Integrand size = 15, antiderivative size = 228 \[ \int \frac {x^{5/2}}{a+c x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} \sqrt [8]{-a} c^{7/8}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} \sqrt [8]{-a} c^{7/8}}+\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{7/8}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{7/8}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{-a}+\sqrt [4]{c} x}\right )}{2 \sqrt {2} \sqrt [8]{-a} c^{7/8}} \] Output:
1/4*arctan(-1+2^(1/2)*c^(1/8)*x^(1/2)/(-a)^(1/8))*2^(1/2)/(-a)^(1/8)/c^(7/ 8)+1/4*arctan(1+2^(1/2)*c^(1/8)*x^(1/2)/(-a)^(1/8))*2^(1/2)/(-a)^(1/8)/c^( 7/8)+1/2*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(1/8)/c^(7/8)-1/2*arctanh (c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(1/8)/c^(7/8)-1/4*arctanh(2^(1/2)*(-a)^( 1/8)*c^(1/8)*x^(1/2)/((-a)^(1/4)+c^(1/4)*x))*2^(1/2)/(-a)^(1/8)/c^(7/8)
Time = 0.65 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.04 \[ \int \frac {x^{5/2}}{a+c x^4} \, dx=-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{4 \sqrt [8]{a} c^{7/8}} \] Input:
Integrate[x^(5/2)/(a + c*x^4),x]
Output:
-1/4*(Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[1 - 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x)) /(a^(1/8)*c^(1/8)*Sqrt[x])] + Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[1 + 1/Sqrt[2] ]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sqrt[x])] + Sqrt[2 + Sqrt[2]]*Ar cTanh[(Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^(1/4)*x)] + Sqrt[2 - Sqrt[2]]*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt[-((-2 + Sqrt[2])*x)])/(a^ (1/4) + c^(1/4)*x)])/(a^(1/8)*c^(7/8))
Time = 0.93 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.42, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.867, Rules used = {851, 830, 826, 827, 218, 221, 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.
| \(\displaystyle \int \frac {x^{5/2}}{a+c x^4} \, dx\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle 2 \int \frac {x^3}{c x^4+a}d\sqrt {x}\) |
| \(\Big \downarrow \) 830 |
| \(\displaystyle 2 \left (\frac {\int \frac {x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt {c}}-\frac {\int \frac {x}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {c}}\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\int \frac {x}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {c}}\right )\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {1}{\sqrt [4]{c} x+\sqrt [4]{-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{c} \left (x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}\right )}{\sqrt [8]{c} \left (x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [8]{-a}-2 \sqrt [8]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}+\sqrt [8]{-a}}{x+\frac {\sqrt {2} \sqrt [8]{-a} \sqrt {x}}{\sqrt [8]{c}}+\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}}d\sqrt {x}}{2 \sqrt [8]{-a} \sqrt [4]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {c}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {c}}\right )\) |
Input:
Int[x^(5/2)/(a + c*x^4),x]
Output:
2*(-1/2*(-1/2*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/((-a)^(1/8)*c^(3/8)) + ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(1/8)*c^(3/8)))/Sqrt[c] + (( -(ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(Sqrt[2]*(-a)^(1/8)*c^( 1/8))) + ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(Sqrt[2]*(-a)^(1 /8)*c^(1/8)))/(2*c^(1/4)) - (-1/2*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1 /8)*Sqrt[x] + c^(1/4)*x]/(Sqrt[2]*(-a)^(1/8)*c^(1/8)) + Log[(-a)^(1/4) + S qrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x]/(2*Sqrt[2]*(-a)^(1/8)*c^(1/ 8)))/(2*c^(1/4)))/(2*Sqrt[c]))
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt [-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[x^(m - n/2)/( r + s*x^(n/2)), x], x] - Simp[s/(2*b) Int[x^(m - n/2)/(r - s*x^(n/2)), x] , x]] /; FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LeQ[n/2, m] && L tQ[m, n] && !GtQ[a/b, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.46 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.13
| method | result | size |
| derivativedivides | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}}{4 c}\) | \(29\) |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}}}{4 c}\) | \(29\) |
Input:
int(x^(5/2)/(c*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/4/c*sum(1/_R*ln(x^(1/2)-_R),_R=RootOf(_Z^8*c+a))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.26 \[ \int \frac {x^{5/2}}{a+c x^4} \, dx=-\left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a c^{6} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a c^{6} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) - \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a c^{6} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a c^{6} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + \frac {1}{4} \, \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \log \left (a c^{6} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) - \frac {1}{4} i \, \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \log \left (i \, a c^{6} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) + \frac {1}{4} i \, \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \log \left (-i \, a c^{6} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) - \frac {1}{4} \, \left (-\frac {1}{a c^{7}}\right )^{\frac {1}{8}} \log \left (-a c^{6} \left (-\frac {1}{a c^{7}}\right )^{\frac {7}{8}} + \sqrt {x}\right ) \] Input:
integrate(x^(5/2)/(c*x^4+a),x, algorithm="fricas")
Output:
-(1/8*I - 1/8)*sqrt(2)*(-1/(a*c^7))^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a*c^6* (-1/(a*c^7))^(7/8) + sqrt(x)) + (1/8*I + 1/8)*sqrt(2)*(-1/(a*c^7))^(1/8)*l og(-(1/2*I - 1/2)*sqrt(2)*a*c^6*(-1/(a*c^7))^(7/8) + sqrt(x)) - (1/8*I + 1 /8)*sqrt(2)*(-1/(a*c^7))^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a*c^6*(-1/(a*c^7) )^(7/8) + sqrt(x)) + (1/8*I - 1/8)*sqrt(2)*(-1/(a*c^7))^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*a*c^6*(-1/(a*c^7))^(7/8) + sqrt(x)) + 1/4*(-1/(a*c^7))^(1/8 )*log(a*c^6*(-1/(a*c^7))^(7/8) + sqrt(x)) - 1/4*I*(-1/(a*c^7))^(1/8)*log(I *a*c^6*(-1/(a*c^7))^(7/8) + sqrt(x)) + 1/4*I*(-1/(a*c^7))^(1/8)*log(-I*a*c ^6*(-1/(a*c^7))^(7/8) + sqrt(x)) - 1/4*(-1/(a*c^7))^(1/8)*log(-a*c^6*(-1/( a*c^7))^(7/8) + sqrt(x))
Time = 18.88 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.27 \[ \int \frac {x^{5/2}}{a+c x^4} \, dx=\begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge c = 0 \\\frac {2 x^{\frac {7}{2}}}{7 a} & \text {for}\: c = 0 \\- \frac {2}{c \sqrt {x}} & \text {for}\: a = 0 \\\frac {\log {\left (\sqrt {x} - \sqrt [8]{- \frac {a}{c}} \right )}}{4 c \sqrt [8]{- \frac {a}{c}}} - \frac {\log {\left (\sqrt {x} + \sqrt [8]{- \frac {a}{c}} \right )}}{4 c \sqrt [8]{- \frac {a}{c}}} + \frac {\sqrt {2} \log {\left (- 4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 c \sqrt [8]{- \frac {a}{c}}} - \frac {\sqrt {2} \log {\left (4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 c \sqrt [8]{- \frac {a}{c}}} + \frac {\operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} \right )}}{2 c \sqrt [8]{- \frac {a}{c}}} + \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} - 1 \right )}}{4 c \sqrt [8]{- \frac {a}{c}}} + \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} + 1 \right )}}{4 c \sqrt [8]{- \frac {a}{c}}} & \text {otherwise} \end {cases} \] Input:
integrate(x**(5/2)/(c*x**4+a),x)
Output:
Piecewise((zoo/sqrt(x), Eq(a, 0) & Eq(c, 0)), (2*x**(7/2)/(7*a), Eq(c, 0)) , (-2/(c*sqrt(x)), Eq(a, 0)), (log(sqrt(x) - (-a/c)**(1/8))/(4*c*(-a/c)**( 1/8)) - log(sqrt(x) + (-a/c)**(1/8))/(4*c*(-a/c)**(1/8)) + sqrt(2)*log(-4* sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/(8*c*(-a/c)**(1/8)) - sqrt(2)*log(4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/(8 *c*(-a/c)**(1/8)) + atan(sqrt(x)/(-a/c)**(1/8))/(2*c*(-a/c)**(1/8)) + sqrt (2)*atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) - 1)/(4*c*(-a/c)**(1/8)) + sqrt(2)* atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) + 1)/(4*c*(-a/c)**(1/8)), True))
\[ \int \frac {x^{5/2}}{a+c x^4} \, dx=\int { \frac {x^{\frac {5}{2}}}{c x^{4} + a} \,d x } \] Input:
integrate(x^(5/2)/(c*x^4+a),x, algorithm="maxima")
Output:
integrate(x^(5/2)/(c*x^4 + a), x)
Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (151) = 302\).
Time = 0.22 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.95 \[ \int \frac {x^{5/2}}{a+c x^4} \, dx=\frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {7}{8}} \log \left (-\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} \] Input:
integrate(x^(5/2)/(c*x^4+a),x, algorithm="giac")
Output:
1/2*(a/c)^(7/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt( sqrt(2) + 2)*(a/c)^(1/8)))/(a*sqrt(-2*sqrt(2) + 4)) + 1/2*(a/c)^(7/8)*arct an(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^ (1/8)))/(a*sqrt(-2*sqrt(2) + 4)) + 1/2*(a/c)^(7/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a*sqrt(2*sq rt(2) + 4)) + 1/2*(a/c)^(7/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*s qrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a*sqrt(2*sqrt(2) + 4)) - 1/4*(a /c)^(7/8)*log(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a* sqrt(-2*sqrt(2) + 4)) + 1/4*(a/c)^(7/8)*log(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/ c)^(1/8) + x + (a/c)^(1/4))/(a*sqrt(-2*sqrt(2) + 4)) - 1/4*(a/c)^(7/8)*log (sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*sqrt(2*sqrt( 2) + 4)) + 1/4*(a/c)^(7/8)*log(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*sqrt(2*sqrt(2) + 4))
Time = 0.17 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.52 \[ \int \frac {x^{5/2}}{a+c x^4} \, dx=\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{2\,{\left (-a\right )}^{1/8}\,c^{7/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{1/8}\,c^{7/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}\,c^{7/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}\,c^{7/8}} \] Input:
int(x^(5/2)/(a + c*x^4),x)
Output:
atan((c^(1/8)*x^(1/2))/(-a)^(1/8))/(2*(-a)^(1/8)*c^(7/8)) + (atan((c^(1/8) *x^(1/2)*1i)/(-a)^(1/8))*1i)/(2*(-a)^(1/8)*c^(7/8)) + (2^(1/2)*atan((2^(1/ 2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)^(1/8))*(1/4 - 1i/4))/((-a)^(1/8)*c^( 7/8)) + (2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 + 1i/2))/(-a)^(1/8))*( 1/4 + 1i/4))/((-a)^(1/8)*c^(7/8))
Time = 0.25 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.41 \[ \int \frac {x^{5/2}}{a+c x^4} \, dx=\frac {-2 \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}-2 \sqrt {x}\, c^{\frac {1}{4}}}{c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}}\right )+2 \sqrt {\sqrt {2}+2}\, \mathit {atan} \left (\frac {c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}+2 \sqrt {x}\, c^{\frac {1}{4}}}{c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}}\right )-2 \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}-2 \sqrt {x}\, c^{\frac {1}{4}}}{c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}}\right )+2 \sqrt {-\sqrt {2}+2}\, \mathit {atan} \left (\frac {c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}+2 \sqrt {x}\, c^{\frac {1}{4}}}{c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}}\right )+\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}+a^{\frac {1}{4}}+c^{\frac {1}{4}} x \right )-\sqrt {-\sqrt {2}+2}\, \mathrm {log}\left (\sqrt {x}\, c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {-\sqrt {2}+2}+a^{\frac {1}{4}}+c^{\frac {1}{4}} x \right )+\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}+a^{\frac {1}{4}}+c^{\frac {1}{4}} x \right )-\sqrt {\sqrt {2}+2}\, \mathrm {log}\left (\sqrt {x}\, c^{\frac {1}{8}} a^{\frac {1}{8}} \sqrt {\sqrt {2}+2}+a^{\frac {1}{4}}+c^{\frac {1}{4}} x \right )}{8 c^{\frac {7}{8}} a^{\frac {1}{8}}} \] Input:
int(x^(5/2)/(c*x^4+a),x)
Output:
(c**(1/8)*a**(7/8)*( - 2*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2))) + 2*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + 2*sq rt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2))) - 2*sqrt( - sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c** (1/8)*a**(1/8)*sqrt( - sqrt(2) + 2))) + 2*sqrt( - sqrt(2) + 2)*atan((c**(1 /8)*a**(1/8)*sqrt(sqrt(2) + 2) + 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sq rt( - sqrt(2) + 2))) + sqrt( - sqrt(2) + 2)*log( - sqrt(x)*c**(1/8)*a**(1/ 8)*sqrt( - sqrt(2) + 2) + a**(1/4) + c**(1/4)*x) - sqrt( - sqrt(2) + 2)*lo g(sqrt(x)*c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + a**(1/4) + c**(1/4)*x) + sqrt(sqrt(2) + 2)*log( - sqrt(x)*c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) + a **(1/4) + c**(1/4)*x) - sqrt(sqrt(2) + 2)*log(sqrt(x)*c**(1/8)*a**(1/8)*sq rt(sqrt(2) + 2) + a**(1/4) + c**(1/4)*x)))/(8*a*c)