Integrand size = 26, antiderivative size = 191 \[ \int \frac {A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )} \, dx=-\frac {2 A}{5 b x^{5/2}}-\frac {2 (b B-A c)}{b^2 \sqrt {x}}+\frac {\sqrt [4]{c} (b B-A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{9/4}}-\frac {\sqrt [4]{c} (b B-A c) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} b^{9/4}}+\frac {\sqrt [4]{c} (b B-A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{\sqrt {2} b^{9/4}} \] Output:
-2/5*A/b/x^(5/2)-2*(-A*c+B*b)/b^2/x^(1/2)+1/2*c^(1/4)*(-A*c+B*b)*arctan(1- 2^(1/2)*c^(1/4)*x^(1/2)/b^(1/4))*2^(1/2)/b^(9/4)-1/2*c^(1/4)*(-A*c+B*b)*ar ctan(1+2^(1/2)*c^(1/4)*x^(1/2)/b^(1/4))*2^(1/2)/b^(9/4)+1/2*c^(1/4)*(-A*c+ B*b)*arctanh(2^(1/2)*b^(1/4)*c^(1/4)*x^(1/2)/(b^(1/2)+c^(1/2)*x))*2^(1/2)/ b^(9/4)
Time = 0.23 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.80 \[ \int \frac {A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )} \, dx=-\frac {2 \left (A b+5 b B x^2-5 A c x^2\right )}{5 b^2 x^{5/2}}+\frac {\sqrt [4]{c} (b B-A c) \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt {2} b^{9/4}}+\frac {\sqrt [4]{c} (b B-A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{\sqrt {2} b^{9/4}} \] Input:
Integrate[(A + B*x^2)/(x^(3/2)*(b*x^2 + c*x^4)),x]
Output:
(-2*(A*b + 5*b*B*x^2 - 5*A*c*x^2))/(5*b^2*x^(5/2)) + (c^(1/4)*(b*B - A*c)* ArcTan[(Sqrt[b] - Sqrt[c]*x)/(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])])/(Sqrt[2]* b^(9/4)) + (c^(1/4)*(b*B - A*c)*ArcTanh[(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])/ (Sqrt[b] + Sqrt[c]*x)])/(Sqrt[2]*b^(9/4))
Time = 0.76 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.34, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {9, 359, 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.
| \(\displaystyle \int \frac {A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \int \frac {A+B x^2}{x^{7/2} \left (b+c x^2\right )}dx\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {(b B-A c) \int \frac {1}{x^{3/2} \left (c x^2+b\right )}dx}{b}-\frac {2 A}{5 b x^{5/2}}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {(b B-A c) \left (-\frac {c \int \frac {\sqrt {x}}{c x^2+b}dx}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2 A}{5 b x^{5/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \int \frac {x}{c x^2+b}d\sqrt {x}}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2 A}{5 b x^{5/2}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {c} x+\sqrt {b}}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2 A}{5 b x^{5/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2 A}{5 b x^{5/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2 A}{5 b x^{5/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2 A}{5 b x^{5/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2 A}{5 b x^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2 A}{5 b x^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2 A}{5 b x^{5/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(b B-A c) \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2 A}{5 b x^{5/2}}\) |
Input:
Int[(A + B*x^2)/(x^(3/2)*(b*x^2 + c*x^4)),x]
Output:
(-2*A)/(5*b*x^(5/2)) + ((b*B - A*c)*(-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[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])))/b))/b
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[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[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 = 0.45 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.73
| method | result | size |
| derivativedivides | \(\frac {\left (A c -B b \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{2} \left (\frac {b}{c}\right )^{\frac {1}{4}}}-\frac {2 A}{5 b \,x^{\frac {5}{2}}}-\frac {2 \left (-A c +B b \right )}{b^{2} \sqrt {x}}\) | \(140\) |
| default | \(\frac {\left (A c -B b \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{2} \left (\frac {b}{c}\right )^{\frac {1}{4}}}-\frac {2 A}{5 b \,x^{\frac {5}{2}}}-\frac {2 \left (-A c +B b \right )}{b^{2} \sqrt {x}}\) | \(140\) |
| risch | \(-\frac {2 \left (-5 A c \,x^{2}+5 B b \,x^{2}+A b \right )}{5 b^{2} x^{\frac {5}{2}}}+\frac {\left (A c -B b \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{2} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(141\) |
Input:
int((B*x^2+A)/x^(3/2)/(c*x^4+b*x^2),x,method=_RETURNVERBOSE)
Output:
1/4*(A*c-B*b)/b^2/(b/c)^(1/4)*2^(1/2)*(ln((x-(b/c)^(1/4)*x^(1/2)*2^(1/2)+( b/c)^(1/2))/(x+(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2)))+2*arctan(2^(1/2)/ (b/c)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1))-2/5*A/b/x^ (5/2)-2*(-A*c+B*b)/b^2/x^(1/2)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 738, normalized size of antiderivative = 3.86 \[ \int \frac {A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )} \, dx=\frac {5 \, b^{2} x^{3} \left (-\frac {B^{4} b^{4} c - 4 \, A B^{3} b^{3} c^{2} + 6 \, A^{2} B^{2} b^{2} c^{3} - 4 \, A^{3} B b c^{4} + A^{4} c^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (b^{7} \left (-\frac {B^{4} b^{4} c - 4 \, A B^{3} b^{3} c^{2} + 6 \, A^{2} B^{2} b^{2} c^{3} - 4 \, A^{3} B b c^{4} + A^{4} c^{5}}{b^{9}}\right )^{\frac {3}{4}} - {\left (B^{3} b^{3} c - 3 \, A B^{2} b^{2} c^{2} + 3 \, A^{2} B b c^{3} - A^{3} c^{4}\right )} \sqrt {x}\right ) - 5 i \, b^{2} x^{3} \left (-\frac {B^{4} b^{4} c - 4 \, A B^{3} b^{3} c^{2} + 6 \, A^{2} B^{2} b^{2} c^{3} - 4 \, A^{3} B b c^{4} + A^{4} c^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (i \, b^{7} \left (-\frac {B^{4} b^{4} c - 4 \, A B^{3} b^{3} c^{2} + 6 \, A^{2} B^{2} b^{2} c^{3} - 4 \, A^{3} B b c^{4} + A^{4} c^{5}}{b^{9}}\right )^{\frac {3}{4}} - {\left (B^{3} b^{3} c - 3 \, A B^{2} b^{2} c^{2} + 3 \, A^{2} B b c^{3} - A^{3} c^{4}\right )} \sqrt {x}\right ) + 5 i \, b^{2} x^{3} \left (-\frac {B^{4} b^{4} c - 4 \, A B^{3} b^{3} c^{2} + 6 \, A^{2} B^{2} b^{2} c^{3} - 4 \, A^{3} B b c^{4} + A^{4} c^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-i \, b^{7} \left (-\frac {B^{4} b^{4} c - 4 \, A B^{3} b^{3} c^{2} + 6 \, A^{2} B^{2} b^{2} c^{3} - 4 \, A^{3} B b c^{4} + A^{4} c^{5}}{b^{9}}\right )^{\frac {3}{4}} - {\left (B^{3} b^{3} c - 3 \, A B^{2} b^{2} c^{2} + 3 \, A^{2} B b c^{3} - A^{3} c^{4}\right )} \sqrt {x}\right ) - 5 \, b^{2} x^{3} \left (-\frac {B^{4} b^{4} c - 4 \, A B^{3} b^{3} c^{2} + 6 \, A^{2} B^{2} b^{2} c^{3} - 4 \, A^{3} B b c^{4} + A^{4} c^{5}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-b^{7} \left (-\frac {B^{4} b^{4} c - 4 \, A B^{3} b^{3} c^{2} + 6 \, A^{2} B^{2} b^{2} c^{3} - 4 \, A^{3} B b c^{4} + A^{4} c^{5}}{b^{9}}\right )^{\frac {3}{4}} - {\left (B^{3} b^{3} c - 3 \, A B^{2} b^{2} c^{2} + 3 \, A^{2} B b c^{3} - A^{3} c^{4}\right )} \sqrt {x}\right ) - 4 \, {\left (5 \, {\left (B b - A c\right )} x^{2} + A b\right )} \sqrt {x}}{10 \, b^{2} x^{3}} \] Input:
integrate((B*x^2+A)/x^(3/2)/(c*x^4+b*x^2),x, algorithm="fricas")
Output:
1/10*(5*b^2*x^3*(-(B^4*b^4*c - 4*A*B^3*b^3*c^2 + 6*A^2*B^2*b^2*c^3 - 4*A^3 *B*b*c^4 + A^4*c^5)/b^9)^(1/4)*log(b^7*(-(B^4*b^4*c - 4*A*B^3*b^3*c^2 + 6* A^2*B^2*b^2*c^3 - 4*A^3*B*b*c^4 + A^4*c^5)/b^9)^(3/4) - (B^3*b^3*c - 3*A*B ^2*b^2*c^2 + 3*A^2*B*b*c^3 - A^3*c^4)*sqrt(x)) - 5*I*b^2*x^3*(-(B^4*b^4*c - 4*A*B^3*b^3*c^2 + 6*A^2*B^2*b^2*c^3 - 4*A^3*B*b*c^4 + A^4*c^5)/b^9)^(1/4 )*log(I*b^7*(-(B^4*b^4*c - 4*A*B^3*b^3*c^2 + 6*A^2*B^2*b^2*c^3 - 4*A^3*B*b *c^4 + A^4*c^5)/b^9)^(3/4) - (B^3*b^3*c - 3*A*B^2*b^2*c^2 + 3*A^2*B*b*c^3 - A^3*c^4)*sqrt(x)) + 5*I*b^2*x^3*(-(B^4*b^4*c - 4*A*B^3*b^3*c^2 + 6*A^2*B ^2*b^2*c^3 - 4*A^3*B*b*c^4 + A^4*c^5)/b^9)^(1/4)*log(-I*b^7*(-(B^4*b^4*c - 4*A*B^3*b^3*c^2 + 6*A^2*B^2*b^2*c^3 - 4*A^3*B*b*c^4 + A^4*c^5)/b^9)^(3/4) - (B^3*b^3*c - 3*A*B^2*b^2*c^2 + 3*A^2*B*b*c^3 - A^3*c^4)*sqrt(x)) - 5*b^ 2*x^3*(-(B^4*b^4*c - 4*A*B^3*b^3*c^2 + 6*A^2*B^2*b^2*c^3 - 4*A^3*B*b*c^4 + A^4*c^5)/b^9)^(1/4)*log(-b^7*(-(B^4*b^4*c - 4*A*B^3*b^3*c^2 + 6*A^2*B^2*b ^2*c^3 - 4*A^3*B*b*c^4 + A^4*c^5)/b^9)^(3/4) - (B^3*b^3*c - 3*A*B^2*b^2*c^ 2 + 3*A^2*B*b*c^3 - A^3*c^4)*sqrt(x)) - 4*(5*(B*b - A*c)*x^2 + A*b)*sqrt(x ))/(b^2*x^3)
Time = 36.03 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.35 \[ \int \frac {A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )} \, dx=A \left (\begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: b = 0 \wedge c = 0 \\- \frac {2}{9 c x^{\frac {9}{2}}} & \text {for}\: b = 0 \\- \frac {2}{5 b x^{\frac {5}{2}}} & \text {for}\: c = 0 \\- \frac {2}{5 b x^{\frac {5}{2}}} + \frac {c \log {\left (\sqrt {x} - \sqrt [4]{- \frac {b}{c}} \right )}}{2 b^{2} \sqrt [4]{- \frac {b}{c}}} - \frac {c \log {\left (\sqrt {x} + \sqrt [4]{- \frac {b}{c}} \right )}}{2 b^{2} \sqrt [4]{- \frac {b}{c}}} + \frac {c \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {b}{c}}} \right )}}{b^{2} \sqrt [4]{- \frac {b}{c}}} + \frac {2 c}{b^{2} \sqrt {x}} & \text {otherwise} \end {cases}\right ) + B \left (\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: b = 0 \wedge c = 0 \\- \frac {2}{5 c x^{\frac {5}{2}}} & \text {for}\: b = 0 \\- \frac {2}{b \sqrt {x}} & \text {for}\: c = 0 \\- \frac {\log {\left (\sqrt {x} - \sqrt [4]{- \frac {b}{c}} \right )}}{2 b \sqrt [4]{- \frac {b}{c}}} + \frac {\log {\left (\sqrt {x} + \sqrt [4]{- \frac {b}{c}} \right )}}{2 b \sqrt [4]{- \frac {b}{c}}} - \frac {\operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {b}{c}}} \right )}}{b \sqrt [4]{- \frac {b}{c}}} - \frac {2}{b \sqrt {x}} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((B*x**2+A)/x**(3/2)/(c*x**4+b*x**2),x)
Output:
A*Piecewise((zoo/x**(9/2), Eq(b, 0) & Eq(c, 0)), (-2/(9*c*x**(9/2)), Eq(b, 0)), (-2/(5*b*x**(5/2)), Eq(c, 0)), (-2/(5*b*x**(5/2)) + c*log(sqrt(x) - (-b/c)**(1/4))/(2*b**2*(-b/c)**(1/4)) - c*log(sqrt(x) + (-b/c)**(1/4))/(2* b**2*(-b/c)**(1/4)) + c*atan(sqrt(x)/(-b/c)**(1/4))/(b**2*(-b/c)**(1/4)) + 2*c/(b**2*sqrt(x)), True)) + B*Piecewise((zoo/x**(5/2), Eq(b, 0) & Eq(c, 0)), (-2/(5*c*x**(5/2)), Eq(b, 0)), (-2/(b*sqrt(x)), Eq(c, 0)), (-log(sqrt (x) - (-b/c)**(1/4))/(2*b*(-b/c)**(1/4)) + log(sqrt(x) + (-b/c)**(1/4))/(2 *b*(-b/c)**(1/4)) - atan(sqrt(x)/(-b/c)**(1/4))/(b*(-b/c)**(1/4)) - 2/(b*s qrt(x)), True))
Time = 0.12 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.12 \[ \int \frac {A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )} \, dx=-\frac {{\left (B b c - A c^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{4 \, b^{2}} - \frac {2 \, {\left (5 \, {\left (B b - A c\right )} x^{2} + A b\right )}}{5 \, b^{2} x^{\frac {5}{2}}} \] Input:
integrate((B*x^2+A)/x^(3/2)/(c*x^4+b*x^2),x, algorithm="maxima")
Output:
-1/4*(B*b*c - A*c^2)*(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)*s qrt(x))/sqrt(sqrt(b)*sqrt(c)))/(sqrt(sqrt(b)*sqrt(c))*sqrt(c)) - sqrt(2)*l og(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)))/b^2 - 2/5*(5*(B*b - A*c)*x^2 + A*b)/(b^2*x^(5/2))
Time = 0.21 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.40 \[ \int \frac {A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )} \, dx=-\frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {3}{4}} B b - \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, b^{3} c^{2}} - \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {3}{4}} B b - \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, b^{3} c^{2}} + \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {3}{4}} B b - \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, b^{3} c^{2}} - \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {3}{4}} B b - \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, b^{3} c^{2}} - \frac {2 \, {\left (5 \, B b x^{2} - 5 \, A c x^{2} + A b\right )}}{5 \, b^{2} x^{\frac {5}{2}}} \] Input:
integrate((B*x^2+A)/x^(3/2)/(c*x^4+b*x^2),x, algorithm="giac")
Output:
-1/2*sqrt(2)*((b*c^3)^(3/4)*B*b - (b*c^3)^(3/4)*A*c)*arctan(1/2*sqrt(2)*(s qrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/(b^3*c^2) - 1/2*sqrt(2)*((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^3*c^2) + 1/4*sqrt(2)*((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^3*c^ 2) - 1/4*sqrt(2)*((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^3*c^2) - 2/5*(5*B*b*x^2 - 5*A*c*x^2 + A*b)/(b^2*x^(5/2))
Time = 8.99 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.47 \[ \int \frac {A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )} \, dx=\frac {{\left (-c\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )\,\left (A\,c-B\,b\right )}{b^{9/4}}-\frac {\frac {2\,A}{5\,b}-\frac {2\,x^2\,\left (A\,c-B\,b\right )}{b^2}}{x^{5/2}}-\frac {{\left (-c\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )\,\left (A\,c-B\,b\right )}{b^{9/4}} \] Input:
int((A + B*x^2)/(x^(3/2)*(b*x^2 + c*x^4)),x)
Output:
((-c)^(1/4)*atan(((-c)^(1/4)*x^(1/2))/b^(1/4))*(A*c - B*b))/b^(9/4) - ((2* A)/(5*b) - (2*x^2*(A*c - B*b))/b^2)/x^(5/2) - ((-c)^(1/4)*atanh(((-c)^(1/4 )*x^(1/2))/b^(1/4))*(A*c - B*b))/b^(9/4)
Time = 0.20 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.83 \[ \int \frac {A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )} \, dx=\frac {-10 \sqrt {x}\, c^{\frac {5}{4}} b^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right ) a \,x^{2}+10 \sqrt {x}\, c^{\frac {1}{4}} b^{\frac {11}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}+10 \sqrt {x}\, c^{\frac {5}{4}} b^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right ) a \,x^{2}-10 \sqrt {x}\, c^{\frac {1}{4}} b^{\frac {11}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}+5 \sqrt {x}\, c^{\frac {5}{4}} b^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right ) a \,x^{2}-5 \sqrt {x}\, c^{\frac {1}{4}} b^{\frac {11}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right ) x^{2}-5 \sqrt {x}\, c^{\frac {5}{4}} b^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right ) a \,x^{2}+5 \sqrt {x}\, c^{\frac {1}{4}} b^{\frac {11}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right ) x^{2}-8 a \,b^{2}+40 a b c \,x^{2}-40 b^{3} x^{2}}{20 \sqrt {x}\, b^{3} x^{2}} \] Input:
int((B*x^2+A)/x^(3/2)/(c*x^4+b*x^2),x)
Output:
( - 10*sqrt(x)*c**(1/4)*b**(3/4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqrt(2)))*a*c*x**2 + 10*sqrt(x)*c** (1/4)*b**(3/4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(c) )/(c**(1/4)*b**(1/4)*sqrt(2)))*b**2*x**2 + 10*sqrt(x)*c**(1/4)*b**(3/4)*sq rt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1 /4)*sqrt(2)))*a*c*x**2 - 10*sqrt(x)*c**(1/4)*b**(3/4)*sqrt(2)*atan((c**(1/ 4)*b**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqrt(2)))*b**2 *x**2 + 5*sqrt(x)*c**(1/4)*b**(3/4)*sqrt(2)*log( - sqrt(x)*c**(1/4)*b**(1/ 4)*sqrt(2) + sqrt(b) + sqrt(c)*x)*a*c*x**2 - 5*sqrt(x)*c**(1/4)*b**(3/4)*s qrt(2)*log( - sqrt(x)*c**(1/4)*b**(1/4)*sqrt(2) + sqrt(b) + sqrt(c)*x)*b** 2*x**2 - 5*sqrt(x)*c**(1/4)*b**(3/4)*sqrt(2)*log(sqrt(x)*c**(1/4)*b**(1/4) *sqrt(2) + sqrt(b) + sqrt(c)*x)*a*c*x**2 + 5*sqrt(x)*c**(1/4)*b**(3/4)*sqr t(2)*log(sqrt(x)*c**(1/4)*b**(1/4)*sqrt(2) + sqrt(b) + sqrt(c)*x)*b**2*x** 2 - 8*a*b**2 + 40*a*b*c*x**2 - 40*b**3*x**2)/(20*sqrt(x)*b**3*x**2)