pq, x, 0], 0]

Rule 1899

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[P
q, 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
, 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^9} \, dx &=-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}-(6 b) \int \frac {\left (-\frac {c}{8}-\frac {d x}{7}-\frac {e x^2}{6}-\frac {f x^3}{5}\right ) \sqrt {a+b x^4}}{x^5} \, dx\\ &=-\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}+\left (12 b^2\right ) \int \frac {\frac {c}{32}+\frac {d x}{21}+\frac {e x^2}{12}+\frac {f x^3}{5}}{x \sqrt {a+b x^4}} \, dx\\ &=-\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}+\left (12 b^2\right ) \int \frac {\frac {d}{21}+\frac {e x}{12}+\frac {f x^2}{5}}{\sqrt {a+b x^4}} \, dx+\frac {1}{8} \left (3 b^2 c\right ) \int \frac {1}{x \sqrt {a+b x^4}} \, dx\\ &=-\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}+\left (12 b^2\right ) \int \left (\frac {e x}{12 \sqrt {a+b x^4}}+\frac {\frac {d}{21}+\frac {f x^2}{5}}{\sqrt {a+b x^4}}\right ) \, dx+\frac {1}{32} \left (3 b^2 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )\\ &=-\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}+\left (12 b^2\right ) \int \frac {\frac {d}{21}+\frac {f x^2}{5}}{\sqrt {a+b x^4}} \, dx+\frac {1}{16} (3 b c) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )+\left (b^2 e\right ) \int \frac {x}{\sqrt {a+b x^4}} \, dx\\ &=-\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}-\frac {3 b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}+\frac {1}{2} \left (b^2 e\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} f\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx+\frac {1}{35} \left (4 b^2 \left (5 d+\frac {21 \sqrt {a} f}{\sqrt {b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx\\ &=-\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}+\frac {12 b^{3/2} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}-\frac {3 b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {12 \sqrt [4]{a} b^{5/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} d+21 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{35 \sqrt [4]{a} \sqrt {a+b x^4}}+\frac {1}{2} \left (b^2 e\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )\\ &=-\frac {1}{560} b \left (\frac {105 c}{x^4}+\frac {160 d}{x^3}+\frac {280 e}{x^2}+\frac {672 f}{x}\right ) \sqrt {a+b x^4}+\frac {12 b^{3/2} f x \sqrt {a+b x^4}}{5 \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {1}{840} \left (\frac {105 c}{x^8}+\frac {120 d}{x^7}+\frac {140 e}{x^6}+\frac {168 f}{x^5}\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{2} b^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3 b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {12 \sqrt [4]{a} b^{5/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 \sqrt {a+b x^4}}+\frac {2 b^{5/4} \left (5 \sqrt {b} d+21 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{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 = 11.77, size = 309, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {a+b x^4} \left (b x^4 \left (525 c+16 x \left (45 d+70 e x+147 f x^2\right )\right )+a (210 c+8 x (30 d+7 x (5 e+6 f x)))\right )}{1680 x^8}+\frac {1}{2} b^{3/2} e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )-\frac {3 b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}}-\frac {12 i a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} b f \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 )}{5 \sqrt {a+b x^4}}-\frac {4 b^{3/2} \left (5 i \sqrt {b} d+21 \sqrt {a} f\right ) \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 )}{35 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \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^9,x]

[Out]

-1/1680*(Sqrt[a + b*x^4]*(b*x^4*(525*c + 16*x*(45*d + 70*e*x + 147*f*x^2)) + a*(210*c + 8*x*(30*d + 7*x*(5*e +
6*f*x)))))/x^8 + (b^(3/2)*e*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a + b*x^4]])/2 - (3*b^2*c*ArcTanh[Sqrt[a + b*x^4]/Sqrt
[a]])/(16*Sqrt[a]) - (((12*I)/5)*a*Sqrt[(I*Sqrt[b])/Sqrt[a]]*b*f*Sqrt[1 + (b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[
(I*Sqrt[b])/Sqrt[a]]*x], -1])/Sqrt[a + b*x^4] - (4*b^(3/2)*((5*I)*Sqrt[b]*d + 21*Sqrt[a]*f)*Sqrt[1 + (b*x^4)/a
]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(35*Sqrt[(I*Sqrt[b])/Sqrt[a]]*Sqrt[a + b*x^4])

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Maple [C] Result contains complex when optimal does not.
time = 0.41, size = 357, normalized size = 0.95

method	result	size
elliptic	\(-\frac {a c \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {a d \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {a e \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {a f \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {5 b c \sqrt {b \,x^{4}+a}}{16 x^{4}}-\frac {3 b d \sqrt {b \,x^{4}+a}}{7 x^{3}}-\frac {2 b e \sqrt {b \,x^{4}+a}}{3 x^{2}}-\frac {7 b f \sqrt {b \,x^{4}+a}}{5 x}+\frac {4 b^{2} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{\frac {3}{2}} e \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{2}+\frac {12 i b^{\frac {3}{2}} f \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b^{2} c \arctanh \left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{16 \sqrt {a}}\)	\(351\)
risch	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (2352 b f \,x^{7}+1120 b e \,x^{6}+720 b d \,x^{5}+525 b c \,x^{4}+336 a f \,x^{3}+280 a e \,x^{2}+240 a d x +210 a c \right )}{1680 x^{8}}+\frac {12 i b^{\frac {3}{2}} f \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {12 i b^{\frac {3}{2}} f \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {b^{\frac {3}{2}} e \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2}+\frac {4 b^{2} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {3 b^{2} c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}\)	\(354\)
default	\(f \left (-\frac {a \sqrt {b \,x^{4}+a}}{5 x^{5}}-\frac {7 b \sqrt {b \,x^{4}+a}}{5 x}+\frac {12 i b^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+e \left (\frac {b^{\frac {3}{2}} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2}-\frac {a \sqrt {b \,x^{4}+a}}{6 x^{6}}-\frac {2 b \sqrt {b \,x^{4}+a}}{3 x^{2}}\right )+c \left (-\frac {a \sqrt {b \,x^{4}+a}}{8 x^{8}}-\frac {5 b \sqrt {b \,x^{4}+a}}{16 x^{4}}-\frac {3 b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}\right )+d \left (-\frac {a \sqrt {b \,x^{4}+a}}{7 x^{7}}-\frac {3 b \sqrt {b \,x^{4}+a}}{7 x^{3}}+\frac {4 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\)	\(357\)

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^9,x,method=_RETURNVERBOSE)

[Out]

f*(-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)))+e*(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*(-1/8*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))+d*(-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)*Ellipti
cF(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^9,x, algorithm="maxima")

[Out]

1/32*(3*b^2*log((sqrt(b*x^4 + a) - sqrt(a))/(sqrt(b*x^4 + a) + sqrt(a)))/sqrt(a) - 2*(5*(b*x^4 + a)^(3/2)*b^2
- 3*sqrt(b*x^4 + a)*a*b^2)/((b*x^4 + a)^2 - 2*(b*x^4 + a)*a + a^2))*c + integrate((b*f*x^6 + b*x^5*e + b*d*x^4
+ a*f*x^2 + a*x*e + a*d)*sqrt(b*x^4 + a)/x^8, x)

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Fricas [F]
time = 0.27, size = 59, normalized size = 0.16 \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^{9}}, 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^9,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^9, x)

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Sympy [C] Result contains complex when optimal does not.
time = 7.11, size = 444, normalized size = 1.18 \begin {gather*} \frac {a^{\frac {3}{2}} d \Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, - \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac {3}{4}\right )} + \frac {a^{\frac {3}{2}} f \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {\sqrt {a} b d \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 e}{2 x^{2} \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b f \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 {a^{2} c}{8 \sqrt {b} x^{10} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {3 a \sqrt {b} c}{16 x^{6} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {a \sqrt {b} e \sqrt {\frac {a}{b x^{4}} + 1}}{6 x^{4}} - \frac {b^{\frac {3}{2}} c \sqrt {\frac {a}{b x^{4}} + 1}}{4 x^{2}} - \frac {b^{\frac {3}{2}} c}{16 x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} - \frac {b^{\frac {3}{2}} e \sqrt {\frac {a}{b x^{4}} + 1}}{6} + \frac {b^{\frac {3}{2}} e \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2} - \frac {3 b^{2} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{16 \sqrt {a}} - \frac {b^{2} e 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**9,x)

[Out]

a**(3/2)*d*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)*
f*gamma(-5/4)*hyper((-5/4, -1/2), (-1/4,), b*x**4*exp_polar(I*pi)/a)/(4*x**5*gamma(-1/4)) + sqrt(a)*b*d*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*e/(2*x**2*sqrt(1 +
b*x**4/a)) + sqrt(a)*b*f*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), b*x**4*exp_polar(I*pi)/a)/(4*x*gamma(3/4)) -
a**2*c/(8*sqrt(b)*x**10*sqrt(a/(b*x**4) + 1)) - 3*a*sqrt(b)*c/(16*x**6*sqrt(a/(b*x**4) + 1)) - a*sqrt(b)*e*sq
rt(a/(b*x**4) + 1)/(6*x**4) - b**(3/2)*c*sqrt(a/(b*x**4) + 1)/(4*x**2) - b**(3/2)*c/(16*x**2*sqrt(a/(b*x**4) +
1)) - b**(3/2)*e*sqrt(a/(b*x**4) + 1)/6 + b**(3/2)*e*asinh(sqrt(b)*x**2/sqrt(a))/2 - 3*b**2*c*asinh(sqrt(a)/(
sqrt(b)*x**2))/(16*sqrt(a)) - b**2*e*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^9,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/2)*(f*x^3 + x^2*e + d*x + c)/x^9, 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^9} \,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^9,x)

[Out]

int(((a + b*x^4)^(3/2)*(c + d*x + e*x^2 + f*x^3))/x^9, x)

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