Integrand size = 26, antiderivative size = 322 \[ \int \frac {x^{21/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {7 (11 b B-3 A c) x^{3/2}}{48 b c^3}-\frac {(b B-A c) x^{11/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(11 b B-3 A c) x^{7/2}}{16 b c^2 \left (b+c x^2\right )}+\frac {7 (11 b B-3 A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} \sqrt [4]{b} c^{15/4}}-\frac {7 (11 b B-3 A c) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} \sqrt [4]{b} c^{15/4}}-\frac {7 (11 b B-3 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} \sqrt [4]{b} c^{15/4}}+\frac {7 (11 b B-3 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} \sqrt [4]{b} c^{15/4}} \]
7/48*(-3*A*c+11*B*b)*x^(3/2)/b/c^3-1/4*(-A*c+B*b)*x^(11/2)/b/c/(c*x^2+b)^2 -1/16*(-3*A*c+11*B*b)*x^(7/2)/b/c^2/(c*x^2+b)+7/64*(-3*A*c+11*B*b)*arctan( 1-c^(1/4)*2^(1/2)*x^(1/2)/b^(1/4))/b^(1/4)/c^(15/4)*2^(1/2)-7/64*(-3*A*c+1 1*B*b)*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/b^(1/4))/b^(1/4)/c^(15/4)*2^(1/2)- 7/128*(-3*A*c+11*B*b)*ln(b^(1/2)+x*c^(1/2)-b^(1/4)*c^(1/4)*2^(1/2)*x^(1/2) )/b^(1/4)/c^(15/4)*2^(1/2)+7/128*(-3*A*c+11*B*b)*ln(b^(1/2)+x*c^(1/2)+b^(1 /4)*c^(1/4)*2^(1/2)*x^(1/2))/b^(1/4)/c^(15/4)*2^(1/2)
Time = 0.94 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.57 \[ \int \frac {x^{21/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {4 c^{3/4} x^{3/2} \left (77 b^2 B-21 A b c+121 b B c x^2-33 A c^2 x^2+32 B c^2 x^4\right )}{\left (b+c x^2\right )^2}+\frac {21 \sqrt {2} (11 b B-3 A c) \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt [4]{b}}+\frac {21 \sqrt {2} (11 b B-3 A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{\sqrt [4]{b}}}{192 c^{15/4}} \]
((4*c^(3/4)*x^(3/2)*(77*b^2*B - 21*A*b*c + 121*b*B*c*x^2 - 33*A*c^2*x^2 + 32*B*c^2*x^4))/(b + c*x^2)^2 + (21*Sqrt[2]*(11*b*B - 3*A*c)*ArcTan[(Sqrt[b ] - Sqrt[c]*x)/(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])])/b^(1/4) + (21*Sqrt[2]*( 11*b*B - 3*A*c)*ArcTanh[(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[b] + Sqrt[ c]*x)])/b^(1/4))/(192*c^(15/4))
Time = 0.51 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.97, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {9, 362, 252, 262, 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.
| \(\displaystyle \int \frac {x^{21/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \int \frac {x^{9/2} \left (A+B x^2\right )}{\left (b+c x^2\right )^3}dx\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {(11 b B-3 A c) \int \frac {x^{9/2}}{\left (c x^2+b\right )^2}dx}{8 b c}-\frac {x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {(11 b B-3 A c) \left (\frac {7 \int \frac {x^{5/2}}{c x^2+b}dx}{4 c}-\frac {x^{7/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {(11 b B-3 A c) \left (\frac {7 \left (\frac {2 x^{3/2}}{3 c}-\frac {b \int \frac {\sqrt {x}}{c x^2+b}dx}{c}\right )}{4 c}-\frac {x^{7/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {(11 b B-3 A c) \left (\frac {7 \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \int \frac {x}{c x^2+b}d\sqrt {x}}{c}\right )}{4 c}-\frac {x^{7/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {(11 b B-3 A c) \left (\frac {7 \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\int \frac {\sqrt {c} x+\sqrt {b}}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{c}\right )}{4 c}-\frac {x^{7/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {(11 b B-3 A c) \left (\frac {7 \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{c}\right )}{4 c}-\frac {x^{7/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(11 b B-3 A c) \left (\frac {7 \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{c}\right )}{4 c}-\frac {x^{7/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(11 b B-3 A c) \left (\frac {7 \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{c}\right )}{4 c}-\frac {x^{7/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {(11 b B-3 A c) \left (\frac {7 \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{c}\right )}{4 c}-\frac {x^{7/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(11 b B-3 A c) \left (\frac {7 \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{c}\right )}{4 c}-\frac {x^{7/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(11 b B-3 A c) \left (\frac {7 \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {c}}\right )}{c}\right )}{4 c}-\frac {x^{7/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(11 b B-3 A c) \left (\frac {7 \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{c}\right )}{4 c}-\frac {x^{7/2}}{2 c \left (b+c x^2\right )}\right )}{8 b c}-\frac {x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2}\) |
-1/4*((b*B - A*c)*x^(11/2))/(b*c*(b + c*x^2)^2) + ((11*b*B - 3*A*c)*(-1/2* x^(7/2)/(c*(b + c*x^2)) + (7*((2*x^(3/2))/(3*c) - (2*b*((-(ArcTan[1 - (Sqr t[2]*c^(1/4)*Sqrt[x])/b^(1/4)]/(Sqrt[2]*b^(1/4)*c^(1/4))) + ArcTan[1 + (Sq rt[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])))/c))/(4*c)))/(8*b*c)
3.3.8.3.1 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] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[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 = 173, normalized size of antiderivative = 0.54
| method | result | size |
| derivativedivides | \(\frac {2 B \,x^{\frac {3}{2}}}{3 c^{3}}+\frac {\frac {2 \left (-\frac {c \left (11 A c -19 B b \right ) x^{\frac {7}{2}}}{32}+\left (-\frac {7}{32} A b c +\frac {15}{32} B \,b^{2}\right ) x^{\frac {3}{2}}\right )}{\left (c \,x^{2}+b \right )^{2}}+\frac {\left (\frac {21 A c}{32}-\frac {77 B b}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}}{c^{3}}\) | \(173\) |
| default | \(\frac {2 B \,x^{\frac {3}{2}}}{3 c^{3}}+\frac {\frac {2 \left (-\frac {c \left (11 A c -19 B b \right ) x^{\frac {7}{2}}}{32}+\left (-\frac {7}{32} A b c +\frac {15}{32} B \,b^{2}\right ) x^{\frac {3}{2}}\right )}{\left (c \,x^{2}+b \right )^{2}}+\frac {\left (\frac {21 A c}{32}-\frac {77 B b}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}}{c^{3}}\) | \(173\) |
| risch | \(\frac {2 B \,x^{\frac {3}{2}}}{3 c^{3}}+\frac {\frac {-\frac {c \left (11 A c -19 B b \right ) x^{\frac {7}{2}}}{16}+2 \left (-\frac {7}{32} A b c +\frac {15}{32} B \,b^{2}\right ) x^{\frac {3}{2}}}{\left (c \,x^{2}+b \right )^{2}}+\frac {\left (\frac {21 A c}{32}-\frac {77 B b}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}}{c^{3}}\) | \(173\) |
2/3*B/c^3*x^(3/2)+2/c^3*((-1/32*c*(11*A*c-19*B*b)*x^(7/2)+(-7/32*A*b*c+15/ 32*B*b^2)*x^(3/2))/(c*x^2+b)^2+1/8*(21/32*A*c-77/32*B*b)/c/(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 = 0.49 (sec) , antiderivative size = 872, normalized size of antiderivative = 2.71 \[ \int \frac {x^{21/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {21 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )} \left (-\frac {14641 \, B^{4} b^{4} - 15972 \, A B^{3} b^{3} c + 6534 \, A^{2} B^{2} b^{2} c^{2} - 1188 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b c^{15}}\right )^{\frac {1}{4}} \log \left (343 \, b c^{11} \left (-\frac {14641 \, B^{4} b^{4} - 15972 \, A B^{3} b^{3} c + 6534 \, A^{2} B^{2} b^{2} c^{2} - 1188 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b c^{15}}\right )^{\frac {3}{4}} - 343 \, {\left (1331 \, B^{3} b^{3} - 1089 \, A B^{2} b^{2} c + 297 \, A^{2} B b c^{2} - 27 \, A^{3} c^{3}\right )} \sqrt {x}\right ) - 21 \, {\left (i \, c^{5} x^{4} + 2 i \, b c^{4} x^{2} + i \, b^{2} c^{3}\right )} \left (-\frac {14641 \, B^{4} b^{4} - 15972 \, A B^{3} b^{3} c + 6534 \, A^{2} B^{2} b^{2} c^{2} - 1188 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b c^{15}}\right )^{\frac {1}{4}} \log \left (343 i \, b c^{11} \left (-\frac {14641 \, B^{4} b^{4} - 15972 \, A B^{3} b^{3} c + 6534 \, A^{2} B^{2} b^{2} c^{2} - 1188 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b c^{15}}\right )^{\frac {3}{4}} - 343 \, {\left (1331 \, B^{3} b^{3} - 1089 \, A B^{2} b^{2} c + 297 \, A^{2} B b c^{2} - 27 \, A^{3} c^{3}\right )} \sqrt {x}\right ) - 21 \, {\left (-i \, c^{5} x^{4} - 2 i \, b c^{4} x^{2} - i \, b^{2} c^{3}\right )} \left (-\frac {14641 \, B^{4} b^{4} - 15972 \, A B^{3} b^{3} c + 6534 \, A^{2} B^{2} b^{2} c^{2} - 1188 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b c^{15}}\right )^{\frac {1}{4}} \log \left (-343 i \, b c^{11} \left (-\frac {14641 \, B^{4} b^{4} - 15972 \, A B^{3} b^{3} c + 6534 \, A^{2} B^{2} b^{2} c^{2} - 1188 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b c^{15}}\right )^{\frac {3}{4}} - 343 \, {\left (1331 \, B^{3} b^{3} - 1089 \, A B^{2} b^{2} c + 297 \, A^{2} B b c^{2} - 27 \, A^{3} c^{3}\right )} \sqrt {x}\right ) - 21 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )} \left (-\frac {14641 \, B^{4} b^{4} - 15972 \, A B^{3} b^{3} c + 6534 \, A^{2} B^{2} b^{2} c^{2} - 1188 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b c^{15}}\right )^{\frac {1}{4}} \log \left (-343 \, b c^{11} \left (-\frac {14641 \, B^{4} b^{4} - 15972 \, A B^{3} b^{3} c + 6534 \, A^{2} B^{2} b^{2} c^{2} - 1188 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b c^{15}}\right )^{\frac {3}{4}} - 343 \, {\left (1331 \, B^{3} b^{3} - 1089 \, A B^{2} b^{2} c + 297 \, A^{2} B b c^{2} - 27 \, A^{3} c^{3}\right )} \sqrt {x}\right ) + 4 \, {\left (32 \, B c^{2} x^{5} + 11 \, {\left (11 \, B b c - 3 \, A c^{2}\right )} x^{3} + 7 \, {\left (11 \, B b^{2} - 3 \, A b c\right )} x\right )} \sqrt {x}}{192 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} \]
1/192*(21*(c^5*x^4 + 2*b*c^4*x^2 + b^2*c^3)*(-(14641*B^4*b^4 - 15972*A*B^3 *b^3*c + 6534*A^2*B^2*b^2*c^2 - 1188*A^3*B*b*c^3 + 81*A^4*c^4)/(b*c^15))^( 1/4)*log(343*b*c^11*(-(14641*B^4*b^4 - 15972*A*B^3*b^3*c + 6534*A^2*B^2*b^ 2*c^2 - 1188*A^3*B*b*c^3 + 81*A^4*c^4)/(b*c^15))^(3/4) - 343*(1331*B^3*b^3 - 1089*A*B^2*b^2*c + 297*A^2*B*b*c^2 - 27*A^3*c^3)*sqrt(x)) - 21*(I*c^5*x ^4 + 2*I*b*c^4*x^2 + I*b^2*c^3)*(-(14641*B^4*b^4 - 15972*A*B^3*b^3*c + 653 4*A^2*B^2*b^2*c^2 - 1188*A^3*B*b*c^3 + 81*A^4*c^4)/(b*c^15))^(1/4)*log(343 *I*b*c^11*(-(14641*B^4*b^4 - 15972*A*B^3*b^3*c + 6534*A^2*B^2*b^2*c^2 - 11 88*A^3*B*b*c^3 + 81*A^4*c^4)/(b*c^15))^(3/4) - 343*(1331*B^3*b^3 - 1089*A* B^2*b^2*c + 297*A^2*B*b*c^2 - 27*A^3*c^3)*sqrt(x)) - 21*(-I*c^5*x^4 - 2*I* b*c^4*x^2 - I*b^2*c^3)*(-(14641*B^4*b^4 - 15972*A*B^3*b^3*c + 6534*A^2*B^2 *b^2*c^2 - 1188*A^3*B*b*c^3 + 81*A^4*c^4)/(b*c^15))^(1/4)*log(-343*I*b*c^1 1*(-(14641*B^4*b^4 - 15972*A*B^3*b^3*c + 6534*A^2*B^2*b^2*c^2 - 1188*A^3*B *b*c^3 + 81*A^4*c^4)/(b*c^15))^(3/4) - 343*(1331*B^3*b^3 - 1089*A*B^2*b^2* c + 297*A^2*B*b*c^2 - 27*A^3*c^3)*sqrt(x)) - 21*(c^5*x^4 + 2*b*c^4*x^2 + b ^2*c^3)*(-(14641*B^4*b^4 - 15972*A*B^3*b^3*c + 6534*A^2*B^2*b^2*c^2 - 1188 *A^3*B*b*c^3 + 81*A^4*c^4)/(b*c^15))^(1/4)*log(-343*b*c^11*(-(14641*B^4*b^ 4 - 15972*A*B^3*b^3*c + 6534*A^2*B^2*b^2*c^2 - 1188*A^3*B*b*c^3 + 81*A^4*c ^4)/(b*c^15))^(3/4) - 343*(1331*B^3*b^3 - 1089*A*B^2*b^2*c + 297*A^2*B*b*c ^2 - 27*A^3*c^3)*sqrt(x)) + 4*(32*B*c^2*x^5 + 11*(11*B*b*c - 3*A*c^2)*x...
Timed out. \[ \int \frac {x^{21/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.80 \[ \int \frac {x^{21/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {{\left (19 \, B b c - 11 \, A c^{2}\right )} x^{\frac {7}{2}} + {\left (15 \, B b^{2} - 7 \, A b c\right )} x^{\frac {3}{2}}}{16 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} + \frac {2 \, B x^{\frac {3}{2}}}{3 \, c^{3}} - \frac {7 \, {\left (11 \, B b - 3 \, A c\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{128 \, c^{3}} \]
1/16*((19*B*b*c - 11*A*c^2)*x^(7/2) + (15*B*b^2 - 7*A*b*c)*x^(3/2))/(c^5*x ^4 + 2*b*c^4*x^2 + b^2*c^3) + 2/3*B*x^(3/2)/c^3 - 7/128*(11*B*b - 3*A*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)) + 2*sqrt(2)*arctan (-1/2*sqrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) - 2*sqrt(c)*sqrt(x))/sqrt(sqrt(b)*s qrt(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)*sqrt(x) + sqrt(c)*x + sqrt(b))/(b^(1/4)*c^(3/4)))/c^3
Time = 0.31 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.94 \[ \int \frac {x^{21/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {2 \, B x^{\frac {3}{2}}}{3 \, c^{3}} + \frac {19 \, B b c x^{\frac {7}{2}} - 11 \, A c^{2} x^{\frac {7}{2}} + 15 \, B b^{2} x^{\frac {3}{2}} - 7 \, A b c x^{\frac {3}{2}}}{16 \, {\left (c x^{2} + b\right )}^{2} c^{3}} - \frac {7 \, \sqrt {2} {\left (11 \, \left (b c^{3}\right )^{\frac {3}{4}} B b - 3 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b c^{6}} - \frac {7 \, \sqrt {2} {\left (11 \, \left (b c^{3}\right )^{\frac {3}{4}} B b - 3 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b c^{6}} + \frac {7 \, \sqrt {2} {\left (11 \, \left (b c^{3}\right )^{\frac {3}{4}} B b - 3 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b c^{6}} - \frac {7 \, \sqrt {2} {\left (11 \, \left (b c^{3}\right )^{\frac {3}{4}} B b - 3 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b c^{6}} \]
2/3*B*x^(3/2)/c^3 + 1/16*(19*B*b*c*x^(7/2) - 11*A*c^2*x^(7/2) + 15*B*b^2*x ^(3/2) - 7*A*b*c*x^(3/2))/((c*x^2 + b)^2*c^3) - 7/64*sqrt(2)*(11*(b*c^3)^( 3/4)*B*b - 3*(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*c^6) - 7/64*sqrt(2)*(11*(b*c^3)^(3/4)*B*b - 3*( 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*c^6) + 7/128*sqrt(2)*(11*(b*c^3)^(3/4)*B*b - 3*(b*c^3)^(3/4) *A*c)*log(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b*c^6) - 7/128*sqr t(2)*(11*(b*c^3)^(3/4)*B*b - 3*(b*c^3)^(3/4)*A*c)*log(-sqrt(2)*sqrt(x)*(b/ c)^(1/4) + x + sqrt(b/c))/(b*c^6)
Time = 9.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.43 \[ \int \frac {x^{21/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {x^{3/2}\,\left (\frac {15\,B\,b^2}{16}-\frac {7\,A\,b\,c}{16}\right )-x^{7/2}\,\left (\frac {11\,A\,c^2}{16}-\frac {19\,B\,b\,c}{16}\right )}{b^2\,c^3+2\,b\,c^4\,x^2+c^5\,x^4}+\frac {2\,B\,x^{3/2}}{3\,c^3}+\frac {7\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )\,\left (3\,A\,c-11\,B\,b\right )}{32\,{\left (-b\right )}^{1/4}\,c^{15/4}}+\frac {\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-b\right )}^{1/4}}\right )\,\left (3\,A\,c-11\,B\,b\right )\,7{}\mathrm {i}}{32\,{\left (-b\right )}^{1/4}\,c^{15/4}} \]
(x^(3/2)*((15*B*b^2)/16 - (7*A*b*c)/16) - x^(7/2)*((11*A*c^2)/16 - (19*B*b *c)/16))/(b^2*c^3 + c^5*x^4 + 2*b*c^4*x^2) + (2*B*x^(3/2))/(3*c^3) + (7*at an((c^(1/4)*x^(1/2))/(-b)^(1/4))*(3*A*c - 11*B*b))/(32*(-b)^(1/4)*c^(15/4) ) + (atan((c^(1/4)*x^(1/2)*1i)/(-b)^(1/4))*(3*A*c - 11*B*b)*7i)/(32*(-b)^( 1/4)*c^(15/4))