Integrand size = 24, antiderivative size = 536 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {c x^{3/2}}{2 d (b c-a d) \left (c+d x^2\right )}-\frac {a^{7/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{3/4} (b c-a d)^2}+\frac {a^{7/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{3/4} (b c-a d)^2}-\frac {c^{3/4} (3 b c-7 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{7/4} (b c-a d)^2}+\frac {c^{3/4} (3 b c-7 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} d^{7/4} (b c-a d)^2}+\frac {a^{7/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{3/4} (b c-a d)^2}-\frac {a^{7/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{3/4} (b c-a d)^2}+\frac {c^{3/4} (3 b c-7 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{7/4} (b c-a d)^2}-\frac {c^{3/4} (3 b c-7 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} d^{7/4} (b c-a d)^2} \]
-1/2*c*x^(3/2)/d/(-a*d+b*c)/(d*x^2+c)-1/2*a^(7/4)*arctan(1-b^(1/4)*2^(1/2) *x^(1/2)/a^(1/4))/b^(3/4)/(-a*d+b*c)^2*2^(1/2)+1/2*a^(7/4)*arctan(1+b^(1/4 )*2^(1/2)*x^(1/2)/a^(1/4))/b^(3/4)/(-a*d+b*c)^2*2^(1/2)-1/8*c^(3/4)*(-7*a* d+3*b*c)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/d^(7/4)/(-a*d+b*c)^2*2^ (1/2)+1/8*c^(3/4)*(-7*a*d+3*b*c)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4)) /d^(7/4)/(-a*d+b*c)^2*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^(3/4)/(-a*d+b*c)^2*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^(3/4)/(-a*d+b*c)^2*2^(1/2)+1/ 16*c^(3/4)*(-7*a*d+3*b*c)*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*2^(1/2)-1/16*c^(3/4)*(-7*a*d+3*b*c)*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*2^(1/2)
Time = 0.85 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.57 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {1}{8} \left (\frac {4 c x^{3/2}}{d (-b c+a d) \left (c+d x^2\right )}-\frac {4 \sqrt {2} a^{7/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{b^{3/4} (b c-a d)^2}-\frac {\sqrt {2} c^{3/4} (3 b c-7 a d) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{d^{7/4} (b c-a d)^2}-\frac {4 \sqrt {2} a^{7/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{b^{3/4} (b c-a d)^2}-\frac {\sqrt {2} c^{3/4} (3 b c-7 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{d^{7/4} (b c-a d)^2}\right ) \]
((4*c*x^(3/2))/(d*(-(b*c) + a*d)*(c + d*x^2)) - (4*Sqrt[2]*a^(7/4)*ArcTan[ (Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(b^(3/4)*(b*c - a*d)^2) - (Sqrt[2]*c^(3/4)*(3*b*c - 7*a*d)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(S qrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(d^(7/4)*(b*c - a*d)^2) - (4*Sqrt[2]*a^( 7/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(b^ (3/4)*(b*c - a*d)^2) - (Sqrt[2]*c^(3/4)*(3*b*c - 7*a*d)*ArcTanh[(Sqrt[2]*c ^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(d^(7/4)*(b*c - a*d)^2))/8
Time = 0.69 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {368, 970, 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.
| \(\displaystyle \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {x^5}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}\) |
| \(\Big \downarrow \) 970 |
| \(\displaystyle 2 \left (\frac {\int \frac {x \left ((3 b c-4 a d) x^2+3 a c\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 d (b c-a d)}-\frac {c x^{3/2}}{4 d \left (c+d x^2\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 \left (\frac {\int \left (\frac {c (3 b c-7 a d) x}{(b c-a d) \left (d x^2+c\right )}-\frac {4 a^2 d x}{(a d-b c) \left (b x^2+a\right )}\right )d\sqrt {x}}{4 d (b c-a d)}-\frac {c x^{3/2}}{4 d \left (c+d x^2\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {-\frac {\sqrt {2} a^{7/4} d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{b^{3/4} (b c-a d)}+\frac {\sqrt {2} a^{7/4} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{b^{3/4} (b c-a d)}+\frac {a^{7/4} d \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt {2} b^{3/4} (b c-a d)}-\frac {a^{7/4} d \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt {2} b^{3/4} (b c-a d)}-\frac {c^{3/4} (3 b c-7 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{2 \sqrt {2} d^{3/4} (b c-a d)}+\frac {c^{3/4} (3 b c-7 a d) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} d^{3/4} (b c-a d)}+\frac {c^{3/4} (3 b c-7 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} d^{3/4} (b c-a d)}-\frac {c^{3/4} (3 b c-7 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} d^{3/4} (b c-a d)}}{4 d (b c-a d)}-\frac {c x^{3/2}}{4 d \left (c+d x^2\right ) (b c-a d)}\right )\) |
2*(-1/4*(c*x^(3/2))/(d*(b*c - a*d)*(c + d*x^2)) + (-((Sqrt[2]*a^(7/4)*d*Ar cTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(b^(3/4)*(b*c - a*d))) + (Sqr t[2]*a^(7/4)*d*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(b^(3/4)*(b* c - a*d)) - (c^(3/4)*(3*b*c - 7*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/ c^(1/4)])/(2*Sqrt[2]*d^(3/4)*(b*c - a*d)) + (c^(3/4)*(3*b*c - 7*a*d)*ArcTa n[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])/( 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])/(Sqrt[2]*b^(3/4)*(b*c - a*d)) + (c^(3/4)*(3*b* c - 7*a*d)*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)) - (c^(3/4)*(3*b*c - 7*a*d)*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))) /(4*d*(b*c - a*d)))
3.5.71.3.1 Defintions of rubi rules used
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[(-a)*e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n) ^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Simp[e^(2*n) /(b*n*(b*c - a*d)*(p + 1)) Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d *x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^ n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && LtQ[p, -1] && 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 = 3.23 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.51
| method | result | size |
| derivativedivides | \(\frac {a^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{2} b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2 c \left (-\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 d \left (d \,x^{2}+c \right )}+\frac {\left (7 a d -3 b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{2}}\) | \(273\) |
| default | \(\frac {a^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{2} b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {2 c \left (-\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 d \left (d \,x^{2}+c \right )}+\frac {\left (7 a d -3 b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 d^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{2}}\) | \(273\) |
1/4*a^2/(a*d-b*c)^2/b/(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*arctan(2^(1 /2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1))-2*c/(a *d-b*c)^2*(-1/4/d*(a*d-b*c)*x^(3/2)/(d*x^2+c)+1/32*(7*a*d-3*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)))
Result contains complex when optimal does not.
Time = 5.67 (sec) , antiderivative size = 3551, normalized size of antiderivative = 6.62 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
-1/8*(4*c*x^(3/2) - 4*(-a^7/(b^11*c^8 - 8*a*b^10*c^7*d + 28*a^2*b^9*c^6*d^ 2 - 56*a^3*b^8*c^5*d^3 + 70*a^4*b^7*c^4*d^4 - 56*a^5*b^6*c^3*d^5 + 28*a^6* b^5*c^2*d^6 - 8*a^7*b^4*c*d^7 + a^8*b^3*d^8))^(1/4)*(b*c^2*d - a*c*d^2 + ( b*c*d^2 - a*d^3)*x^2)*log(a^5*sqrt(x) + (b^8*c^6 - 6*a*b^7*c^5*d + 15*a^2* b^6*c^4*d^2 - 20*a^3*b^5*c^3*d^3 + 15*a^4*b^4*c^2*d^4 - 6*a^5*b^3*c*d^5 + a^6*b^2*d^6)*(-a^7/(b^11*c^8 - 8*a*b^10*c^7*d + 28*a^2*b^9*c^6*d^2 - 56*a^ 3*b^8*c^5*d^3 + 70*a^4*b^7*c^4*d^4 - 56*a^5*b^6*c^3*d^5 + 28*a^6*b^5*c^2*d ^6 - 8*a^7*b^4*c*d^7 + a^8*b^3*d^8))^(3/4)) + 4*(-a^7/(b^11*c^8 - 8*a*b^10 *c^7*d + 28*a^2*b^9*c^6*d^2 - 56*a^3*b^8*c^5*d^3 + 70*a^4*b^7*c^4*d^4 - 56 *a^5*b^6*c^3*d^5 + 28*a^6*b^5*c^2*d^6 - 8*a^7*b^4*c*d^7 + a^8*b^3*d^8))^(1 /4)*(b*c^2*d - a*c*d^2 + (b*c*d^2 - a*d^3)*x^2)*log(a^5*sqrt(x) - (b^8*c^6 - 6*a*b^7*c^5*d + 15*a^2*b^6*c^4*d^2 - 20*a^3*b^5*c^3*d^3 + 15*a^4*b^4*c^ 2*d^4 - 6*a^5*b^3*c*d^5 + a^6*b^2*d^6)*(-a^7/(b^11*c^8 - 8*a*b^10*c^7*d + 28*a^2*b^9*c^6*d^2 - 56*a^3*b^8*c^5*d^3 + 70*a^4*b^7*c^4*d^4 - 56*a^5*b^6* c^3*d^5 + 28*a^6*b^5*c^2*d^6 - 8*a^7*b^4*c*d^7 + a^8*b^3*d^8))^(3/4)) - 4* (-a^7/(b^11*c^8 - 8*a*b^10*c^7*d + 28*a^2*b^9*c^6*d^2 - 56*a^3*b^8*c^5*d^3 + 70*a^4*b^7*c^4*d^4 - 56*a^5*b^6*c^3*d^5 + 28*a^6*b^5*c^2*d^6 - 8*a^7*b^ 4*c*d^7 + a^8*b^3*d^8))^(1/4)*(I*b*c^2*d - I*a*c*d^2 + I*(b*c*d^2 - a*d^3) *x^2)*log(a^5*sqrt(x) - (I*b^8*c^6 - 6*I*a*b^7*c^5*d + 15*I*a^2*b^6*c^4*d^ 2 - 20*I*a^3*b^5*c^3*d^3 + 15*I*a^4*b^4*c^2*d^4 - 6*I*a^5*b^3*c*d^5 + I...
Timed out. \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 450, normalized size of antiderivative = 0.84 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {a^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac {c x^{\frac {3}{2}}}{2 \, {\left (b c^{2} d - a c d^{2} + {\left (b c d^{2} - a d^{3}\right )} x^{2}\right )}} + \frac {{\left (3 \, b c^{2} - 7 \, a c d\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{16 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}} \]
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^2 - 2*a*b*c*d + a^2*d^2) - 1/2*c*x^(3/2)/(b*c^2*d - a*c*d^2 + ( b*c*d^2 - a*d^3)*x^2) + 1/16*(3*b*c^2 - 7*a*c*d)*(2*sqrt(2)*arctan(1/2*sqr t(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)*s qrt(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 + sqrt(c))/(c^(1/4)*d^(3/4)))/(b^2*c^2*d - 2*a*b*c*d^2 + a^ 2*d^3)
Time = 0.41 (sec) , antiderivative size = 681, normalized size of antiderivative = 1.27 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, 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^{2} - 2 \, \sqrt {2} a b^{4} c d + \sqrt {2} a^{2} b^{3} d^{2}} + \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^{2} - 2 \, \sqrt {2} a b^{4} c d + \sqrt {2} a^{2} b^{3} d^{2}} - \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^{2} - 2 \, \sqrt {2} a b^{4} c d + \sqrt {2} a^{2} b^{3} d^{2}\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^{2} - 2 \, \sqrt {2} a b^{4} c d + \sqrt {2} a^{2} b^{3} d^{2}\right )}} + \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 7 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \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 )}{4 \, {\left (\sqrt {2} b^{2} c^{2} d^{4} - 2 \, \sqrt {2} a b c d^{5} + \sqrt {2} a^{2} d^{6}\right )}} + \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 7 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \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 )}{4 \, {\left (\sqrt {2} b^{2} c^{2} d^{4} - 2 \, \sqrt {2} a b c d^{5} + \sqrt {2} a^{2} d^{6}\right )}} - \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 7 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{2} d^{4} - 2 \, \sqrt {2} a b c d^{5} + \sqrt {2} a^{2} d^{6}\right )}} + \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 7 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{2} d^{4} - 2 \, \sqrt {2} a b c d^{5} + \sqrt {2} a^{2} d^{6}\right )}} - \frac {c x^{\frac {3}{2}}}{2 \, {\left (b c d - a d^{2}\right )} {\left (d x^{2} + c\right )}} \]
(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^2 - 2*sqrt(2)*a*b^4*c*d + sqrt(2)*a^2*b^3*d^2) + (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^2 - 2*sqrt(2)*a*b^4*c*d + sqrt(2)*a^2*b^3*d^2) - 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^2 - 2*sqrt(2)*a*b^4*c*d + sqrt(2)*a^2*b^3*d^2) + 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^2 - 2*s qrt(2)*a*b^4*c*d + sqrt(2)*a^2*b^3*d^2) + 1/4*(3*(c*d^3)^(3/4)*b*c - 7*(c* d^3)^(3/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d) ^(1/4))/(sqrt(2)*b^2*c^2*d^4 - 2*sqrt(2)*a*b*c*d^5 + sqrt(2)*a^2*d^6) + 1/ 4*(3*(c*d^3)^(3/4)*b*c - 7*(c*d^3)^(3/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2) *(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^2*d^4 - 2*sqrt(2)*a* b*c*d^5 + sqrt(2)*a^2*d^6) - 1/8*(3*(c*d^3)^(3/4)*b*c - 7*(c*d^3)^(3/4)*a* d)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^2*d^4 - 2*sqrt(2)*a*b*c*d^5 + sqrt(2)*a^2*d^6) + 1/8*(3*(c*d^3)^(3/4)*b*c - 7*(c* d^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2) *b^2*c^2*d^4 - 2*sqrt(2)*a*b*c*d^5 + sqrt(2)*a^2*d^6) - 1/2*c*x^(3/2)/((b* c*d - a*d^2)*(d*x^2 + c))
Time = 7.75 (sec) , antiderivative size = 19871, normalized size of antiderivative = 37.07 \[ \int \frac {x^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
2*atan(((-a^7/(16*b^11*c^8 + 16*a^8*b^3*d^8 - 128*a^7*b^4*c*d^7 + 448*a^2* b^9*c^6*d^2 - 896*a^3*b^8*c^5*d^3 + 1120*a^4*b^7*c^4*d^4 - 896*a^5*b^6*c^3 *d^5 + 448*a^6*b^5*c^2*d^6 - 128*a*b^10*c^7*d))^(1/4)*((-a^7/(16*b^11*c^8 + 16*a^8*b^3*d^8 - 128*a^7*b^4*c*d^7 + 448*a^2*b^9*c^6*d^2 - 896*a^3*b^8*c ^5*d^3 + 1120*a^4*b^7*c^4*d^4 - 896*a^5*b^6*c^3*d^5 + 448*a^6*b^5*c^2*d^6 - 128*a*b^10*c^7*d))^(3/4)*(((864*a^3*b^14*c^14*d^3 - 12096*a^4*b^13*c^13* d^4 + 74592*a^5*b^12*c^12*d^5 - 267008*a^6*b^11*c^11*d^6 + 617152*a^7*b^10 *c^10*d^7 - 968576*a^8*b^9*c^9*d^8 + 1054144*a^9*b^8*c^8*d^9 - 795392*a^10 *b^7*c^7*d^10 + 407008*a^11*b^6*c^6*d^11 - 133952*a^12*b^5*c^5*d^12 + 2531 2*a^13*b^4*c^4*d^13 - 2048*a^14*b^3*c^3*d^14)*1i)/(a^7*d^10 - b^7*c^7*d^3 + 7*a*b^6*c^6*d^4 - 21*a^2*b^5*c^5*d^5 + 35*a^3*b^4*c^4*d^6 - 35*a^4*b^3*c ^3*d^7 + 21*a^5*b^2*c^2*d^8 - 7*a^6*b*c*d^9) + (x^(1/2)*(-a^7/(16*b^11*c^8 + 16*a^8*b^3*d^8 - 128*a^7*b^4*c*d^7 + 448*a^2*b^9*c^6*d^2 - 896*a^3*b^8* c^5*d^3 + 1120*a^4*b^7*c^4*d^4 - 896*a^5*b^6*c^3*d^5 + 448*a^6*b^5*c^2*d^6 - 128*a*b^10*c^7*d))^(1/4)*(2304*a^3*b^14*c^13*d^5 - 29184*a^4*b^13*c^12* d^6 + 167168*a^5*b^12*c^11*d^7 - 563200*a^6*b^11*c^10*d^8 + 1229312*a^7*b^ 10*c^9*d^9 - 1813504*a^8*b^9*c^8*d^10 + 1831424*a^9*b^8*c^7*d^11 - 1251328 *a^10*b^7*c^6*d^12 + 554240*a^11*b^6*c^5*d^13 - 143872*a^12*b^5*c^4*d^14 + 16640*a^13*b^4*c^3*d^15))/(a^6*d^9 + b^6*c^6*d^3 - 6*a*b^5*c^5*d^4 + 15*a ^2*b^4*c^4*d^5 - 20*a^3*b^3*c^3*d^6 + 15*a^4*b^2*c^2*d^7 - 6*a^5*b*c*d^...