Integrand size = 22, antiderivative size = 421 \[ \int \frac {x^2}{\sqrt {c+d x} \left (a+b x^2\right )} \, dx=\frac {2 \sqrt {c+d x}}{b d}+\frac {a d \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{\sqrt {2} b^{5/4} \sqrt {b c^2+a d^2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}-\frac {a d \arctan \left (\frac {\sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}\right )}{\sqrt {2} b^{5/4} \sqrt {b c^2+a d^2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a d^2}}}-\frac {a d \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}} \sqrt {c+d x}}{\sqrt {b c^2+a d^2}+\sqrt {b} (c+d x)}\right )}{\sqrt {2} b^{5/4} \sqrt {b c^2+a d^2} \sqrt {\sqrt {b} c+\sqrt {b c^2+a d^2}}} \] Output:
2*(d*x+c)^(1/2)/b/d+1/2*a*d*arctan(((b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)- 2^(1/2)*b^(1/4)*(d*x+c)^(1/2))/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2))*2^( 1/2)/b^(5/4)/(a*d^2+b*c^2)^(1/2)/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)-1/ 2*a*d*arctan(((b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)+2^(1/2)*b^(1/4)*(d*x+c )^(1/2))/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2))*2^(1/2)/b^(5/4)/(a*d^2+b* c^2)^(1/2)/(-b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)-1/2*a*d*arctanh(2^(1/2)* b^(1/4)*(b^(1/2)*c+(a*d^2+b*c^2)^(1/2))^(1/2)*(d*x+c)^(1/2)/((a*d^2+b*c^2) ^(1/2)+b^(1/2)*(d*x+c)))*2^(1/2)/b^(5/4)/(a*d^2+b*c^2)^(1/2)/(b^(1/2)*c+(a *d^2+b*c^2)^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.46 \[ \int \frac {x^2}{\sqrt {c+d x} \left (a+b x^2\right )} \, dx=\frac {\frac {2 \sqrt {c+d x}}{d}-\frac {i \sqrt {a} \arctan \left (\frac {\sqrt {-b c-i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c+i \sqrt {a} d}\right )}{\sqrt {-b c-i \sqrt {a} \sqrt {b} d}}+\frac {i \sqrt {a} \arctan \left (\frac {\sqrt {-b c+i \sqrt {a} \sqrt {b} d} \sqrt {c+d x}}{\sqrt {b} c-i \sqrt {a} d}\right )}{\sqrt {-b c+i \sqrt {a} \sqrt {b} d}}}{b} \] Input:
Integrate[x^2/(Sqrt[c + d*x]*(a + b*x^2)),x]
Output:
((2*Sqrt[c + d*x])/d - (I*Sqrt[a]*ArcTan[(Sqrt[-(b*c) - I*Sqrt[a]*Sqrt[b]* d]*Sqrt[c + d*x])/(Sqrt[b]*c + I*Sqrt[a]*d)])/Sqrt[-(b*c) - I*Sqrt[a]*Sqrt [b]*d] + (I*Sqrt[a]*ArcTan[(Sqrt[-(b*c) + I*Sqrt[a]*Sqrt[b]*d]*Sqrt[c + d* x])/(Sqrt[b]*c - I*Sqrt[a]*d)])/Sqrt[-(b*c) + I*Sqrt[a]*Sqrt[b]*d])/b
Time = 1.42 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.37, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {561, 27, 1484, 2009}
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.
| \(\displaystyle \int \frac {x^2}{\left (a+b x^2\right ) \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 561 |
| \(\displaystyle \frac {2 \int \frac {x^2}{\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {d^2 x^2}{\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}d\sqrt {c+d x}}{d^3}\) |
| \(\Big \downarrow \) 1484 |
| \(\displaystyle \frac {2 \int \left (\frac {d^2}{b}-\frac {a d^2}{b \left (\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a\right )}\right )d\sqrt {c+d x}}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (-\frac {a d^4 \text {arctanh}\left (\frac {\sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a d^2+b c^2}}}\right )}{2 \sqrt {2} b^{5/4} \sqrt {a d^2+b c^2} \sqrt {\sqrt {b} c-\sqrt {a d^2+b c^2}}}+\frac {a d^4 \text {arctanh}\left (\frac {\sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}+\sqrt {2} \sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c-\sqrt {a d^2+b c^2}}}\right )}{2 \sqrt {2} b^{5/4} \sqrt {a d^2+b c^2} \sqrt {\sqrt {b} c-\sqrt {a d^2+b c^2}}}+\frac {a d^4 \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt {c+d x} \sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}+\sqrt {a d^2+b c^2}+\sqrt {b} (c+d x)\right )}{4 \sqrt {2} b^{5/4} \sqrt {a d^2+b c^2} \sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}}-\frac {a d^4 \log \left (\sqrt {2} \sqrt [4]{b} \sqrt {c+d x} \sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}+\sqrt {a d^2+b c^2}+\sqrt {b} (c+d x)\right )}{4 \sqrt {2} b^{5/4} \sqrt {a d^2+b c^2} \sqrt {\sqrt {a d^2+b c^2}+\sqrt {b} c}}+\frac {d^2 \sqrt {c+d x}}{b}\right )}{d^3}\) |
Input:
Int[x^2/(Sqrt[c + d*x]*(a + b*x^2)),x]
Output:
(2*((d^2*Sqrt[c + d*x])/b - (a*d^4*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]] - Sqrt[2]*b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d ^2]]])/(2*Sqrt[2]*b^(5/4)*Sqrt[b*c^2 + a*d^2]*Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]]) + (a*d^4*ArcTanh[(Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]] + Sqrt[ 2]*b^(1/4)*Sqrt[c + d*x])/Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]]])/(2*Sqrt[ 2]*b^(5/4)*Sqrt[b*c^2 + a*d^2]*Sqrt[Sqrt[b]*c - Sqrt[b*c^2 + a*d^2]]) + (a *d^4*Log[Sqrt[b*c^2 + a*d^2] - Sqrt[2]*b^(1/4)*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]*Sqrt[c + d*x] + Sqrt[b]*(c + d*x)])/(4*Sqrt[2]*b^(5/4)*Sqrt[b*c ^2 + a*d^2]*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]) - (a*d^4*Log[Sqrt[b*c^2 + a*d^2] + Sqrt[2]*b^(1/4)*Sqrt[Sqrt[b]*c + Sqrt[b*c^2 + a*d^2]]*Sqrt[c + d*x] + Sqrt[b]*(c + d*x)])/(4*Sqrt[2]*b^(5/4)*Sqrt[b*c^2 + a*d^2]*Sqrt[Sq rt[b]*c + Sqrt[b*c^2 + a*d^2]])))/d^3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k/d Subst[Int[x^(k*(n + 1) - 1)*(-c /d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac tionQ[n] && IntegerQ[p] && IntegerQ[m]
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
Time = 0.72 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.40
| method | result | size |
| pseudoelliptic | \(\frac {\frac {\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \left (-b c +\sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )-\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{4}-\frac {\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}\, \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \left (-b c +\sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{4}+\left (2 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {d x +c}\, \sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}+d^{2} \left (\arctan \left (\frac {-2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )-\arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )\right ) a \right ) b}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}\, \sqrt {a \,d^{2}+b \,c^{2}}\, b^{2} d}\) | \(590\) |
| risch | \(\frac {2 \sqrt {d x +c}}{b d}-\frac {2 a d \left (\frac {\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (2 a \,d^{2} \sqrt {b}-\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}\, a \,d^{2}}+\frac {\frac {\left (\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c -\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )-\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (2 a \,d^{2} \sqrt {b}+\frac {\left (\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c -\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}-\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}\, a \,d^{2}}\right )}{b}\) | \(816\) |
| derivativedivides | \(\frac {\frac {2 \sqrt {d x +c}}{b}-\frac {2 a \,d^{2} \left (\frac {\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (2 a \,d^{2} \sqrt {b}-\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}\, a \,d^{2}}+\frac {\frac {\left (\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c -\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )-\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (2 a \,d^{2} \sqrt {b}+\frac {\left (\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c -\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}-\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}\, a \,d^{2}}\right )}{b}}{d}\) | \(819\) |
| default | \(\frac {\frac {2 \sqrt {d x +c}}{b}-\frac {2 a \,d^{2} \left (\frac {\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )+\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (2 a \,d^{2} \sqrt {b}-\frac {\left (-\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c +\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}+\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}\, a \,d^{2}}+\frac {\frac {\left (\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c -\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \ln \left (\sqrt {b}\, \left (d x +c \right )-\sqrt {d x +c}\, \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}+\sqrt {a \,d^{2}+b \,c^{2}}\right )}{2 \sqrt {b}}+\frac {2 \left (2 a \,d^{2} \sqrt {b}+\frac {\left (\sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\, b c -\sqrt {a b \,d^{2}+b^{2} c^{2}}\, \sqrt {2 \sqrt {a b \,d^{2}+b^{2} c^{2}}+2 b c}\right ) \sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{2 \sqrt {b}}\right ) \arctan \left (\frac {2 \sqrt {b}\, \sqrt {d x +c}-\sqrt {2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}+2 b c}}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}\right )}{\sqrt {4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}-2 \sqrt {\left (a \,d^{2}+b \,c^{2}\right ) b}-2 b c}}}{4 \sqrt {a \,d^{2}+b \,c^{2}}\, \sqrt {b}\, a \,d^{2}}\right )}{b}}{d}\) | \(819\) |
Input:
int(x^2/(d*x+c)^(1/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2)/(a *d^2+b*c^2)^(1/2)*(1/4*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2)*(4*(a*d^2+b *c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2)*(-b*c+((a*d^2+b *c^2)*b)^(1/2))*ln(b^(1/2)*(d*x+c)-(d*x+c)^(1/2)*(2*((a*d^2+b*c^2)*b)^(1/2 )+2*b*c)^(1/2)+(a*d^2+b*c^2)^(1/2))-1/4*(2*((a*d^2+b*c^2)*b)^(1/2)+2*b*c)^ (1/2)*(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b*c)^(1/2 )*(-b*c+((a*d^2+b*c^2)*b)^(1/2))*ln(b^(1/2)*(d*x+c)+(d*x+c)^(1/2)*(2*((a*d ^2+b*c^2)*b)^(1/2)+2*b*c)^(1/2)+(a*d^2+b*c^2)^(1/2))+(2*(a*d^2+b*c^2)^(1/2 )*(d*x+c)^(1/2)*(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2 *b*c)^(1/2)+d^2*(arctan((-2*b^(1/2)*(d*x+c)^(1/2)+(2*((a*d^2+b*c^2)*b)^(1/ 2)+2*b*c)^(1/2))/(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)- 2*b*c)^(1/2))-arctan((2*b^(1/2)*(d*x+c)^(1/2)+(2*((a*d^2+b*c^2)*b)^(1/2)+2 *b*c)^(1/2))/(4*(a*d^2+b*c^2)^(1/2)*b^(1/2)-2*((a*d^2+b*c^2)*b)^(1/2)-2*b* c)^(1/2)))*a)*b)/b^2/d
Leaf count of result is larger than twice the leaf count of optimal. 1056 vs. \(2 (334) = 668\).
Time = 0.10 (sec) , antiderivative size = 1056, normalized size of antiderivative = 2.51 \[ \int \frac {x^2}{\sqrt {c+d x} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate(x^2/(d*x+c)^(1/2)/(b*x^2+a),x, algorithm="fricas")
Output:
-1/2*(b*d*sqrt(-(a*c + (b^3*c^2 + a*b^2*d^2)*sqrt(-a^3*d^2/(b^7*c^4 + 2*a* b^6*c^2*d^2 + a^2*b^5*d^4)))/(b^3*c^2 + a*b^2*d^2))*log(sqrt(d*x + c)*a^2* d + (a^2*b*d^2 + (b^5*c^3 + a*b^4*c*d^2)*sqrt(-a^3*d^2/(b^7*c^4 + 2*a*b^6* c^2*d^2 + a^2*b^5*d^4)))*sqrt(-(a*c + (b^3*c^2 + a*b^2*d^2)*sqrt(-a^3*d^2/ (b^7*c^4 + 2*a*b^6*c^2*d^2 + a^2*b^5*d^4)))/(b^3*c^2 + a*b^2*d^2))) - b*d* sqrt(-(a*c + (b^3*c^2 + a*b^2*d^2)*sqrt(-a^3*d^2/(b^7*c^4 + 2*a*b^6*c^2*d^ 2 + a^2*b^5*d^4)))/(b^3*c^2 + a*b^2*d^2))*log(sqrt(d*x + c)*a^2*d - (a^2*b *d^2 + (b^5*c^3 + a*b^4*c*d^2)*sqrt(-a^3*d^2/(b^7*c^4 + 2*a*b^6*c^2*d^2 + a^2*b^5*d^4)))*sqrt(-(a*c + (b^3*c^2 + a*b^2*d^2)*sqrt(-a^3*d^2/(b^7*c^4 + 2*a*b^6*c^2*d^2 + a^2*b^5*d^4)))/(b^3*c^2 + a*b^2*d^2))) + b*d*sqrt(-(a*c - (b^3*c^2 + a*b^2*d^2)*sqrt(-a^3*d^2/(b^7*c^4 + 2*a*b^6*c^2*d^2 + a^2*b^ 5*d^4)))/(b^3*c^2 + a*b^2*d^2))*log(sqrt(d*x + c)*a^2*d + (a^2*b*d^2 - (b^ 5*c^3 + a*b^4*c*d^2)*sqrt(-a^3*d^2/(b^7*c^4 + 2*a*b^6*c^2*d^2 + a^2*b^5*d^ 4)))*sqrt(-(a*c - (b^3*c^2 + a*b^2*d^2)*sqrt(-a^3*d^2/(b^7*c^4 + 2*a*b^6*c ^2*d^2 + a^2*b^5*d^4)))/(b^3*c^2 + a*b^2*d^2))) - b*d*sqrt(-(a*c - (b^3*c^ 2 + a*b^2*d^2)*sqrt(-a^3*d^2/(b^7*c^4 + 2*a*b^6*c^2*d^2 + a^2*b^5*d^4)))/( b^3*c^2 + a*b^2*d^2))*log(sqrt(d*x + c)*a^2*d - (a^2*b*d^2 - (b^5*c^3 + a* b^4*c*d^2)*sqrt(-a^3*d^2/(b^7*c^4 + 2*a*b^6*c^2*d^2 + a^2*b^5*d^4)))*sqrt( -(a*c - (b^3*c^2 + a*b^2*d^2)*sqrt(-a^3*d^2/(b^7*c^4 + 2*a*b^6*c^2*d^2 + a ^2*b^5*d^4)))/(b^3*c^2 + a*b^2*d^2))) - 4*sqrt(d*x + c))/(b*d)
\[ \int \frac {x^2}{\sqrt {c+d x} \left (a+b x^2\right )} \, dx=\int \frac {x^{2}}{\left (a + b x^{2}\right ) \sqrt {c + d x}}\, dx \] Input:
integrate(x**2/(d*x+c)**(1/2)/(b*x**2+a),x)
Output:
Integral(x**2/((a + b*x**2)*sqrt(c + d*x)), x)
\[ \int \frac {x^2}{\sqrt {c+d x} \left (a+b x^2\right )} \, dx=\int { \frac {x^{2}}{{\left (b x^{2} + a\right )} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^2/(d*x+c)^(1/2)/(b*x^2+a),x, algorithm="maxima")
Output:
integrate(x^2/((b*x^2 + a)*sqrt(d*x + c)), x)
Time = 0.16 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.59 \[ \int \frac {x^2}{\sqrt {c+d x} \left (a+b x^2\right )} \, dx=\frac {\frac {2 \, \sqrt {d x + c} d}{b} + \frac {{\left (a d^{3} {\left | b \right |} {\left | d \right |} + \sqrt {-a b} b c d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{2} c + \sqrt {b^{4} c^{2} - {\left (b^{2} c^{2} + a b d^{2}\right )} b^{2}}}{b^{2}}}}\right )}{{\left (b^{2} c - \sqrt {-a b} b d\right )} \sqrt {-b^{2} c - \sqrt {-a b} b d} {\left | d \right |}} + \frac {{\left (a d^{3} {\left | b \right |} {\left | d \right |} - \sqrt {-a b} b c d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-\frac {b^{2} c - \sqrt {b^{4} c^{2} - {\left (b^{2} c^{2} + a b d^{2}\right )} b^{2}}}{b^{2}}}}\right )}{{\left (b^{2} c + \sqrt {-a b} b d\right )} \sqrt {-b^{2} c + \sqrt {-a b} b d} {\left | d \right |}}}{d^{2}} \] Input:
integrate(x^2/(d*x+c)^(1/2)/(b*x^2+a),x, algorithm="giac")
Output:
(2*sqrt(d*x + c)*d/b + (a*d^3*abs(b)*abs(d) + sqrt(-a*b)*b*c*d^3)*arctan(s qrt(d*x + c)/sqrt(-(b^2*c + sqrt(b^4*c^2 - (b^2*c^2 + a*b*d^2)*b^2))/b^2)) /((b^2*c - sqrt(-a*b)*b*d)*sqrt(-b^2*c - sqrt(-a*b)*b*d)*abs(d)) + (a*d^3* abs(b)*abs(d) - sqrt(-a*b)*b*c*d^3)*arctan(sqrt(d*x + c)/sqrt(-(b^2*c - sq rt(b^4*c^2 - (b^2*c^2 + a*b*d^2)*b^2))/b^2))/((b^2*c + sqrt(-a*b)*b*d)*sqr t(-b^2*c + sqrt(-a*b)*b*d)*abs(d)))/d^2
Time = 0.43 (sec) , antiderivative size = 1341, normalized size of antiderivative = 3.19 \[ \int \frac {x^2}{\sqrt {c+d x} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
int(x^2/((a + b*x^2)*(c + d*x)^(1/2)),x)
Output:
(2*(c + d*x)^(1/2))/(b*d) - 2*atanh((32*a^2*b*d^2*((d*(-a^3*b^5)^(1/2))/(4 *(b^6*c^2 + a*b^5*d^2)) - (a*b^3*c)/(4*(b^6*c^2 + a*b^5*d^2)))^(1/2)*(c + d*x)^(1/2))/((16*a^2*b^2*d^4*(-a^3*b^5)^(1/2))/(b^6*c^2 + a*b^5*d^2) - (16 *a^3*b^5*c*d^3)/(b^6*c^2 + a*b^5*d^2)) + (32*a^2*b^7*c^2*d^2*((d*(-a^3*b^5 )^(1/2))/(4*(b^6*c^2 + a*b^5*d^2)) - (a*b^3*c)/(4*(b^6*c^2 + a*b^5*d^2)))^ (1/2)*(c + d*x)^(1/2))/((16*a^3*b^11*c^3*d^3)/(b^6*c^2 + a*b^5*d^2) - (16* a^3*b^7*d^6*(-a^3*b^5)^(1/2))/(b^6*c^2 + a*b^5*d^2) + (16*a^4*b^10*c*d^5)/ (b^6*c^2 + a*b^5*d^2) - (16*a^2*b^8*c^2*d^4*(-a^3*b^5)^(1/2))/(b^6*c^2 + a *b^5*d^2)) - (32*a*b^4*c*d^3*((d*(-a^3*b^5)^(1/2))/(4*(b^6*c^2 + a*b^5*d^2 )) - (a*b^3*c)/(4*(b^6*c^2 + a*b^5*d^2)))^(1/2)*(-a^3*b^5)^(1/2)*(c + d*x) ^(1/2))/((16*a^3*b^11*c^3*d^3)/(b^6*c^2 + a*b^5*d^2) - (16*a^3*b^7*d^6*(-a ^3*b^5)^(1/2))/(b^6*c^2 + a*b^5*d^2) + (16*a^4*b^10*c*d^5)/(b^6*c^2 + a*b^ 5*d^2) - (16*a^2*b^8*c^2*d^4*(-a^3*b^5)^(1/2))/(b^6*c^2 + a*b^5*d^2)))*((d *(-a^3*b^5)^(1/2) - a*b^3*c)/(4*(b^6*c^2 + a*b^5*d^2)))^(1/2) - 2*atanh((3 2*a^2*b^7*c^2*d^2*(- (d*(-a^3*b^5)^(1/2))/(4*(b^6*c^2 + a*b^5*d^2)) - (a*b ^3*c)/(4*(b^6*c^2 + a*b^5*d^2)))^(1/2)*(c + d*x)^(1/2))/((16*a^3*b^11*c^3* d^3)/(b^6*c^2 + a*b^5*d^2) + (16*a^3*b^7*d^6*(-a^3*b^5)^(1/2))/(b^6*c^2 + a*b^5*d^2) + (16*a^4*b^10*c*d^5)/(b^6*c^2 + a*b^5*d^2) + (16*a^2*b^8*c^2*d ^4*(-a^3*b^5)^(1/2))/(b^6*c^2 + a*b^5*d^2)) - (32*a^2*b*d^2*(- (d*(-a^3*b^ 5)^(1/2))/(4*(b^6*c^2 + a*b^5*d^2)) - (a*b^3*c)/(4*(b^6*c^2 + a*b^5*d^2...
Time = 0.20 (sec) , antiderivative size = 1137, normalized size of antiderivative = 2.70 \[ \int \frac {x^2}{\sqrt {c+d x} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:
int(x^2/(d*x+c)^(1/2)/(b*x^2+a),x)
Output:
(2*sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2) *atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) - 2*sqrt(b)*sqrt( c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*b*c + 2*sqr t(b)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(sqrt(b)* sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) - 2*sqrt(b)*sqrt(c + d*x))/(sqrt(sqrt (b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*a*d**2 + 2*sqrt(b)*sqrt(sqrt(b) *sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c **2) + b*c)*sqrt(2) - 2*sqrt(b)*sqrt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*b*c**2 - 2*sqrt(a*d**2 + b*c**2)*sqrt(sqrt(b)*sq rt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2 ) + b*c)*sqrt(2) + 2*sqrt(b)*sqrt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b* c**2) - b*c)*sqrt(2)))*b*c - 2*sqrt(b)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*atan((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) + 2 *sqrt(b)*sqrt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2) ))*a*d**2 - 2*sqrt(b)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)*at an((sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) + 2*sqrt(b)*sqrt(c + d*x))/(sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) - b*c)*sqrt(2)))*b*c**2 - sqrt( a*d**2 + b*c**2)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2)*log( - sqrt(c + d*x)*sqrt(sqrt(b)*sqrt(a*d**2 + b*c**2) + b*c)*sqrt(2) + sqrt(a*d **2 + b*c**2) + sqrt(b)*c + sqrt(b)*d*x)*b*c + sqrt(a*d**2 + b*c**2)*sq...