Integrand size = 19, antiderivative size = 226 \[ \int \frac {\sqrt {x}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {221}{144 b^3 x^{9/2}}+\frac {221 c}{80 b^4 x^{5/2}}-\frac {221 c^2}{16 b^5 \sqrt {x}}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}+\frac {17}{16 b^2 x^{9/2} \left (b+c x^2\right )}+\frac {221 c^{9/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{21/4}}-\frac {221 c^{9/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{21/4}}+\frac {221 c^{9/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{32 \sqrt {2} b^{21/4}} \] Output:
-221/144/b^3/x^(9/2)+221/80*c/b^4/x^(5/2)-221/16*c^2/b^5/x^(1/2)+1/4/b/x^( 9/2)/(c*x^2+b)^2+17/16/b^2/x^(9/2)/(c*x^2+b)+221/64*c^(9/4)*arctan(1-2^(1/ 2)*c^(1/4)*x^(1/2)/b^(1/4))*2^(1/2)/b^(21/4)-221/64*c^(9/4)*arctan(1+2^(1/ 2)*c^(1/4)*x^(1/2)/b^(1/4))*2^(1/2)/b^(21/4)+221/64*c^(9/4)*arctanh(2^(1/2 )*b^(1/4)*c^(1/4)*x^(1/2)/(b^(1/2)+c^(1/2)*x))*2^(1/2)/b^(21/4)
Time = 0.44 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {x}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{b} \left (160 b^4-544 b^3 c x^2+7072 b^2 c^2 x^4+17901 b c^3 x^6+9945 c^4 x^8\right )}{x^{9/2} \left (b+c x^2\right )^2}+9945 \sqrt {2} c^{9/4} \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )+9945 \sqrt {2} c^{9/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{2880 b^{21/4}} \] Input:
Integrate[Sqrt[x]/(b*x^2 + c*x^4)^3,x]
Output:
((-4*b^(1/4)*(160*b^4 - 544*b^3*c*x^2 + 7072*b^2*c^2*x^4 + 17901*b*c^3*x^6 + 9945*c^4*x^8))/(x^(9/2)*(b + c*x^2)^2) + 9945*Sqrt[2]*c^(9/4)*ArcTan[(S qrt[b] - Sqrt[c]*x)/(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])] + 9945*Sqrt[2]*c^(9 /4)*ArcTanh[(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[b] + Sqrt[c]*x)])/(288 0*b^(21/4))
Time = 0.83 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.44, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {9, 253, 253, 264, 264, 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.
| \(\displaystyle \int \frac {\sqrt {x}}{\left (b x^2+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \int \frac {1}{x^{11/2} \left (b+c x^2\right )^3}dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {17 \int \frac {1}{x^{11/2} \left (c x^2+b\right )^2}dx}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {17 \left (\frac {13 \int \frac {1}{x^{11/2} \left (c x^2+b\right )}dx}{4 b}+\frac {1}{2 b x^{9/2} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (-\frac {c \int \frac {1}{x^{7/2} \left (c x^2+b\right )}dx}{b}-\frac {2}{9 b x^{9/2}}\right )}{4 b}+\frac {1}{2 b x^{9/2} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (-\frac {c \left (-\frac {c \int \frac {1}{x^{3/2} \left (c x^2+b\right )}dx}{b}-\frac {2}{5 b x^{5/2}}\right )}{b}-\frac {2}{9 b x^{9/2}}\right )}{4 b}+\frac {1}{2 b x^{9/2} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (-\frac {c \left (-\frac {c \left (-\frac {c \int \frac {\sqrt {x}}{c x^2+b}dx}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{b}-\frac {2}{9 b x^{9/2}}\right )}{4 b}+\frac {1}{2 b x^{9/2} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (-\frac {c \left (-\frac {c \left (-\frac {2 c \int \frac {x}{c x^2+b}d\sqrt {x}}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{b}-\frac {2}{9 b x^{9/2}}\right )}{4 b}+\frac {1}{2 b x^{9/2} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (-\frac {c \left (-\frac {c \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {c} x+\sqrt {b}}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{b}-\frac {2}{9 b x^{9/2}}\right )}{4 b}+\frac {1}{2 b x^{9/2} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (-\frac {c \left (-\frac {c \left (-\frac {2 c \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{b}-\frac {2}{9 b x^{9/2}}\right )}{4 b}+\frac {1}{2 b x^{9/2} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (-\frac {c \left (-\frac {c \left (-\frac {2 c \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{b}-\frac {2}{9 b x^{9/2}}\right )}{4 b}+\frac {1}{2 b x^{9/2} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (-\frac {c \left (-\frac {c \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{b}-\frac {2}{9 b x^{9/2}}\right )}{4 b}+\frac {1}{2 b x^{9/2} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (-\frac {c \left (-\frac {c \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{b}-\frac {2}{9 b x^{9/2}}\right )}{4 b}+\frac {1}{2 b x^{9/2} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (-\frac {c \left (-\frac {c \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{b}-\frac {2}{9 b x^{9/2}}\right )}{4 b}+\frac {1}{2 b x^{9/2} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (-\frac {c \left (-\frac {c \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{b}-\frac {2}{9 b x^{9/2}}\right )}{4 b}+\frac {1}{2 b x^{9/2} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {17 \left (\frac {13 \left (-\frac {c \left (-\frac {c \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{b}-\frac {2}{9 b x^{9/2}}\right )}{4 b}+\frac {1}{2 b x^{9/2} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b x^{9/2} \left (b+c x^2\right )^2}\) |
Input:
Int[Sqrt[x]/(b*x^2 + c*x^4)^3,x]
Output:
1/(4*b*x^(9/2)*(b + c*x^2)^2) + (17*(1/(2*b*x^(9/2)*(b + c*x^2)) + (13*(-2 /(9*b*x^(9/2)) - (c*(-2/(5*b*x^(5/2)) - (c*(-2/(b*Sqrt[x]) - (2*c*((-(ArcT an[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]/(Sqrt[2]*b^(1/4)*c^(1/4))) + Arc Tan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]/(Sqrt[2]*b^(1/4)*c^(1/4)))/(2*S qrt[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))/b))/(4*b) ))/(8*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)/(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[(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 = 0.23 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(-\frac {2 c^{3} \left (\frac {\frac {29 c \,x^{\frac {7}{2}}}{32}+\frac {33 b \,x^{\frac {3}{2}}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {221 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{5}}-\frac {2}{9 b^{3} x^{\frac {9}{2}}}-\frac {12 c^{2}}{b^{5} \sqrt {x}}+\frac {6 c}{5 b^{4} x^{\frac {5}{2}}}\) | \(167\) |
| default | \(-\frac {2 c^{3} \left (\frac {\frac {29 c \,x^{\frac {7}{2}}}{32}+\frac {33 b \,x^{\frac {3}{2}}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {221 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{5}}-\frac {2}{9 b^{3} x^{\frac {9}{2}}}-\frac {12 c^{2}}{b^{5} \sqrt {x}}+\frac {6 c}{5 b^{4} x^{\frac {5}{2}}}\) | \(167\) |
| risch | \(-\frac {2 \left (270 c^{2} x^{4}-27 b c \,x^{2}+5 b^{2}\right )}{45 b^{5} x^{\frac {9}{2}}}-\frac {c^{3} \left (\frac {\frac {29 c \,x^{\frac {7}{2}}}{16}+\frac {33 b \,x^{\frac {3}{2}}}{16}}{\left (c \,x^{2}+b \right )^{2}}+\frac {221 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{5}}\) | \(169\) |
Input:
int(x^(1/2)/(c*x^4+b*x^2)^3,x,method=_RETURNVERBOSE)
Output:
-2/b^5*c^3*((29/32*c*x^(7/2)+33/32*b*x^(3/2))/(c*x^2+b)^2+221/256/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)))-2/9/b^3/x^(9/2)-12* c^2/b^5/x^(1/2)+6/5*c/b^4/x^(5/2)
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {x}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {9945 \, {\left (b^{5} c^{2} x^{9} + 2 \, b^{6} c x^{7} + b^{7} x^{5}\right )} \left (-\frac {c^{9}}{b^{21}}\right )^{\frac {1}{4}} \log \left (10793861 \, b^{16} \left (-\frac {c^{9}}{b^{21}}\right )^{\frac {3}{4}} + 10793861 \, c^{7} \sqrt {x}\right ) + 9945 \, {\left (-i \, b^{5} c^{2} x^{9} - 2 i \, b^{6} c x^{7} - i \, b^{7} x^{5}\right )} \left (-\frac {c^{9}}{b^{21}}\right )^{\frac {1}{4}} \log \left (10793861 i \, b^{16} \left (-\frac {c^{9}}{b^{21}}\right )^{\frac {3}{4}} + 10793861 \, c^{7} \sqrt {x}\right ) + 9945 \, {\left (i \, b^{5} c^{2} x^{9} + 2 i \, b^{6} c x^{7} + i \, b^{7} x^{5}\right )} \left (-\frac {c^{9}}{b^{21}}\right )^{\frac {1}{4}} \log \left (-10793861 i \, b^{16} \left (-\frac {c^{9}}{b^{21}}\right )^{\frac {3}{4}} + 10793861 \, c^{7} \sqrt {x}\right ) - 9945 \, {\left (b^{5} c^{2} x^{9} + 2 \, b^{6} c x^{7} + b^{7} x^{5}\right )} \left (-\frac {c^{9}}{b^{21}}\right )^{\frac {1}{4}} \log \left (-10793861 \, b^{16} \left (-\frac {c^{9}}{b^{21}}\right )^{\frac {3}{4}} + 10793861 \, c^{7} \sqrt {x}\right ) + 4 \, {\left (9945 \, c^{4} x^{8} + 17901 \, b c^{3} x^{6} + 7072 \, b^{2} c^{2} x^{4} - 544 \, b^{3} c x^{2} + 160 \, b^{4}\right )} \sqrt {x}}{2880 \, {\left (b^{5} c^{2} x^{9} + 2 \, b^{6} c x^{7} + b^{7} x^{5}\right )}} \] Input:
integrate(x^(1/2)/(c*x^4+b*x^2)^3,x, algorithm="fricas")
Output:
-1/2880*(9945*(b^5*c^2*x^9 + 2*b^6*c*x^7 + b^7*x^5)*(-c^9/b^21)^(1/4)*log( 10793861*b^16*(-c^9/b^21)^(3/4) + 10793861*c^7*sqrt(x)) + 9945*(-I*b^5*c^2 *x^9 - 2*I*b^6*c*x^7 - I*b^7*x^5)*(-c^9/b^21)^(1/4)*log(10793861*I*b^16*(- c^9/b^21)^(3/4) + 10793861*c^7*sqrt(x)) + 9945*(I*b^5*c^2*x^9 + 2*I*b^6*c* x^7 + I*b^7*x^5)*(-c^9/b^21)^(1/4)*log(-10793861*I*b^16*(-c^9/b^21)^(3/4) + 10793861*c^7*sqrt(x)) - 9945*(b^5*c^2*x^9 + 2*b^6*c*x^7 + b^7*x^5)*(-c^9 /b^21)^(1/4)*log(-10793861*b^16*(-c^9/b^21)^(3/4) + 10793861*c^7*sqrt(x)) + 4*(9945*c^4*x^8 + 17901*b*c^3*x^6 + 7072*b^2*c^2*x^4 - 544*b^3*c*x^2 + 1 60*b^4)*sqrt(x))/(b^5*c^2*x^9 + 2*b^6*c*x^7 + b^7*x^5)
Timed out. \[ \int \frac {\sqrt {x}}{\left (b x^2+c x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**(1/2)/(c*x**4+b*x**2)**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {x}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {9945 \, c^{4} x^{8} + 17901 \, b c^{3} x^{6} + 7072 \, b^{2} c^{2} x^{4} - 544 \, b^{3} c x^{2} + 160 \, b^{4}}{720 \, {\left (b^{5} c^{2} x^{\frac {17}{2}} + 2 \, b^{6} c x^{\frac {13}{2}} + b^{7} x^{\frac {9}{2}}\right )}} - \frac {221 \, c^{3} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{128 \, b^{5}} \] Input:
integrate(x^(1/2)/(c*x^4+b*x^2)^3,x, algorithm="maxima")
Output:
-1/720*(9945*c^4*x^8 + 17901*b*c^3*x^6 + 7072*b^2*c^2*x^4 - 544*b^3*c*x^2 + 160*b^4)/(b^5*c^2*x^(17/2) + 2*b^6*c*x^(13/2) + b^7*x^(9/2)) - 221/128*c ^3*(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)*ar ctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) - 2*sqrt(c)*sqrt(x))/sqrt(sqrt( b)*sqrt(c)))/(sqrt(sqrt(b)*sqrt(c))*sqrt(c)) - sqrt(2)*log(sqrt(2)*b^(1/4) *c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(b))/(b^(1/4)*c^(3/4)) + sqrt(2)*log(-s qrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(b))/(b^(1/4)*c^(3/4)))/b ^5
Time = 0.15 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {x}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {221 \, \sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{6}} - \frac {221 \, \sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{6}} + \frac {221 \, \sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{6}} - \frac {221 \, \sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{6}} - \frac {29 \, c^{4} x^{\frac {7}{2}} + 33 \, b c^{3} x^{\frac {3}{2}}}{16 \, {\left (c x^{2} + b\right )}^{2} b^{5}} - \frac {2 \, {\left (270 \, c^{2} x^{4} - 27 \, b c x^{2} + 5 \, b^{2}\right )}}{45 \, b^{5} x^{\frac {9}{2}}} \] Input:
integrate(x^(1/2)/(c*x^4+b*x^2)^3,x, algorithm="giac")
Output:
-221/64*sqrt(2)*(b*c^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2* sqrt(x))/(b/c)^(1/4))/b^6 - 221/64*sqrt(2)*(b*c^3)^(3/4)*arctan(-1/2*sqrt( 2)*(sqrt(2)*(b/c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/b^6 + 221/128*sqrt(2)*(b *c^3)^(3/4)*log(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^6 - 221/128 *sqrt(2)*(b*c^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b ^6 - 1/16*(29*c^4*x^(7/2) + 33*b*c^3*x^(3/2))/((c*x^2 + b)^2*b^5) - 2/45*( 270*c^2*x^4 - 27*b*c*x^2 + 5*b^2)/(b^5*x^(9/2))
Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt {x}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {221\,{\left (-c\right )}^{9/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{32\,b^{21/4}}-\frac {221\,{\left (-c\right )}^{9/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{32\,b^{21/4}}-\frac {\frac {2}{9\,b}-\frac {34\,c\,x^2}{45\,b^2}+\frac {442\,c^2\,x^4}{45\,b^3}+\frac {1989\,c^3\,x^6}{80\,b^4}+\frac {221\,c^4\,x^8}{16\,b^5}}{b^2\,x^{9/2}+c^2\,x^{17/2}+2\,b\,c\,x^{13/2}} \] Input:
int(x^(1/2)/(b*x^2 + c*x^4)^3,x)
Output:
(221*(-c)^(9/4)*atanh(((-c)^(1/4)*x^(1/2))/b^(1/4)))/(32*b^(21/4)) - (221* (-c)^(9/4)*atan(((-c)^(1/4)*x^(1/2))/b^(1/4)))/(32*b^(21/4)) - (2/(9*b) - (34*c*x^2)/(45*b^2) + (442*c^2*x^4)/(45*b^3) + (1989*c^3*x^6)/(80*b^4) + ( 221*c^4*x^8)/(16*b^5))/(b^2*x^(9/2) + c^2*x^(17/2) + 2*b*c*x^(13/2))
Time = 0.17 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.40 \[ \int \frac {\sqrt {x}}{\left (b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int(x^(1/2)/(c*x^4+b*x^2)^3,x)
Output:
(19890*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*c**2*x**4 + 39780*sq rt(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*c**3*x**6 + 19890*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)))*c**4*x**8 - 19890*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)))*b**2*c**2*x**4 - 39780*sqrt(x)*c**(1/4)*b**(3/4)*sqrt(2)*ata n((c**(1/4)*b**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqrt( 2)))*b*c**3*x**6 - 19890*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)))*c**4*x* *8 - 9945*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)*b**2*c**2*x**4 - 19890*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)*b*c**3*x**6 - 9945*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)*c**4*x**8 + 9945*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)*b**2*c**2*x**4 + 19890*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)*b*c**3*x**6 + 99 45*sqrt(x)*c**(1/4)*b**(3/4)*sqrt(2)*log(sqrt(x)*c**(1/4)*b**(1/4)*sqrt...