Integrand size = 26, antiderivative size = 316 \[ \int \frac {x^{9/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {5 (b B-9 A c)}{16 b^3 c \sqrt {x}}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}-\frac {b B-9 A c}{16 b^2 c \sqrt {x} \left (b+c x^2\right )}-\frac {5 (b B-9 A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{13/4} c^{3/4}}+\frac {5 (b B-9 A c) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{13/4} c^{3/4}}+\frac {5 (b B-9 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{13/4} c^{3/4}}-\frac {5 (b B-9 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{13/4} c^{3/4}} \]
-5/64*(-9*A*c+B*b)*arctan(1-c^(1/4)*2^(1/2)*x^(1/2)/b^(1/4))/b^(13/4)/c^(3 /4)*2^(1/2)+5/64*(-9*A*c+B*b)*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/b^(1/4))/b^ (13/4)/c^(3/4)*2^(1/2)+5/128*(-9*A*c+B*b)*ln(b^(1/2)+x*c^(1/2)-b^(1/4)*c^( 1/4)*2^(1/2)*x^(1/2))/b^(13/4)/c^(3/4)*2^(1/2)-5/128*(-9*A*c+B*b)*ln(b^(1/ 2)+x*c^(1/2)+b^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))/b^(13/4)/c^(3/4)*2^(1/2)+5/1 6*(-9*A*c+B*b)/b^3/c/x^(1/2)+1/4*(A*c-B*b)/b/c/(c*x^2+b)^2/x^(1/2)+1/16*(9 *A*c-B*b)/b^2/c/(c*x^2+b)/x^(1/2)
Time = 1.00 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.59 \[ \int \frac {x^{9/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{b} \left (-b B x^2 \left (9 b+5 c x^2\right )+A \left (32 b^2+81 b c x^2+45 c^2 x^4\right )\right )}{\sqrt {x} \left (b+c x^2\right )^2}+\frac {5 \sqrt {2} (-b B+9 A c) \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{c^{3/4}}+\frac {5 \sqrt {2} (-b B+9 A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{c^{3/4}}}{64 b^{13/4}} \]
((-4*b^(1/4)*(-(b*B*x^2*(9*b + 5*c*x^2)) + A*(32*b^2 + 81*b*c*x^2 + 45*c^2 *x^4)))/(Sqrt[x]*(b + c*x^2)^2) + (5*Sqrt[2]*(-(b*B) + 9*A*c)*ArcTan[(Sqrt [b] - Sqrt[c]*x)/(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])])/c^(3/4) + (5*Sqrt[2]* (-(b*B) + 9*A*c)*ArcTanh[(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[b] + Sqrt [c]*x)])/c^(3/4))/(64*b^(13/4))
Time = 0.51 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.98, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {9, 362, 253, 264, 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.
| \(\displaystyle \int \frac {x^{9/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \int \frac {A+B x^2}{x^{3/2} \left (b+c x^2\right )^3}dx\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle -\frac {(b B-9 A c) \int \frac {1}{x^{3/2} \left (c x^2+b\right )^2}dx}{8 b c}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle -\frac {(b B-9 A c) \left (\frac {5 \int \frac {1}{x^{3/2} \left (c x^2+b\right )}dx}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -\frac {(b B-9 A c) \left (\frac {5 \left (-\frac {c \int \frac {\sqrt {x}}{c x^2+b}dx}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {(b B-9 A c) \left (\frac {5 \left (-\frac {2 c \int \frac {x}{c x^2+b}d\sqrt {x}}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {(b B-9 A c) \left (\frac {5 \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {c} x+\sqrt {b}}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {(b B-9 A c) \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {(b B-9 A c) \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {(b B-9 A c) \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {(b B-9 A c) \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {(b B-9 A c) \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {(b B-9 A c) \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {(b B-9 A c) \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}\) |
-1/4*(b*B - A*c)/(b*c*Sqrt[x]*(b + c*x^2)^2) - ((b*B - 9*A*c)*(1/(2*b*Sqrt [x]*(b + c*x^2)) + (5*(-2/(b*Sqrt[x]) - (2*c*((-(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[S qrt[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])))/b))/(4*b)))/(8*b*c)
3.3.14.3.1 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*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && 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 = 0.55
| method | result | size |
| derivativedivides | \(-\frac {2 \left (\frac {\left (\frac {13}{32} A \,c^{2}-\frac {5}{32} B b c \right ) x^{\frac {7}{2}}+\frac {b \left (17 A c -9 B b \right ) x^{\frac {3}{2}}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {\left (\frac {45 A c}{32}-\frac {5 B b}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{3}}-\frac {2 A}{b^{3} \sqrt {x}}\) | \(173\) |
| default | \(-\frac {2 \left (\frac {\left (\frac {13}{32} A \,c^{2}-\frac {5}{32} B b c \right ) x^{\frac {7}{2}}+\frac {b \left (17 A c -9 B b \right ) x^{\frac {3}{2}}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {\left (\frac {45 A c}{32}-\frac {5 B b}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{3}}-\frac {2 A}{b^{3} \sqrt {x}}\) | \(173\) |
| risch | \(-\frac {2 A}{b^{3} \sqrt {x}}-\frac {\frac {2 \left (\frac {13}{32} A \,c^{2}-\frac {5}{32} B b c \right ) x^{\frac {7}{2}}+\frac {b \left (17 A c -9 B b \right ) x^{\frac {3}{2}}}{16}}{\left (c \,x^{2}+b \right )^{2}}+\frac {\left (\frac {45 A c}{32}-\frac {5 B b}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}}{b^{3}}\) | \(174\) |
-2/b^3*(((13/32*A*c^2-5/32*B*b*c)*x^(7/2)+1/32*b*(17*A*c-9*B*b)*x^(3/2))/( c*x^2+b)^2+1/8*(45/32*A*c-5/32*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)))-2*A/b^3/x^(1/2)
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 870, normalized size of antiderivative = 2.75 \[ \int \frac {x^{9/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {5 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {1}{4}} \log \left (125 \, b^{10} c^{2} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {3}{4}} - 125 \, {\left (B^{3} b^{3} - 27 \, A B^{2} b^{2} c + 243 \, A^{2} B b c^{2} - 729 \, A^{3} c^{3}\right )} \sqrt {x}\right ) + 5 \, {\left (-i \, b^{3} c^{2} x^{5} - 2 i \, b^{4} c x^{3} - i \, b^{5} x\right )} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {1}{4}} \log \left (125 i \, b^{10} c^{2} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {3}{4}} - 125 \, {\left (B^{3} b^{3} - 27 \, A B^{2} b^{2} c + 243 \, A^{2} B b c^{2} - 729 \, A^{3} c^{3}\right )} \sqrt {x}\right ) + 5 \, {\left (i \, b^{3} c^{2} x^{5} + 2 i \, b^{4} c x^{3} + i \, b^{5} x\right )} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {1}{4}} \log \left (-125 i \, b^{10} c^{2} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {3}{4}} - 125 \, {\left (B^{3} b^{3} - 27 \, A B^{2} b^{2} c + 243 \, A^{2} B b c^{2} - 729 \, A^{3} c^{3}\right )} \sqrt {x}\right ) - 5 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {1}{4}} \log \left (-125 \, b^{10} c^{2} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {3}{4}} - 125 \, {\left (B^{3} b^{3} - 27 \, A B^{2} b^{2} c + 243 \, A^{2} B b c^{2} - 729 \, A^{3} c^{3}\right )} \sqrt {x}\right ) - 4 \, {\left (5 \, {\left (B b c - 9 \, A c^{2}\right )} x^{4} - 32 \, A b^{2} + 9 \, {\left (B b^{2} - 9 \, A b c\right )} x^{2}\right )} \sqrt {x}}{64 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )}} \]
-1/64*(5*(b^3*c^2*x^5 + 2*b^4*c*x^3 + b^5*x)*(-(B^4*b^4 - 36*A*B^3*b^3*c + 486*A^2*B^2*b^2*c^2 - 2916*A^3*B*b*c^3 + 6561*A^4*c^4)/(b^13*c^3))^(1/4)* log(125*b^10*c^2*(-(B^4*b^4 - 36*A*B^3*b^3*c + 486*A^2*B^2*b^2*c^2 - 2916* A^3*B*b*c^3 + 6561*A^4*c^4)/(b^13*c^3))^(3/4) - 125*(B^3*b^3 - 27*A*B^2*b^ 2*c + 243*A^2*B*b*c^2 - 729*A^3*c^3)*sqrt(x)) + 5*(-I*b^3*c^2*x^5 - 2*I*b^ 4*c*x^3 - I*b^5*x)*(-(B^4*b^4 - 36*A*B^3*b^3*c + 486*A^2*B^2*b^2*c^2 - 291 6*A^3*B*b*c^3 + 6561*A^4*c^4)/(b^13*c^3))^(1/4)*log(125*I*b^10*c^2*(-(B^4* b^4 - 36*A*B^3*b^3*c + 486*A^2*B^2*b^2*c^2 - 2916*A^3*B*b*c^3 + 6561*A^4*c ^4)/(b^13*c^3))^(3/4) - 125*(B^3*b^3 - 27*A*B^2*b^2*c + 243*A^2*B*b*c^2 - 729*A^3*c^3)*sqrt(x)) + 5*(I*b^3*c^2*x^5 + 2*I*b^4*c*x^3 + I*b^5*x)*(-(B^4 *b^4 - 36*A*B^3*b^3*c + 486*A^2*B^2*b^2*c^2 - 2916*A^3*B*b*c^3 + 6561*A^4* c^4)/(b^13*c^3))^(1/4)*log(-125*I*b^10*c^2*(-(B^4*b^4 - 36*A*B^3*b^3*c + 4 86*A^2*B^2*b^2*c^2 - 2916*A^3*B*b*c^3 + 6561*A^4*c^4)/(b^13*c^3))^(3/4) - 125*(B^3*b^3 - 27*A*B^2*b^2*c + 243*A^2*B*b*c^2 - 729*A^3*c^3)*sqrt(x)) - 5*(b^3*c^2*x^5 + 2*b^4*c*x^3 + b^5*x)*(-(B^4*b^4 - 36*A*B^3*b^3*c + 486*A^ 2*B^2*b^2*c^2 - 2916*A^3*B*b*c^3 + 6561*A^4*c^4)/(b^13*c^3))^(1/4)*log(-12 5*b^10*c^2*(-(B^4*b^4 - 36*A*B^3*b^3*c + 486*A^2*B^2*b^2*c^2 - 2916*A^3*B* b*c^3 + 6561*A^4*c^4)/(b^13*c^3))^(3/4) - 125*(B^3*b^3 - 27*A*B^2*b^2*c + 243*A^2*B*b*c^2 - 729*A^3*c^3)*sqrt(x)) - 4*(5*(B*b*c - 9*A*c^2)*x^4 - 32* A*b^2 + 9*(B*b^2 - 9*A*b*c)*x^2)*sqrt(x))/(b^3*c^2*x^5 + 2*b^4*c*x^3 + ...
Timed out. \[ \int \frac {x^{9/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.81 \[ \int \frac {x^{9/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {5 \, {\left (B b c - 9 \, A c^{2}\right )} x^{4} - 32 \, A b^{2} + 9 \, {\left (B b^{2} - 9 \, A b c\right )} x^{2}}{16 \, {\left (b^{3} c^{2} x^{\frac {9}{2}} + 2 \, b^{4} c x^{\frac {5}{2}} + b^{5} \sqrt {x}\right )}} + \frac {5 \, {\left (B b - 9 \, A c\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{128 \, b^{3}} \]
1/16*(5*(B*b*c - 9*A*c^2)*x^4 - 32*A*b^2 + 9*(B*b^2 - 9*A*b*c)*x^2)/(b^3*c ^2*x^(9/2) + 2*b^4*c*x^(5/2) + b^5*sqrt(x)) + 5/128*(B*b - 9*A*c)*(2*sqrt( 2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) + 2*sqrt(c)*sqrt(x))/sqrt(s qrt(b)*sqrt(c)))/(sqrt(sqrt(b)*sqrt(c))*sqrt(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)) - sqrt(2)*log(sqrt(2)*b^(1/4)*c^(1/4)*sq rt(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)))/b^3
Time = 0.30 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.95 \[ \int \frac {x^{9/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {2 \, A}{b^{3} \sqrt {x}} + \frac {5 \, B b c x^{\frac {7}{2}} - 13 \, A c^{2} x^{\frac {7}{2}} + 9 \, B b^{2} x^{\frac {3}{2}} - 17 \, A b c x^{\frac {3}{2}}}{16 \, {\left (c x^{2} + b\right )}^{2} b^{3}} + \frac {5 \, \sqrt {2} {\left (\left (b c^{3}\right )^{\frac {3}{4}} B b - 9 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{4} c^{3}} + \frac {5 \, \sqrt {2} {\left (\left (b c^{3}\right )^{\frac {3}{4}} B b - 9 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{4} c^{3}} - \frac {5 \, \sqrt {2} {\left (\left (b c^{3}\right )^{\frac {3}{4}} B b - 9 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{4} c^{3}} + \frac {5 \, \sqrt {2} {\left (\left (b c^{3}\right )^{\frac {3}{4}} B b - 9 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{4} c^{3}} \]
-2*A/(b^3*sqrt(x)) + 1/16*(5*B*b*c*x^(7/2) - 13*A*c^2*x^(7/2) + 9*B*b^2*x^ (3/2) - 17*A*b*c*x^(3/2))/((c*x^2 + b)^2*b^3) + 5/64*sqrt(2)*((b*c^3)^(3/4 )*B*b - 9*(b*c^3)^(3/4)*A*c)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*s qrt(x))/(b/c)^(1/4))/(b^4*c^3) + 5/64*sqrt(2)*((b*c^3)^(3/4)*B*b - 9*(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^4*c^3) - 5/128*sqrt(2)*((b*c^3)^(3/4)*B*b - 9*(b*c^3)^(3/4)*A*c) *log(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^4*c^3) + 5/128*sqrt(2 )*((b*c^3)^(3/4)*B*b - 9*(b*c^3)^(3/4)*A*c)*log(-sqrt(2)*sqrt(x)*(b/c)^(1/ 4) + x + sqrt(b/c))/(b^4*c^3)
Time = 9.21 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.42 \[ \int \frac {x^{9/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {5\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )\,\left (9\,A\,c-B\,b\right )}{32\,{\left (-b\right )}^{13/4}\,c^{3/4}}-\frac {\frac {2\,A}{b}+\frac {9\,x^2\,\left (9\,A\,c-B\,b\right )}{16\,b^2}+\frac {5\,c\,x^4\,\left (9\,A\,c-B\,b\right )}{16\,b^3}}{b^2\,\sqrt {x}+c^2\,x^{9/2}+2\,b\,c\,x^{5/2}}-\frac {5\,\mathrm {atanh}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )\,\left (9\,A\,c-B\,b\right )}{32\,{\left (-b\right )}^{13/4}\,c^{3/4}} \]