Integrand size = 26, antiderivative size = 293 \[ \int \frac {x^{17/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {(b B-A c) x^{7/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(7 b B+A c) x^{3/2}}{16 b c^2 \left (b+c x^2\right )}-\frac {3 (7 b B+A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{5/4} c^{11/4}}+\frac {3 (7 b B+A c) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{5/4} c^{11/4}}+\frac {3 (7 b B+A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{5/4} c^{11/4}}-\frac {3 (7 b B+A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{5/4} c^{11/4}} \]
-1/4*(-A*c+B*b)*x^(7/2)/b/c/(c*x^2+b)^2-1/16*(A*c+7*B*b)*x^(3/2)/b/c^2/(c* x^2+b)-3/64*(A*c+7*B*b)*arctan(1-c^(1/4)*2^(1/2)*x^(1/2)/b^(1/4))/b^(5/4)/ c^(11/4)*2^(1/2)+3/64*(A*c+7*B*b)*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/b^(1/4) )/b^(5/4)/c^(11/4)*2^(1/2)+3/128*(A*c+7*B*b)*ln(b^(1/2)+x*c^(1/2)-b^(1/4)* c^(1/4)*2^(1/2)*x^(1/2))/b^(5/4)/c^(11/4)*2^(1/2)-3/128*(A*c+7*B*b)*ln(b^( 1/2)+x*c^(1/2)+b^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))/b^(5/4)/c^(11/4)*2^(1/2)
Time = 0.95 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.58 \[ \int \frac {x^{17/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{b} c^{3/4} x^{3/2} \left (7 b^2 B-3 A c^2 x^2+b c \left (A+11 B x^2\right )\right )}{\left (b+c x^2\right )^2}-3 \sqrt {2} (7 b B+A c) \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )-3 \sqrt {2} (7 b B+A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{64 b^{5/4} c^{11/4}} \]
((-4*b^(1/4)*c^(3/4)*x^(3/2)*(7*b^2*B - 3*A*c^2*x^2 + b*c*(A + 11*B*x^2))) /(b + c*x^2)^2 - 3*Sqrt[2]*(7*b*B + A*c)*ArcTan[(Sqrt[b] - Sqrt[c]*x)/(Sqr t[2]*b^(1/4)*c^(1/4)*Sqrt[x])] - 3*Sqrt[2]*(7*b*B + A*c)*ArcTanh[(Sqrt[2]* b^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[b] + Sqrt[c]*x)])/(64*b^(5/4)*c^(11/4))
Time = 0.48 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {9, 362, 252, 266, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^{17/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (b+c x^2\right )^3}dx\) |
\(\Big \downarrow \) 362 |
\(\displaystyle \frac {(A c+7 b B) \int \frac {x^{5/2}}{\left (c x^2+b\right )^2}dx}{8 b c}-\frac {x^{7/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {(A c+7 b B) \left (\frac {3 \int \frac {\sqrt {x}}{c x^2+b}dx}{4 c}-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{7/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {(A c+7 b B) \left (\frac {3 \int \frac {x}{c x^2+b}d\sqrt {x}}{2 c}-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{7/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 826 |
\(\displaystyle \frac {(A c+7 b B) \left (\frac {3 \left (\frac {\int \frac {\sqrt {c} x+\sqrt {b}}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{2 c}-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{7/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {(A c+7 b B) \left (\frac {3 \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{2 c}-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{7/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {(A c+7 b B) \left (\frac {3 \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{2 c}-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{7/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {(A c+7 b B) \left (\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{2 c}-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{7/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {(A c+7 b B) \left (\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{2 c}-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{7/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {(A c+7 b B) \left (\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{2 c}-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{7/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(A c+7 b B) \left (\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {c}}\right )}{2 c}-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{7/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {(A c+7 b B) \left (\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{2 c}-\frac {x^{3/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{7/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
-1/4*((b*B - A*c)*x^(7/2))/(b*c*(b + c*x^2)^2) + ((7*b*B + A*c)*(-1/2*x^(3 /2)/(c*(b + c*x^2)) + (3*((-(ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] /(Sqrt[2]*b^(1/4)*c^(1/4))) + ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4) ]/(Sqrt[2]*b^(1/4)*c^(1/4)))/(2*Sqrt[c]) - (-1/2*Log[Sqrt[b] - Sqrt[2]*b^( 1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]/(Sqrt[2]*b^(1/4)*c^(1/4)) + Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]/(2*Sqrt[2]*b^(1/4)*c^(1/4)) )/(2*Sqrt[c])))/(2*c)))/(8*b*c)
3.3.10.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 1.84 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.57
method | result | size |
derivativedivides | \(\frac {\frac {\left (3 A c -11 B b \right ) x^{\frac {7}{2}}}{16 b c}-\frac {\left (A c +7 B b \right ) x^{\frac {3}{2}}}{16 c^{2}}}{\left (c \,x^{2}+b \right )^{2}}+\frac {3 \left (A c +7 B b \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c^{3} b \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(166\) |
default | \(\frac {\frac {\left (3 A c -11 B b \right ) x^{\frac {7}{2}}}{16 b c}-\frac {\left (A c +7 B b \right ) x^{\frac {3}{2}}}{16 c^{2}}}{\left (c \,x^{2}+b \right )^{2}}+\frac {3 \left (A c +7 B b \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c^{3} b \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(166\) |
2*(1/32*(3*A*c-11*B*b)/b/c*x^(7/2)-1/32*(A*c+7*B*b)/c^2*x^(3/2))/(c*x^2+b) ^2+3/128*(A*c+7*B*b)/c^3/b/(1/c*b)^(1/4)*2^(1/2)*(ln((x-(1/c*b)^(1/4)*x^(1 /2)*2^(1/2)+(1/c*b)^(1/2))/(x+(1/c*b)^(1/4)*x^(1/2)*2^(1/2)+(1/c*b)^(1/2)) )+2*arctan(2^(1/2)/(1/c*b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(1/c*b)^(1/4) *x^(1/2)-1))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 871, normalized size of antiderivative = 2.97 \[ \int \frac {x^{17/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {3 \, {\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )} \left (-\frac {2401 \, B^{4} b^{4} + 1372 \, A B^{3} b^{3} c + 294 \, A^{2} B^{2} b^{2} c^{2} + 28 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{11}}\right )^{\frac {1}{4}} \log \left (27 \, b^{4} c^{8} \left (-\frac {2401 \, B^{4} b^{4} + 1372 \, A B^{3} b^{3} c + 294 \, A^{2} B^{2} b^{2} c^{2} + 28 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{11}}\right )^{\frac {3}{4}} + 27 \, {\left (343 \, B^{3} b^{3} + 147 \, A B^{2} b^{2} c + 21 \, A^{2} B b c^{2} + A^{3} c^{3}\right )} \sqrt {x}\right ) - 3 \, {\left (i \, b c^{4} x^{4} + 2 i \, b^{2} c^{3} x^{2} + i \, b^{3} c^{2}\right )} \left (-\frac {2401 \, B^{4} b^{4} + 1372 \, A B^{3} b^{3} c + 294 \, A^{2} B^{2} b^{2} c^{2} + 28 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{11}}\right )^{\frac {1}{4}} \log \left (27 i \, b^{4} c^{8} \left (-\frac {2401 \, B^{4} b^{4} + 1372 \, A B^{3} b^{3} c + 294 \, A^{2} B^{2} b^{2} c^{2} + 28 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{11}}\right )^{\frac {3}{4}} + 27 \, {\left (343 \, B^{3} b^{3} + 147 \, A B^{2} b^{2} c + 21 \, A^{2} B b c^{2} + A^{3} c^{3}\right )} \sqrt {x}\right ) - 3 \, {\left (-i \, b c^{4} x^{4} - 2 i \, b^{2} c^{3} x^{2} - i \, b^{3} c^{2}\right )} \left (-\frac {2401 \, B^{4} b^{4} + 1372 \, A B^{3} b^{3} c + 294 \, A^{2} B^{2} b^{2} c^{2} + 28 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{11}}\right )^{\frac {1}{4}} \log \left (-27 i \, b^{4} c^{8} \left (-\frac {2401 \, B^{4} b^{4} + 1372 \, A B^{3} b^{3} c + 294 \, A^{2} B^{2} b^{2} c^{2} + 28 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{11}}\right )^{\frac {3}{4}} + 27 \, {\left (343 \, B^{3} b^{3} + 147 \, A B^{2} b^{2} c + 21 \, A^{2} B b c^{2} + A^{3} c^{3}\right )} \sqrt {x}\right ) - 3 \, {\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )} \left (-\frac {2401 \, B^{4} b^{4} + 1372 \, A B^{3} b^{3} c + 294 \, A^{2} B^{2} b^{2} c^{2} + 28 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{11}}\right )^{\frac {1}{4}} \log \left (-27 \, b^{4} c^{8} \left (-\frac {2401 \, B^{4} b^{4} + 1372 \, A B^{3} b^{3} c + 294 \, A^{2} B^{2} b^{2} c^{2} + 28 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{11}}\right )^{\frac {3}{4}} + 27 \, {\left (343 \, B^{3} b^{3} + 147 \, A B^{2} b^{2} c + 21 \, A^{2} B b c^{2} + A^{3} c^{3}\right )} \sqrt {x}\right ) - 4 \, {\left ({\left (11 \, B b c - 3 \, A c^{2}\right )} x^{3} + {\left (7 \, B b^{2} + A b c\right )} x\right )} \sqrt {x}}{64 \, {\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )}} \]
1/64*(3*(b*c^4*x^4 + 2*b^2*c^3*x^2 + b^3*c^2)*(-(2401*B^4*b^4 + 1372*A*B^3 *b^3*c + 294*A^2*B^2*b^2*c^2 + 28*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^11))^(1/4) *log(27*b^4*c^8*(-(2401*B^4*b^4 + 1372*A*B^3*b^3*c + 294*A^2*B^2*b^2*c^2 + 28*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^11))^(3/4) + 27*(343*B^3*b^3 + 147*A*B^2 *b^2*c + 21*A^2*B*b*c^2 + A^3*c^3)*sqrt(x)) - 3*(I*b*c^4*x^4 + 2*I*b^2*c^3 *x^2 + I*b^3*c^2)*(-(2401*B^4*b^4 + 1372*A*B^3*b^3*c + 294*A^2*B^2*b^2*c^2 + 28*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^11))^(1/4)*log(27*I*b^4*c^8*(-(2401*B^ 4*b^4 + 1372*A*B^3*b^3*c + 294*A^2*B^2*b^2*c^2 + 28*A^3*B*b*c^3 + A^4*c^4) /(b^5*c^11))^(3/4) + 27*(343*B^3*b^3 + 147*A*B^2*b^2*c + 21*A^2*B*b*c^2 + A^3*c^3)*sqrt(x)) - 3*(-I*b*c^4*x^4 - 2*I*b^2*c^3*x^2 - I*b^3*c^2)*(-(2401 *B^4*b^4 + 1372*A*B^3*b^3*c + 294*A^2*B^2*b^2*c^2 + 28*A^3*B*b*c^3 + A^4*c ^4)/(b^5*c^11))^(1/4)*log(-27*I*b^4*c^8*(-(2401*B^4*b^4 + 1372*A*B^3*b^3*c + 294*A^2*B^2*b^2*c^2 + 28*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^11))^(3/4) + 27* (343*B^3*b^3 + 147*A*B^2*b^2*c + 21*A^2*B*b*c^2 + A^3*c^3)*sqrt(x)) - 3*(b *c^4*x^4 + 2*b^2*c^3*x^2 + b^3*c^2)*(-(2401*B^4*b^4 + 1372*A*B^3*b^3*c + 2 94*A^2*B^2*b^2*c^2 + 28*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^11))^(1/4)*log(-27*b ^4*c^8*(-(2401*B^4*b^4 + 1372*A*B^3*b^3*c + 294*A^2*B^2*b^2*c^2 + 28*A^3*B *b*c^3 + A^4*c^4)/(b^5*c^11))^(3/4) + 27*(343*B^3*b^3 + 147*A*B^2*b^2*c + 21*A^2*B*b*c^2 + A^3*c^3)*sqrt(x)) - 4*((11*B*b*c - 3*A*c^2)*x^3 + (7*B*b^ 2 + A*b*c)*x)*sqrt(x))/(b*c^4*x^4 + 2*b^2*c^3*x^2 + b^3*c^2)
Timed out. \[ \int \frac {x^{17/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.86 \[ \int \frac {x^{17/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {{\left (11 \, B b c - 3 \, A c^{2}\right )} x^{\frac {7}{2}} + {\left (7 \, B b^{2} + A b c\right )} x^{\frac {3}{2}}}{16 \, {\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )}} + \frac {3 \, {\left (7 \, B b + A c\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{128 \, b c^{2}} \]
-1/16*((11*B*b*c - 3*A*c^2)*x^(7/2) + (7*B*b^2 + A*b*c)*x^(3/2))/(b*c^4*x^ 4 + 2*b^2*c^3*x^2 + b^3*c^2) + 3/128*(7*B*b + A*c)*(2*sqrt(2)*arctan(1/2*s qrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) + 2*sqrt(c)*sqrt(x))/sqrt(sqrt(b)*sqrt(c)) )/(sqrt(sqrt(b)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) *b^(1/4)*c^(1/4) - 2*sqrt(c)*sqrt(x))/sqrt(sqrt(b)*sqrt(c)))/(sqrt(sqrt(b) *sqrt(c))*sqrt(c)) - sqrt(2)*log(sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c) *x + sqrt(b))/(b^(1/4)*c^(3/4)) + sqrt(2)*log(-sqrt(2)*b^(1/4)*c^(1/4)*sqr t(x) + sqrt(c)*x + sqrt(b))/(b^(1/4)*c^(3/4)))/(b*c^2)
Time = 0.30 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00 \[ \int \frac {x^{17/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {11 \, B b c x^{\frac {7}{2}} - 3 \, A c^{2} x^{\frac {7}{2}} + 7 \, B b^{2} x^{\frac {3}{2}} + A b c x^{\frac {3}{2}}}{16 \, {\left (c x^{2} + b\right )}^{2} b c^{2}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {3}{4}} B b + \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{2} c^{5}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {3}{4}} B b + \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{2} c^{5}} - \frac {3 \, \sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {3}{4}} B b + \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{2} c^{5}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {3}{4}} B b + \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{2} c^{5}} \]
-1/16*(11*B*b*c*x^(7/2) - 3*A*c^2*x^(7/2) + 7*B*b^2*x^(3/2) + A*b*c*x^(3/2 ))/((c*x^2 + b)^2*b*c^2) + 3/64*sqrt(2)*(7*(b*c^3)^(3/4)*B*b + (b*c^3)^(3/ 4)*A*c)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/ (b^2*c^5) + 3/64*sqrt(2)*(7*(b*c^3)^(3/4)*B*b + (b*c^3)^(3/4)*A*c)*arctan( -1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/(b^2*c^5) - 3/ 128*sqrt(2)*(7*(b*c^3)^(3/4)*B*b + (b*c^3)^(3/4)*A*c)*log(sqrt(2)*sqrt(x)* (b/c)^(1/4) + x + sqrt(b/c))/(b^2*c^5) + 3/128*sqrt(2)*(7*(b*c^3)^(3/4)*B* b + (b*c^3)^(3/4)*A*c)*log(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/( b^2*c^5)
Time = 0.18 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.42 \[ \int \frac {x^{17/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {3\,\mathrm {atanh}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )\,\left (A\,c+7\,B\,b\right )}{32\,{\left (-b\right )}^{5/4}\,c^{11/4}}-\frac {3\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )\,\left (A\,c+7\,B\,b\right )}{32\,{\left (-b\right )}^{5/4}\,c^{11/4}}-\frac {\frac {x^{3/2}\,\left (A\,c+7\,B\,b\right )}{16\,c^2}-\frac {x^{7/2}\,\left (3\,A\,c-11\,B\,b\right )}{16\,b\,c}}{b^2+2\,b\,c\,x^2+c^2\,x^4} \]