Integrand size = 26, antiderivative size = 289 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx=\frac {3 b B-7 A c}{6 b^2 c x^{3/2}}-\frac {b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )}-\frac {(3 b B-7 A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {(3 b B-7 A c) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{11/4} \sqrt [4]{c}}-\frac {(3 b B-7 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{11/4} \sqrt [4]{c}}+\frac {(3 b B-7 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{11/4} \sqrt [4]{c}} \]
1/6*(-7*A*c+3*B*b)/b^2/c/x^(3/2)+1/2*(A*c-B*b)/b/c/x^(3/2)/(c*x^2+b)-1/8*( -7*A*c+3*B*b)*arctan(1-c^(1/4)*2^(1/2)*x^(1/2)/b^(1/4))/b^(11/4)/c^(1/4)*2 ^(1/2)+1/8*(-7*A*c+3*B*b)*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/b^(1/4))/b^(11/ 4)/c^(1/4)*2^(1/2)-1/16*(-7*A*c+3*B*b)*ln(b^(1/2)+x*c^(1/2)-b^(1/4)*c^(1/4 )*2^(1/2)*x^(1/2))/b^(11/4)/c^(1/4)*2^(1/2)+1/16*(-7*A*c+3*B*b)*ln(b^(1/2) +x*c^(1/2)+b^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))/b^(11/4)/c^(1/4)*2^(1/2)
Time = 0.82 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.57 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx=\frac {\frac {4 b^{3/4} \left (-4 A b+3 b B x^2-7 A c x^2\right )}{x^{3/2} \left (b+c x^2\right )}+\frac {3 \sqrt {2} (-3 b B+7 A c) \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt [4]{c}}+\frac {3 \sqrt {2} (3 b B-7 A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{\sqrt [4]{c}}}{24 b^{11/4}} \]
((4*b^(3/4)*(-4*A*b + 3*b*B*x^2 - 7*A*c*x^2))/(x^(3/2)*(b + c*x^2)) + (3*S qrt[2]*(-3*b*B + 7*A*c)*ArcTan[(Sqrt[b] - Sqrt[c]*x)/(Sqrt[2]*b^(1/4)*c^(1 /4)*Sqrt[x])])/c^(1/4) + (3*Sqrt[2]*(3*b*B - 7*A*c)*ArcTanh[(Sqrt[2]*b^(1/ 4)*c^(1/4)*Sqrt[x])/(Sqrt[b] + Sqrt[c]*x)])/c^(1/4))/(24*b^(11/4))
Time = 0.48 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {9, 362, 264, 266, 755, 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.
\(\displaystyle \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {A+B x^2}{x^{5/2} \left (b+c x^2\right )^2}dx\) |
\(\Big \downarrow \) 362 |
\(\displaystyle -\frac {(3 b B-7 A c) \int \frac {1}{x^{5/2} \left (c x^2+b\right )}dx}{4 b c}-\frac {b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle -\frac {(3 b B-7 A c) \left (-\frac {c \int \frac {1}{\sqrt {x} \left (c x^2+b\right )}dx}{b}-\frac {2}{3 b x^{3/2}}\right )}{4 b c}-\frac {b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle -\frac {(3 b B-7 A c) \left (-\frac {2 c \int \frac {1}{c x^2+b}d\sqrt {x}}{b}-\frac {2}{3 b x^{3/2}}\right )}{4 b c}-\frac {b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )}\) |
\(\Big \downarrow \) 755 |
\(\displaystyle -\frac {(3 b B-7 A c) \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {c} x+\sqrt {b}}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{4 b c}-\frac {b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle -\frac {(3 b B-7 A c) \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{4 b c}-\frac {b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle -\frac {(3 b B-7 A c) \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{4 b c}-\frac {b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {(3 b B-7 A c) \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{4 b c}-\frac {b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle -\frac {(3 b B-7 A c) \left (-\frac {2 c \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{4 b c}-\frac {b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {(3 b B-7 A c) \left (-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{4 b c}-\frac {b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {(3 b B-7 A c) \left (-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{4 b c}-\frac {b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle -\frac {(3 b B-7 A c) \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{4 b c}-\frac {b B-A c}{2 b c x^{3/2} \left (b+c x^2\right )}\) |
-1/2*(b*B - A*c)/(b*c*x^(3/2)*(b + c*x^2)) - ((3*b*B - 7*A*c)*(-2/(3*b*x^( 3/2)) - (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[b]) + (-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[b] )))/b))/(4*b*c)
3.3.3.3.1 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)/(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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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 = 1.81 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.53
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {\left (\frac {A c}{4}-\frac {B b}{4}\right ) \sqrt {x}}{c \,x^{2}+b}+\frac {\left (7 A c -3 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b}\right )}{b^{2}}-\frac {2 A}{3 b^{2} x^{\frac {3}{2}}}\) | \(153\) |
default | \(-\frac {2 \left (\frac {\left (\frac {A c}{4}-\frac {B b}{4}\right ) \sqrt {x}}{c \,x^{2}+b}+\frac {\left (7 A c -3 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b}\right )}{b^{2}}-\frac {2 A}{3 b^{2} x^{\frac {3}{2}}}\) | \(153\) |
risch | \(-\frac {2 A}{3 b^{2} x^{\frac {3}{2}}}-\frac {\frac {2 \left (\frac {A c}{4}-\frac {B b}{4}\right ) \sqrt {x}}{c \,x^{2}+b}+\frac {\left (7 A c -3 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b}}{b^{2}}\) | \(154\) |
-2/b^2*((1/4*A*c-1/4*B*b)*x^(1/2)/(c*x^2+b)+1/32*(7*A*c-3*B*b)*(1/c*b)^(1/ 4)/b*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)))-2/3*A/b^2/x^(3/2)
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 691, normalized size of antiderivative = 2.39 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx=-\frac {3 \, {\left (b^{2} c x^{4} + b^{3} x^{2}\right )} \left (-\frac {81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac {1}{4}} \log \left (b^{3} \left (-\frac {81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac {1}{4}} - {\left (3 \, B b - 7 \, A c\right )} \sqrt {x}\right ) + 3 \, {\left (i \, b^{2} c x^{4} + i \, b^{3} x^{2}\right )} \left (-\frac {81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac {1}{4}} \log \left (i \, b^{3} \left (-\frac {81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac {1}{4}} - {\left (3 \, B b - 7 \, A c\right )} \sqrt {x}\right ) + 3 \, {\left (-i \, b^{2} c x^{4} - i \, b^{3} x^{2}\right )} \left (-\frac {81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac {1}{4}} \log \left (-i \, b^{3} \left (-\frac {81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac {1}{4}} - {\left (3 \, B b - 7 \, A c\right )} \sqrt {x}\right ) - 3 \, {\left (b^{2} c x^{4} + b^{3} x^{2}\right )} \left (-\frac {81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac {1}{4}} \log \left (-b^{3} \left (-\frac {81 \, B^{4} b^{4} - 756 \, A B^{3} b^{3} c + 2646 \, A^{2} B^{2} b^{2} c^{2} - 4116 \, A^{3} B b c^{3} + 2401 \, A^{4} c^{4}}{b^{11} c}\right )^{\frac {1}{4}} - {\left (3 \, B b - 7 \, A c\right )} \sqrt {x}\right ) - 4 \, {\left ({\left (3 \, B b - 7 \, A c\right )} x^{2} - 4 \, A b\right )} \sqrt {x}}{24 \, {\left (b^{2} c x^{4} + b^{3} x^{2}\right )}} \]
-1/24*(3*(b^2*c*x^4 + b^3*x^2)*(-(81*B^4*b^4 - 756*A*B^3*b^3*c + 2646*A^2* B^2*b^2*c^2 - 4116*A^3*B*b*c^3 + 2401*A^4*c^4)/(b^11*c))^(1/4)*log(b^3*(-( 81*B^4*b^4 - 756*A*B^3*b^3*c + 2646*A^2*B^2*b^2*c^2 - 4116*A^3*B*b*c^3 + 2 401*A^4*c^4)/(b^11*c))^(1/4) - (3*B*b - 7*A*c)*sqrt(x)) + 3*(I*b^2*c*x^4 + I*b^3*x^2)*(-(81*B^4*b^4 - 756*A*B^3*b^3*c + 2646*A^2*B^2*b^2*c^2 - 4116* A^3*B*b*c^3 + 2401*A^4*c^4)/(b^11*c))^(1/4)*log(I*b^3*(-(81*B^4*b^4 - 756* A*B^3*b^3*c + 2646*A^2*B^2*b^2*c^2 - 4116*A^3*B*b*c^3 + 2401*A^4*c^4)/(b^1 1*c))^(1/4) - (3*B*b - 7*A*c)*sqrt(x)) + 3*(-I*b^2*c*x^4 - I*b^3*x^2)*(-(8 1*B^4*b^4 - 756*A*B^3*b^3*c + 2646*A^2*B^2*b^2*c^2 - 4116*A^3*B*b*c^3 + 24 01*A^4*c^4)/(b^11*c))^(1/4)*log(-I*b^3*(-(81*B^4*b^4 - 756*A*B^3*b^3*c + 2 646*A^2*B^2*b^2*c^2 - 4116*A^3*B*b*c^3 + 2401*A^4*c^4)/(b^11*c))^(1/4) - ( 3*B*b - 7*A*c)*sqrt(x)) - 3*(b^2*c*x^4 + b^3*x^2)*(-(81*B^4*b^4 - 756*A*B^ 3*b^3*c + 2646*A^2*B^2*b^2*c^2 - 4116*A^3*B*b*c^3 + 2401*A^4*c^4)/(b^11*c) )^(1/4)*log(-b^3*(-(81*B^4*b^4 - 756*A*B^3*b^3*c + 2646*A^2*B^2*b^2*c^2 - 4116*A^3*B*b*c^3 + 2401*A^4*c^4)/(b^11*c))^(1/4) - (3*B*b - 7*A*c)*sqrt(x) ) - 4*((3*B*b - 7*A*c)*x^2 - 4*A*b)*sqrt(x))/(b^2*c*x^4 + b^3*x^2)
Timed out. \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.87 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx=\frac {{\left (3 \, B b - 7 \, A c\right )} x^{2} - 4 \, A b}{6 \, {\left (b^{2} c x^{\frac {7}{2}} + b^{3} x^{\frac {3}{2}}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, B b - 7 \, A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} {\left (3 \, B b - 7 \, A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} {\left (3 \, B b - 7 \, A c\right )} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (3 \, B b - 7 \, A c\right )} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}}}{16 \, b^{2}} \]
1/6*((3*B*b - 7*A*c)*x^2 - 4*A*b)/(b^2*c*x^(7/2) + b^3*x^(3/2)) + 1/16*(2* sqrt(2)*(3*B*b - 7*A*c)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) + 2*sq rt(c)*sqrt(x))/sqrt(sqrt(b)*sqrt(c)))/(sqrt(b)*sqrt(sqrt(b)*sqrt(c))) + 2* sqrt(2)*(3*B*b - 7*A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) - 2*s qrt(c)*sqrt(x))/sqrt(sqrt(b)*sqrt(c)))/(sqrt(b)*sqrt(sqrt(b)*sqrt(c))) + s qrt(2)*(3*B*b - 7*A*c)*log(sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + s qrt(b))/(b^(3/4)*c^(1/4)) - sqrt(2)*(3*B*b - 7*A*c)*log(-sqrt(2)*b^(1/4)*c ^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(b))/(b^(3/4)*c^(1/4)))/b^2
Time = 0.36 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.98 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx=\frac {\sqrt {2} {\left (3 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 7 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{8 \, b^{3} c} + \frac {\sqrt {2} {\left (3 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 7 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{8 \, b^{3} c} + \frac {\sqrt {2} {\left (3 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 7 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{16 \, b^{3} c} - \frac {\sqrt {2} {\left (3 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 7 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{16 \, b^{3} c} + \frac {B b \sqrt {x} - A c \sqrt {x}}{2 \, {\left (c x^{2} + b\right )} b^{2}} - \frac {2 \, A}{3 \, b^{2} x^{\frac {3}{2}}} \]
1/8*sqrt(2)*(3*(b*c^3)^(1/4)*B*b - 7*(b*c^3)^(1/4)*A*c)*arctan(1/2*sqrt(2) *(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/(b^3*c) + 1/8*sqrt(2)*(3*( b*c^3)^(1/4)*B*b - 7*(b*c^3)^(1/4)*A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c) ^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/(b^3*c) + 1/16*sqrt(2)*(3*(b*c^3)^(1/4)*B *b - 7*(b*c^3)^(1/4)*A*c)*log(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c)) /(b^3*c) - 1/16*sqrt(2)*(3*(b*c^3)^(1/4)*B*b - 7*(b*c^3)^(1/4)*A*c)*log(-s qrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^3*c) + 1/2*(B*b*sqrt(x) - A *c*sqrt(x))/((c*x^2 + b)*b^2) - 2/3*A/(b^2*x^(3/2))
Time = 9.33 (sec) , antiderivative size = 859, normalized size of antiderivative = 2.97 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx=-\frac {\frac {2\,A}{3\,b}+\frac {x^2\,\left (7\,A\,c-3\,B\,b\right )}{6\,b^2}}{b\,x^{3/2}+c\,x^{7/2}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,b^6\,c^5-1344\,A\,B\,b^7\,c^4+288\,B^2\,b^8\,c^3\right )-\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (1792\,A\,b^9\,c^4-768\,B\,b^{10}\,c^3\right )}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}+\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,b^6\,c^5-1344\,A\,B\,b^7\,c^4+288\,B^2\,b^8\,c^3\right )+\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (1792\,A\,b^9\,c^4-768\,B\,b^{10}\,c^3\right )}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}}{\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,b^6\,c^5-1344\,A\,B\,b^7\,c^4+288\,B^2\,b^8\,c^3\right )-\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (1792\,A\,b^9\,c^4-768\,B\,b^{10}\,c^3\right )}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}\right )}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}-\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,b^6\,c^5-1344\,A\,B\,b^7\,c^4+288\,B^2\,b^8\,c^3\right )+\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (1792\,A\,b^9\,c^4-768\,B\,b^{10}\,c^3\right )}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}\right )}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}}\right )\,\left (7\,A\,c-3\,B\,b\right )\,1{}\mathrm {i}}{4\,{\left (-b\right )}^{11/4}\,c^{1/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,b^6\,c^5-1344\,A\,B\,b^7\,c^4+288\,B^2\,b^8\,c^3\right )-\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (1792\,A\,b^9\,c^4-768\,B\,b^{10}\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}\right )}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}+\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,b^6\,c^5-1344\,A\,B\,b^7\,c^4+288\,B^2\,b^8\,c^3\right )+\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (1792\,A\,b^9\,c^4-768\,B\,b^{10}\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}\right )}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}}{\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,b^6\,c^5-1344\,A\,B\,b^7\,c^4+288\,B^2\,b^8\,c^3\right )-\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (1792\,A\,b^9\,c^4-768\,B\,b^{10}\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}-\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (\sqrt {x}\,\left (1568\,A^2\,b^6\,c^5-1344\,A\,B\,b^7\,c^4+288\,B^2\,b^8\,c^3\right )+\frac {\left (7\,A\,c-3\,B\,b\right )\,\left (1792\,A\,b^9\,c^4-768\,B\,b^{10}\,c^3\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{11/4}\,c^{1/4}}}\right )\,\left (7\,A\,c-3\,B\,b\right )}{4\,{\left (-b\right )}^{11/4}\,c^{1/4}} \]
- ((2*A)/(3*b) + (x^2*(7*A*c - 3*B*b))/(6*b^2))/(b*x^(3/2) + c*x^(7/2)) - (atan((((7*A*c - 3*B*b)*(x^(1/2)*(1568*A^2*b^6*c^5 + 288*B^2*b^8*c^3 - 134 4*A*B*b^7*c^4) - ((7*A*c - 3*B*b)*(1792*A*b^9*c^4 - 768*B*b^10*c^3))/(8*(- b)^(11/4)*c^(1/4)))*1i)/(8*(-b)^(11/4)*c^(1/4)) + ((7*A*c - 3*B*b)*(x^(1/2 )*(1568*A^2*b^6*c^5 + 288*B^2*b^8*c^3 - 1344*A*B*b^7*c^4) + ((7*A*c - 3*B* b)*(1792*A*b^9*c^4 - 768*B*b^10*c^3))/(8*(-b)^(11/4)*c^(1/4)))*1i)/(8*(-b) ^(11/4)*c^(1/4)))/(((7*A*c - 3*B*b)*(x^(1/2)*(1568*A^2*b^6*c^5 + 288*B^2*b ^8*c^3 - 1344*A*B*b^7*c^4) - ((7*A*c - 3*B*b)*(1792*A*b^9*c^4 - 768*B*b^10 *c^3))/(8*(-b)^(11/4)*c^(1/4))))/(8*(-b)^(11/4)*c^(1/4)) - ((7*A*c - 3*B*b )*(x^(1/2)*(1568*A^2*b^6*c^5 + 288*B^2*b^8*c^3 - 1344*A*B*b^7*c^4) + ((7*A *c - 3*B*b)*(1792*A*b^9*c^4 - 768*B*b^10*c^3))/(8*(-b)^(11/4)*c^(1/4))))/( 8*(-b)^(11/4)*c^(1/4))))*(7*A*c - 3*B*b)*1i)/(4*(-b)^(11/4)*c^(1/4)) - (at an((((7*A*c - 3*B*b)*(x^(1/2)*(1568*A^2*b^6*c^5 + 288*B^2*b^8*c^3 - 1344*A *B*b^7*c^4) - ((7*A*c - 3*B*b)*(1792*A*b^9*c^4 - 768*B*b^10*c^3)*1i)/(8*(- b)^(11/4)*c^(1/4))))/(8*(-b)^(11/4)*c^(1/4)) + ((7*A*c - 3*B*b)*(x^(1/2)*( 1568*A^2*b^6*c^5 + 288*B^2*b^8*c^3 - 1344*A*B*b^7*c^4) + ((7*A*c - 3*B*b)* (1792*A*b^9*c^4 - 768*B*b^10*c^3)*1i)/(8*(-b)^(11/4)*c^(1/4))))/(8*(-b)^(1 1/4)*c^(1/4)))/(((7*A*c - 3*B*b)*(x^(1/2)*(1568*A^2*b^6*c^5 + 288*B^2*b^8* c^3 - 1344*A*B*b^7*c^4) - ((7*A*c - 3*B*b)*(1792*A*b^9*c^4 - 768*B*b^10*c^ 3)*1i)/(8*(-b)^(11/4)*c^(1/4)))*1i)/(8*(-b)^(11/4)*c^(1/4)) - ((7*A*c -...