Integrand size = 24, antiderivative size = 478 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {2 x^{3/2}}{3 b d}-\frac {a^{7/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{7/4} (b c-a d)}+\frac {a^{7/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{7/4} (b c-a d)}+\frac {c^{7/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{7/4} (b c-a d)}-\frac {c^{7/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{7/4} (b c-a d)}+\frac {a^{7/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{7/4} (b c-a d)}-\frac {a^{7/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{7/4} (b c-a d)}-\frac {c^{7/4} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{7/4} (b c-a d)}+\frac {c^{7/4} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{7/4} (b c-a d)} \]
2/3*x^(3/2)/b/d-1/2*a^(7/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/b^(7 /4)/(-a*d+b*c)*2^(1/2)+1/2*a^(7/4)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4 ))/b^(7/4)/(-a*d+b*c)*2^(1/2)+1/2*c^(7/4)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2) /c^(1/4))/d^(7/4)/(-a*d+b*c)*2^(1/2)-1/2*c^(7/4)*arctan(1+d^(1/4)*2^(1/2)* x^(1/2)/c^(1/4))/d^(7/4)/(-a*d+b*c)*2^(1/2)+1/4*a^(7/4)*ln(a^(1/2)+x*b^(1/ 2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/b^(7/4)/(-a*d+b*c)*2^(1/2)-1/4*a^(7/4) *ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/b^(7/4)/(-a*d+b*c)* 2^(1/2)-1/4*c^(7/4)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/ d^(7/4)/(-a*d+b*c)*2^(1/2)+1/4*c^(7/4)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4 )*2^(1/2)*x^(1/2))/d^(7/4)/(-a*d+b*c)*2^(1/2)
Time = 0.49 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.52 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {-\frac {4 a x^{3/2}}{b}+\frac {4 c x^{3/2}}{d}-\frac {3 \sqrt {2} a^{7/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{b^{7/4}}+\frac {3 \sqrt {2} c^{7/4} \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{d^{7/4}}-\frac {3 \sqrt {2} a^{7/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{b^{7/4}}+\frac {3 \sqrt {2} c^{7/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{d^{7/4}}}{6 b c-6 a d} \]
((-4*a*x^(3/2))/b + (4*c*x^(3/2))/d - (3*Sqrt[2]*a^(7/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/b^(7/4) + (3*Sqrt[2]*c^(7/ 4)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/d^(7/4 ) - (3*Sqrt[2]*a^(7/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/b^(7/4) + (3*Sqrt[2]*c^(7/4)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/ 4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/d^(7/4))/(6*b*c - 6*a*d)
Time = 0.67 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {368, 979, 27, 1054, 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.
| \(\displaystyle \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {x^5}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}\) |
| \(\Big \downarrow \) 979 |
| \(\displaystyle 2 \left (\frac {x^{3/2}}{3 b d}-\frac {\int \frac {3 x \left ((b c+a d) x^2+a c\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{3 b d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {x^{3/2}}{3 b d}-\frac {\int \frac {x \left ((b c+a d) x^2+a c\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{b d}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 \left (\frac {x^{3/2}}{3 b d}-\frac {\int \left (\frac {d x a^2}{(a d-b c) \left (b x^2+a\right )}+\frac {b c^2 x}{(b c-a d) \left (d x^2+c\right )}\right )d\sqrt {x}}{b d}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {x^{3/2}}{3 b d}-\frac {\frac {a^{7/4} d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{2 \sqrt {2} b^{3/4} (b c-a d)}-\frac {a^{7/4} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} b^{3/4} (b c-a d)}-\frac {a^{7/4} d \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} b^{3/4} (b c-a d)}+\frac {a^{7/4} d \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{4 \sqrt {2} b^{3/4} (b c-a d)}-\frac {b c^{7/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{2 \sqrt {2} d^{3/4} (b c-a d)}+\frac {b c^{7/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} d^{3/4} (b c-a d)}+\frac {b c^{7/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} d^{3/4} (b c-a d)}-\frac {b c^{7/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} d^{3/4} (b c-a d)}}{b d}\right )\) |
2*(x^(3/2)/(3*b*d) - ((a^(7/4)*d*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1 /4)])/(2*Sqrt[2]*b^(3/4)*(b*c - a*d)) - (a^(7/4)*d*ArcTan[1 + (Sqrt[2]*b^( 1/4)*Sqrt[x])/a^(1/4)])/(2*Sqrt[2]*b^(3/4)*(b*c - a*d)) - (b*c^(7/4)*ArcTa n[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(2*Sqrt[2]*d^(3/4)*(b*c - a*d)) + (b*c^(7/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(2*Sqrt[2]*d^( 3/4)*(b*c - a*d)) - (a^(7/4)*d*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[ x] + Sqrt[b]*x])/(4*Sqrt[2]*b^(3/4)*(b*c - a*d)) + (a^(7/4)*d*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(4*Sqrt[2]*b^(3/4)*(b*c - a*d)) + (b*c^(7/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d] *x])/(4*Sqrt[2]*d^(3/4)*(b*c - a*d)) - (b*c^(7/4)*Log[Sqrt[c] + Sqrt[2]*c^ (1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(4*Sqrt[2]*d^(3/4)*(b*c - a*d)))/(b*d) )
3.5.61.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q) + 1))), x] - Simp[e^(2*n)/(b*d *(m + n*(p + q) + 1)) Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Sim p[a*c*(m - 2*n + 1) + (a*d*(m + n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x ^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && I GtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x ]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Time = 2.87 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.52
| method | result | size |
| derivativedivides | \(\frac {2 x^{\frac {3}{2}}}{3 b d}+\frac {c^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {a^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(249\) |
| default | \(\frac {2 x^{\frac {3}{2}}}{3 b d}+\frac {c^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {a^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(249\) |
| risch | \(\frac {2 x^{\frac {3}{2}}}{3 b d}-\frac {\frac {a^{2} d \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {b \,c^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{b d}\) | \(260\) |
2/3*x^(3/2)/b/d+1/4*c^2/(a*d-b*c)/d^2/(c/d)^(1/4)*2^(1/2)*(ln((x-(c/d)^(1/ 4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2) ))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^ (1/2)-1))-1/4*a^2/(a*d-b*c)/b^2/(a/b)^(1/4)*2^(1/2)*(ln((x-(a/b)^(1/4)*x^( 1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*a rctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)- 1))
Result contains complex when optimal does not.
Time = 1.35 (sec) , antiderivative size = 1472, normalized size of antiderivative = 3.08 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Too large to display} \]
1/6*(3*(-a^7/(b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a^3*b^8*c* d^3 + a^4*b^7*d^4))^(1/4)*b*d*log(a^5*sqrt(x) + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*(-a^7/(b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b ^9*c^2*d^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d^4))^(3/4)) - 3*(-a^7/(b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d^4))^(1/4) *b*d*log(a^5*sqrt(x) - (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^ 5*d^3)*(-a^7/(b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a^3*b^8*c* d^3 + a^4*b^7*d^4))^(3/4)) + 3*I*(-a^7/(b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2* b^9*c^2*d^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d^4))^(1/4)*b*d*log(a^5*sqrt(x) - (I*b^8*c^3 - 3*I*a*b^7*c^2*d + 3*I*a^2*b^6*c*d^2 - I*a^3*b^5*d^3)*(-a^7/(b ^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d ^4))^(3/4)) - 3*I*(-a^7/(b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4 *a^3*b^8*c*d^3 + a^4*b^7*d^4))^(1/4)*b*d*log(a^5*sqrt(x) - (-I*b^8*c^3 + 3 *I*a*b^7*c^2*d - 3*I*a^2*b^6*c*d^2 + I*a^3*b^5*d^3)*(-a^7/(b^11*c^4 - 4*a* b^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d^4))^(3/4)) - 3*(-c^7/(b^4*c^4*d^7 - 4*a*b^3*c^3*d^8 + 6*a^2*b^2*c^2*d^9 - 4*a^3*b*c*d^1 0 + a^4*d^11))^(1/4)*b*d*log(c^5*sqrt(x) + (b^3*c^3*d^5 - 3*a*b^2*c^2*d^6 + 3*a^2*b*c*d^7 - a^3*d^8)*(-c^7/(b^4*c^4*d^7 - 4*a*b^3*c^3*d^8 + 6*a^2*b^ 2*c^2*d^9 - 4*a^3*b*c*d^10 + a^4*d^11))^(3/4)) + 3*(-c^7/(b^4*c^4*d^7 - 4* a*b^3*c^3*d^8 + 6*a^2*b^2*c^2*d^9 - 4*a^3*b*c*d^10 + a^4*d^11))^(1/4)*b...
Timed out. \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.82 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {a^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (b^{2} c - a b d\right )}} - \frac {c^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{4 \, {\left (b c d - a d^{2}\right )}} + \frac {2 \, x^{\frac {3}{2}}}{3 \, b d} \]
1/4*a^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b) *sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt( 2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt( sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^ (1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*l og(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4 )))/(b^2*c - a*b*d) - 1/4*c^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/ 4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt( d))*sqrt(d)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2* sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4 )*d^(3/4)) + sqrt(2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sq rt(c))/(c^(1/4)*d^(3/4)))/(b*c*d - a*d^2) + 2/3*x^(3/2)/(b*d)
Time = 0.36 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.00 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\left (a b^{3}\right )^{\frac {3}{4}} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b^{5} c - \sqrt {2} a b^{4} d} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b^{5} c - \sqrt {2} a b^{4} d} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} c \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b c d^{4} - \sqrt {2} a d^{5}} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} c \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b c d^{4} - \sqrt {2} a d^{5}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} a \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{5} c - \sqrt {2} a b^{4} d\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} a \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{5} c - \sqrt {2} a b^{4} d\right )}} + \frac {\left (c d^{3}\right )^{\frac {3}{4}} c \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c d^{4} - \sqrt {2} a d^{5}\right )}} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} c \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c d^{4} - \sqrt {2} a d^{5}\right )}} + \frac {2 \, x^{\frac {3}{2}}}{3 \, b d} \]
(a*b^3)^(3/4)*a*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b) ^(1/4))/(sqrt(2)*b^5*c - sqrt(2)*a*b^4*d) + (a*b^3)^(3/4)*a*arctan(-1/2*sq rt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^5*c - sqrt (2)*a*b^4*d) - (c*d^3)^(3/4)*c*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2 *sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b*c*d^4 - sqrt(2)*a*d^5) - (c*d^3)^(3/4)*c *arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt( 2)*b*c*d^4 - sqrt(2)*a*d^5) - 1/2*(a*b^3)^(3/4)*a*log(sqrt(2)*sqrt(x)*(a/b )^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^5*c - sqrt(2)*a*b^4*d) + 1/2*(a*b^3)^( 3/4)*a*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^5*c - sqrt(2)*a*b^4*d) + 1/2*(c*d^3)^(3/4)*c*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b*c*d^4 - sqrt(2)*a*d^5) - 1/2*(c*d^3)^(3/4)*c*log( -sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b*c*d^4 - sqrt(2)*a *d^5) + 2/3*x^(3/2)/(b*d)
Time = 7.35 (sec) , antiderivative size = 7892, normalized size of antiderivative = 16.51 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Too large to display} \]
atan(((-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2* c^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*((-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 6 4*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(3/4)*((128*(16*a ^3*b^10*c^10*d^3 - 48*a^4*b^9*c^9*d^4 + 48*a^5*b^8*c^8*d^5 - 16*a^6*b^7*c^ 7*d^6 - 16*a^7*b^6*c^6*d^7 + 48*a^8*b^5*c^5*d^8 - 48*a^9*b^4*c^4*d^9 + 16* a^10*b^3*c^3*d^10))/(b^3*d^3) - (256*x^(1/2)*(-c^7/(16*a^4*d^11 + 16*b^4*c ^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*( 16*a^3*b^11*c^9*d^5 - 64*a^4*b^10*c^8*d^6 + 112*a^5*b^9*c^7*d^7 - 128*a^6* b^8*c^6*d^8 + 112*a^7*b^7*c^5*d^9 - 64*a^8*b^6*c^4*d^10 + 16*a^9*b^5*c^3*d ^11))/(b^3*d^3)) - (256*x^(1/2)*(a^5*b^5*c^10 + a^10*c^5*d^5))/(b^3*d^3))* 1i - (-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c ^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*((-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64 *a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(3/4)*((128*(16*a^ 3*b^10*c^10*d^3 - 48*a^4*b^9*c^9*d^4 + 48*a^5*b^8*c^8*d^5 - 16*a^6*b^7*c^7 *d^6 - 16*a^7*b^6*c^6*d^7 + 48*a^8*b^5*c^5*d^8 - 48*a^9*b^4*c^4*d^9 + 16*a ^10*b^3*c^3*d^10))/(b^3*d^3) + (256*x^(1/2)*(-c^7/(16*a^4*d^11 + 16*b^4*c^ 4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*(1 6*a^3*b^11*c^9*d^5 - 64*a^4*b^10*c^8*d^6 + 112*a^5*b^9*c^7*d^7 - 128*a^6*b ^8*c^6*d^8 + 112*a^7*b^7*c^5*d^9 - 64*a^8*b^6*c^4*d^10 + 16*a^9*b^5*c^3*d^ 11))/(b^3*d^3)) + (256*x^(1/2)*(a^5*b^5*c^10 + a^10*c^5*d^5))/(b^3*d^3)...