Integrand size = 19, antiderivative size = 251 \[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {45}{16 b^3 \sqrt {x}}+\frac {1}{4 b \sqrt {x} \left (b+c x^2\right )^2}+\frac {9}{16 b^2 \sqrt {x} \left (b+c x^2\right )}+\frac {45 \sqrt [4]{c} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{13/4}}-\frac {45 \sqrt [4]{c} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{13/4}}-\frac {45 \sqrt [4]{c} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{13/4}}+\frac {45 \sqrt [4]{c} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{13/4}} \]
45/64*c^(1/4)*arctan(1-c^(1/4)*2^(1/2)*x^(1/2)/b^(1/4))/b^(13/4)*2^(1/2)-4 5/64*c^(1/4)*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/b^(1/4))/b^(13/4)*2^(1/2)-45 /128*c^(1/4)*ln(b^(1/2)+x*c^(1/2)-b^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))/b^(13/4 )*2^(1/2)+45/128*c^(1/4)*ln(b^(1/2)+x*c^(1/2)+b^(1/4)*c^(1/4)*2^(1/2)*x^(1 /2))/b^(13/4)*2^(1/2)-45/16/b^3/x^(1/2)+1/4/b/(c*x^2+b)^2/x^(1/2)+9/16/b^2 /(c*x^2+b)/x^(1/2)
Time = 0.39 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.59 \[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{b} \left (32 b^2+81 b c x^2+45 c^2 x^4\right )}{\sqrt {x} \left (b+c x^2\right )^2}+45 \sqrt {2} \sqrt [4]{c} \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )+45 \sqrt {2} \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{64 b^{13/4}} \]
((-4*b^(1/4)*(32*b^2 + 81*b*c*x^2 + 45*c^2*x^4))/(Sqrt[x]*(b + c*x^2)^2) + 45*Sqrt[2]*c^(1/4)*ArcTan[(Sqrt[b] - Sqrt[c]*x)/(Sqrt[2]*b^(1/4)*c^(1/4)* Sqrt[x])] + 45*Sqrt[2]*c^(1/4)*ArcTanh[(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])/( Sqrt[b] + Sqrt[c]*x)])/(64*b^(13/4))
Time = 0.47 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {9, 253, 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.
\(\displaystyle \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {1}{x^{3/2} \left (b+c x^2\right )^3}dx\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {9 \int \frac {1}{x^{3/2} \left (c x^2+b\right )^2}dx}{8 b}+\frac {1}{4 b \sqrt {x} \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {9 \left (\frac {5 \int \frac {1}{x^{3/2} \left (c x^2+b\right )}dx}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b \sqrt {x} \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {c \int \frac {\sqrt {x}}{c x^2+b}dx}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b \sqrt {x} \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 c \int \frac {x}{c x^2+b}d\sqrt {x}}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b \sqrt {x} \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 826 |
\(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {c} x+\sqrt {b}}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b \sqrt {x} \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b \sqrt {x} \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b \sqrt {x} \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b \sqrt {x} \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b \sqrt {x} \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b \sqrt {x} \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b \sqrt {x} \left (b+c x^2\right )^2}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{b}-\frac {2}{b \sqrt {x}}\right )}{4 b}+\frac {1}{2 b \sqrt {x} \left (b+c x^2\right )}\right )}{8 b}+\frac {1}{4 b \sqrt {x} \left (b+c x^2\right )^2}\) |
1/(4*b*Sqrt[x]*(b + c*x^2)^2) + (9*(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[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*Sq rt[c])))/b))/(4*b)))/(8*b)
3.4.46.3.1 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[(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 = 0.15 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.58
method | result | size |
derivativedivides | \(-\frac {2}{b^{3} \sqrt {x}}-\frac {2 c \left (\frac {\frac {13 c \,x^{\frac {7}{2}}}{32}+\frac {17 b \,x^{\frac {3}{2}}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{3}}\) | \(145\) |
default | \(-\frac {2}{b^{3} \sqrt {x}}-\frac {2 c \left (\frac {\frac {13 c \,x^{\frac {7}{2}}}{32}+\frac {17 b \,x^{\frac {3}{2}}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{3}}\) | \(145\) |
risch | \(-\frac {2}{b^{3} \sqrt {x}}-\frac {c \left (\frac {\frac {13 c \,x^{\frac {7}{2}}}{16}+\frac {17 b \,x^{\frac {3}{2}}}{16}}{\left (c \,x^{2}+b \right )^{2}}+\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{b^{3}}\) | \(146\) |
-2/b^3/x^(1/2)-2/b^3*c*((13/32*c*x^(7/2)+17/32*b*x^(3/2))/(c*x^2+b)^2+45/2 56/c/(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)))
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.13 \[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {45 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \left (-\frac {c}{b^{13}}\right )^{\frac {1}{4}} \log \left (91125 \, b^{10} \left (-\frac {c}{b^{13}}\right )^{\frac {3}{4}} + 91125 \, c \sqrt {x}\right ) + 45 \, {\left (-i \, b^{3} c^{2} x^{5} - 2 i \, b^{4} c x^{3} - i \, b^{5} x\right )} \left (-\frac {c}{b^{13}}\right )^{\frac {1}{4}} \log \left (91125 i \, b^{10} \left (-\frac {c}{b^{13}}\right )^{\frac {3}{4}} + 91125 \, c \sqrt {x}\right ) + 45 \, {\left (i \, b^{3} c^{2} x^{5} + 2 i \, b^{4} c x^{3} + i \, b^{5} x\right )} \left (-\frac {c}{b^{13}}\right )^{\frac {1}{4}} \log \left (-91125 i \, b^{10} \left (-\frac {c}{b^{13}}\right )^{\frac {3}{4}} + 91125 \, c \sqrt {x}\right ) - 45 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \left (-\frac {c}{b^{13}}\right )^{\frac {1}{4}} \log \left (-91125 \, b^{10} \left (-\frac {c}{b^{13}}\right )^{\frac {3}{4}} + 91125 \, c \sqrt {x}\right ) + 4 \, {\left (45 \, c^{2} x^{4} + 81 \, b c x^{2} + 32 \, b^{2}\right )} \sqrt {x}}{64 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )}} \]
-1/64*(45*(b^3*c^2*x^5 + 2*b^4*c*x^3 + b^5*x)*(-c/b^13)^(1/4)*log(91125*b^ 10*(-c/b^13)^(3/4) + 91125*c*sqrt(x)) + 45*(-I*b^3*c^2*x^5 - 2*I*b^4*c*x^3 - I*b^5*x)*(-c/b^13)^(1/4)*log(91125*I*b^10*(-c/b^13)^(3/4) + 91125*c*sqr t(x)) + 45*(I*b^3*c^2*x^5 + 2*I*b^4*c*x^3 + I*b^5*x)*(-c/b^13)^(1/4)*log(- 91125*I*b^10*(-c/b^13)^(3/4) + 91125*c*sqrt(x)) - 45*(b^3*c^2*x^5 + 2*b^4* c*x^3 + b^5*x)*(-c/b^13)^(1/4)*log(-91125*b^10*(-c/b^13)^(3/4) + 91125*c*s qrt(x)) + 4*(45*c^2*x^4 + 81*b*c*x^2 + 32*b^2)*sqrt(x))/(b^3*c^2*x^5 + 2*b ^4*c*x^3 + b^5*x)
Timed out. \[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.92 \[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {45 \, c^{2} x^{4} + 81 \, b c x^{2} + 32 \, b^{2}}{16 \, {\left (b^{3} c^{2} x^{\frac {9}{2}} + 2 \, b^{4} c x^{\frac {5}{2}} + b^{5} \sqrt {x}\right )}} - \frac {45 \, c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{128 \, b^{3}} \]
-1/16*(45*c^2*x^4 + 81*b*c*x^2 + 32*b^2)/(b^3*c^2*x^(9/2) + 2*b^4*c*x^(5/2 ) + b^5*sqrt(x)) - 45/128*c*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) *c^(1/4) + 2*sqrt(c)*sqrt(x))/sqrt(sqrt(b)*sqrt(c)))/(sqrt(sqrt(b)*sqrt(c) )*sqrt(c)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) - 2*sq rt(c)*sqrt(x))/sqrt(sqrt(b)*sqrt(c)))/(sqrt(sqrt(b)*sqrt(c))*sqrt(c)) - sq rt(2)*log(sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(b))/(b^(1/4)* c^(3/4)) + sqrt(2)*log(-sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt (b))/(b^(1/4)*c^(3/4)))/b^3
Time = 0.27 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.88 \[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {2}{b^{3} \sqrt {x}} - \frac {45 \, \sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{4} c^{2}} - \frac {45 \, \sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{4} c^{2}} + \frac {45 \, \sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{4} c^{2}} - \frac {45 \, \sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{4} c^{2}} - \frac {13 \, c^{2} x^{\frac {7}{2}} + 17 \, b c x^{\frac {3}{2}}}{16 \, {\left (c x^{2} + b\right )}^{2} b^{3}} \]
-2/(b^3*sqrt(x)) - 45/64*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^4*c^2) - 45/64*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 ^4*c^2) + 45/128*sqrt(2)*(b*c^3)^(3/4)*log(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^4*c^2) - 45/128*sqrt(2)*(b*c^3)^(3/4)*log(-sqrt(2)*sqrt(x )*(b/c)^(1/4) + x + sqrt(b/c))/(b^4*c^2) - 1/16*(13*c^2*x^(7/2) + 17*b*c*x ^(3/2))/((c*x^2 + b)^2*b^3)
Time = 12.96 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.39 \[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {45\,{\left (-c\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{32\,b^{13/4}}-\frac {45\,{\left (-c\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )}{32\,b^{13/4}}-\frac {\frac {2}{b}+\frac {81\,c\,x^2}{16\,b^2}+\frac {45\,c^2\,x^4}{16\,b^3}}{b^2\,\sqrt {x}+c^2\,x^{9/2}+2\,b\,c\,x^{5/2}} \]