Integrand size = 19, antiderivative size = 162 \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=-\frac {2 b x^{3/2}}{3 c^2}+\frac {2 x^{7/2}}{7 c}-\frac {b^{7/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{11/4}}+\frac {b^{7/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{11/4}}-\frac {b^{7/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{\sqrt {2} c^{11/4}} \] Output:
-2/3*b*x^(3/2)/c^2+2/7*x^(7/2)/c-1/2*b^(7/4)*arctan(1-2^(1/2)*c^(1/4)*x^(1 /2)/b^(1/4))*2^(1/2)/c^(11/4)+1/2*b^(7/4)*arctan(1+2^(1/2)*c^(1/4)*x^(1/2) /b^(1/4))*2^(1/2)/c^(11/4)-1/2*b^(7/4)*arctanh(2^(1/2)*b^(1/4)*c^(1/4)*x^( 1/2)/(b^(1/2)+c^(1/2)*x))*2^(1/2)/c^(11/4)
Time = 0.17 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.80 \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=\frac {4 c^{3/4} x^{3/2} \left (-7 b+3 c x^2\right )-21 \sqrt {2} b^{7/4} \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )-21 \sqrt {2} b^{7/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{42 c^{11/4}} \] Input:
Integrate[x^(13/2)/(b*x^2 + c*x^4),x]
Output:
(4*c^(3/4)*x^(3/2)*(-7*b + 3*c*x^2) - 21*Sqrt[2]*b^(7/4)*ArcTan[(Sqrt[b] - Sqrt[c]*x)/(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])] - 21*Sqrt[2]*b^(7/4)*ArcTan h[(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[b] + Sqrt[c]*x)])/(42*c^(11/4))
Time = 0.76 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.54, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {9, 262, 262, 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.
| \(\displaystyle \int \frac {x^{13/2}}{b x^2+c x^4} \, dx\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle \int \frac {x^{9/2}}{b+c x^2}dx\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {2 x^{7/2}}{7 c}-\frac {b \int \frac {x^{5/2}}{c x^2+b}dx}{c}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {2 x^{7/2}}{7 c}-\frac {b \left (\frac {2 x^{3/2}}{3 c}-\frac {b \int \frac {\sqrt {x}}{c x^2+b}dx}{c}\right )}{c}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 x^{7/2}}{7 c}-\frac {b \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \int \frac {x}{c x^2+b}d\sqrt {x}}{c}\right )}{c}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {2 x^{7/2}}{7 c}-\frac {b \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\int \frac {\sqrt {c} x+\sqrt {b}}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 x^{7/2}}{7 c}-\frac {b \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 x^{7/2}}{7 c}-\frac {b \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 x^{7/2}}{7 c}-\frac {b \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {c}}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 x^{7/2}}{7 c}-\frac {b \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 x^{7/2}}{7 c}-\frac {b \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 x^{7/2}}{7 c}-\frac {b \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {c}}\right )}{c}\right )}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 x^{7/2}}{7 c}-\frac {b \left (\frac {2 x^{3/2}}{3 c}-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {c}}\right )}{c}\right )}{c}\) |
Input:
Int[x^(13/2)/(b*x^2 + c*x^4),x]
Output:
(2*x^(7/2))/(7*c) - (b*((2*x^(3/2))/(3*c) - (2*b*((-(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*L og[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])))/c))/c
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*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && 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 = 0.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(-\frac {2 \left (-\frac {c \,x^{\frac {7}{2}}}{7}+\frac {b \,x^{\frac {3}{2}}}{3}\right )}{c^{2}}+\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{3} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(128\) |
| default | \(-\frac {2 \left (-\frac {c \,x^{\frac {7}{2}}}{7}+\frac {b \,x^{\frac {3}{2}}}{3}\right )}{c^{2}}+\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{3} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(128\) |
| risch | \(-\frac {2 x^{\frac {3}{2}} \left (-3 c \,x^{2}+7 b \right )}{21 c^{2}}+\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{3} \left (\frac {b}{c}\right )^{\frac {1}{4}}}\) | \(128\) |
Input:
int(x^(13/2)/(c*x^4+b*x^2),x,method=_RETURNVERBOSE)
Output:
-2/c^2*(-1/7*c*x^(7/2)+1/3*b*x^(3/2))+1/4/c^3*b^2/(1/c*b)^(1/4)*2^(1/2)*(l n((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*arcta n(2^(1/2)/(1/c*b)^(1/4)*x^(1/2)-1))
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.09 \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=\frac {21 \, c^{2} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {1}{4}} \log \left (c^{8} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {3}{4}} + b^{5} \sqrt {x}\right ) - 21 i \, c^{2} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {1}{4}} \log \left (i \, c^{8} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {3}{4}} + b^{5} \sqrt {x}\right ) + 21 i \, c^{2} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {1}{4}} \log \left (-i \, c^{8} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {3}{4}} + b^{5} \sqrt {x}\right ) - 21 \, c^{2} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {1}{4}} \log \left (-c^{8} \left (-\frac {b^{7}}{c^{11}}\right )^{\frac {3}{4}} + b^{5} \sqrt {x}\right ) + 4 \, {\left (3 \, c x^{3} - 7 \, b x\right )} \sqrt {x}}{42 \, c^{2}} \] Input:
integrate(x^(13/2)/(c*x^4+b*x^2),x, algorithm="fricas")
Output:
1/42*(21*c^2*(-b^7/c^11)^(1/4)*log(c^8*(-b^7/c^11)^(3/4) + b^5*sqrt(x)) - 21*I*c^2*(-b^7/c^11)^(1/4)*log(I*c^8*(-b^7/c^11)^(3/4) + b^5*sqrt(x)) + 21 *I*c^2*(-b^7/c^11)^(1/4)*log(-I*c^8*(-b^7/c^11)^(3/4) + b^5*sqrt(x)) - 21* c^2*(-b^7/c^11)^(1/4)*log(-c^8*(-b^7/c^11)^(3/4) + b^5*sqrt(x)) + 4*(3*c*x ^3 - 7*b*x)*sqrt(x))/c^2
Timed out. \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=\text {Timed out} \] Input:
integrate(x**(13/2)/(c*x**4+b*x**2),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.22 \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=\frac {b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{4 \, c^{2}} + \frac {2 \, {\left (3 \, c x^{\frac {7}{2}} - 7 \, b x^{\frac {3}{2}}\right )}}{21 \, c^{2}} \] Input:
integrate(x^(13/2)/(c*x^4+b*x^2),x, algorithm="maxima")
Output:
1/4*b^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)*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)*l og(-sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(b))/(b^(1/4)*c^(3/4 )))/c^2 + 2/21*(3*c*x^(7/2) - 7*b*x^(3/2))/c^2
Time = 0.16 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.22 \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=\frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{5}} + \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{5}} - \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} b \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, c^{5}} + \frac {\sqrt {2} \left (b c^{3}\right )^{\frac {3}{4}} b \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, c^{5}} + \frac {2 \, {\left (3 \, c^{6} x^{\frac {7}{2}} - 7 \, b c^{5} x^{\frac {3}{2}}\right )}}{21 \, c^{7}} \] Input:
integrate(x^(13/2)/(c*x^4+b*x^2),x, algorithm="giac")
Output:
1/2*sqrt(2)*(b*c^3)^(3/4)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sq rt(x))/(b/c)^(1/4))/c^5 + 1/2*sqrt(2)*(b*c^3)^(3/4)*b*arctan(-1/2*sqrt(2)* (sqrt(2)*(b/c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/c^5 - 1/4*sqrt(2)*(b*c^3)^( 3/4)*b*log(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/c^5 + 1/4*sqrt(2)* (b*c^3)^(3/4)*b*log(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/c^5 + 2/ 21*(3*c^6*x^(7/2) - 7*b*c^5*x^(3/2))/c^7
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.41 \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=\frac {2\,x^{7/2}}{7\,c}-\frac {2\,b\,x^{3/2}}{3\,c^2}+\frac {{\left (-b\right )}^{7/4}\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )}{c^{11/4}}+\frac {{\left (-b\right )}^{7/4}\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-b\right )}^{1/4}}\right )\,1{}\mathrm {i}}{c^{11/4}} \] Input:
int(x^(13/2)/(b*x^2 + c*x^4),x)
Output:
(2*x^(7/2))/(7*c) - (2*b*x^(3/2))/(3*c^2) + ((-b)^(7/4)*atan((c^(1/4)*x^(1 /2))/(-b)^(1/4)))/c^(11/4) + ((-b)^(7/4)*atan((c^(1/4)*x^(1/2)*1i)/(-b)^(1 /4))*1i)/c^(11/4)
Time = 0.16 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.98 \[ \int \frac {x^{13/2}}{b x^2+c x^4} \, dx=\frac {-42 c^{\frac {1}{4}} b^{\frac {7}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right )+42 c^{\frac {1}{4}} b^{\frac {7}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right )+21 c^{\frac {1}{4}} b^{\frac {7}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right )-21 c^{\frac {1}{4}} b^{\frac {7}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right )-56 \sqrt {x}\, b c x +24 \sqrt {x}\, c^{2} x^{3}}{84 c^{3}} \] Input:
int(x^(13/2)/(c*x^4+b*x^2),x)
Output:
( - 42*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 + 42*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)*sq rt(2)))*b + 21*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 - 21*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 - 56*sqrt(x)*b*c*x + 24*sqrt(x)*c**2*x**3)/(84*c**3)