Integrand size = 24, antiderivative size = 633 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=-\frac {d x^{3/2}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (13 b c-5 a d) x^{3/2}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b^{9/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^3}+\frac {b^{9/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^3}+\frac {\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^3}-\frac {\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {b^{9/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {b^{9/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^3}-\frac {\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^3}+\frac {\sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^3} \]
-1/4*d*x^(3/2)/c/(-a*d+b*c)/(d*x^2+c)^2-1/16*d*(-5*a*d+13*b*c)*x^(3/2)/c^2 /(-a*d+b*c)^2/(d*x^2+c)-1/2*b^(9/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/ 4))/a^(1/4)/(-a*d+b*c)^3*2^(1/2)+1/2*b^(9/4)*arctan(1+b^(1/4)*2^(1/2)*x^(1 /2)/a^(1/4))/a^(1/4)/(-a*d+b*c)^3*2^(1/2)+1/64*d^(1/4)*(5*a^2*d^2-18*a*b*c *d+45*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(9/4)/(-a*d+b*c )^3*2^(1/2)-1/64*d^(1/4)*(5*a^2*d^2-18*a*b*c*d+45*b^2*c^2)*arctan(1+d^(1/4 )*2^(1/2)*x^(1/2)/c^(1/4))/c^(9/4)/(-a*d+b*c)^3*2^(1/2)+1/4*b^(9/4)*ln(a^( 1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(1/4)/(-a*d+b*c)^3*2^(1/ 2)-1/4*b^(9/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(1/ 4)/(-a*d+b*c)^3*2^(1/2)-1/128*d^(1/4)*(5*a^2*d^2-18*a*b*c*d+45*b^2*c^2)*ln (c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(9/4)/(-a*d+b*c)^3*2 ^(1/2)+1/128*d^(1/4)*(5*a^2*d^2-18*a*b*c*d+45*b^2*c^2)*ln(c^(1/2)+x*d^(1/2 )+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(9/4)/(-a*d+b*c)^3*2^(1/2)
Time = 1.27 (sec) , antiderivative size = 361, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {1}{64} \left (\frac {4 d x^{3/2} \left (a d \left (9 c+5 d x^2\right )-b c \left (17 c+13 d x^2\right )\right )}{c^2 (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {32 \sqrt {2} b^{9/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{a} (-b c+a d)^3}+\frac {\sqrt {2} \sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{9/4} (b c-a d)^3}+\frac {32 \sqrt {2} b^{9/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{a} (-b c+a d)^3}+\frac {\sqrt {2} \sqrt [4]{d} \left (45 b^2 c^2-18 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{9/4} (b c-a d)^3}\right ) \]
((4*d*x^(3/2)*(a*d*(9*c + 5*d*x^2) - b*c*(17*c + 13*d*x^2)))/(c^2*(b*c - a *d)^2*(c + d*x^2)^2) + (32*Sqrt[2]*b^(9/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(S qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(1/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^ (1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/( Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(9/4)*(b*c - a*d)^3) + (32*Sqrt[2]*b ^(9/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/( a^(1/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5* a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)]) /(c^(9/4)*(b*c - a*d)^3))/64
Time = 0.83 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {368, 972, 1049, 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.
| \(\displaystyle \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {x}{\left (b x^2+a\right ) \left (d x^2+c\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle 2 \left (\frac {\int \frac {x \left (-5 b d x^2+8 b c-5 a d\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{8 c (b c-a d)}-\frac {d x^{3/2}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {x \left (32 b^2 c^2-13 a b d c+5 a^2 d^2-b d (13 b c-5 a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 c (b c-a d)}-\frac {d x^{3/2} (13 b c-5 a d)}{4 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d x^{3/2}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 \left (\frac {\frac {\int \left (\frac {32 b^3 c^2 x}{(b c-a d) \left (b x^2+a\right )}-\frac {d \left (45 b^2 c^2-18 a b d c+5 a^2 d^2\right ) x}{(b c-a d) \left (d x^2+c\right )}\right )d\sqrt {x}}{4 c (b c-a d)}-\frac {d x^{3/2} (13 b c-5 a d)}{4 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d x^{3/2}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {8 \sqrt {2} b^{9/4} c^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} (b c-a d)}+\frac {8 \sqrt {2} b^{9/4} c^2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a} (b c-a d)}+\frac {4 \sqrt {2} b^{9/4} c^2 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{a} (b c-a d)}-\frac {4 \sqrt {2} b^{9/4} c^2 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt [4]{a} (b c-a d)}}{4 c (b c-a d)}-\frac {d x^{3/2} (13 b c-5 a d)}{4 c \left (c+d x^2\right ) (b c-a d)}}{8 c (b c-a d)}-\frac {d x^{3/2}}{8 c \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
2*(-1/8*(d*x^(3/2))/(c*(b*c - a*d)*(c + d*x^2)^2) + (-1/4*(d*(13*b*c - 5*a *d)*x^(3/2))/(c*(b*c - a*d)*(c + d*x^2)) + ((-8*Sqrt[2]*b^(9/4)*c^2*ArcTan [1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(b*c - a*d)) + (8*Sqrt[2 ]*b^(9/4)*c^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(b*c - a*d)) + (d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt [2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(2*Sqrt[2]*c^(1/4)*(b*c - a*d)) - (d^(1/4)* (45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x]) /c^(1/4)])/(2*Sqrt[2]*c^(1/4)*(b*c - a*d)) + (4*Sqrt[2]*b^(9/4)*c^2*Log[Sq rt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*(b*c - a*d) ) - (4*Sqrt[2]*b^(9/4)*c^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*(b*c - a*d)) - (d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5 *a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(4*S qrt[2]*c^(1/4)*(b*c - a*d)) + (d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^ 2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(4*Sqrt[2]* c^(1/4)*(b*c - a*d)))/(4*c*(b*c - a*d)))/(8*c*(b*c - a*d)))
3.5.83.3.1 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[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x ^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] & & IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1))) , x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^n)^(p + 1)*( c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
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 = 2.75 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.53
| method | result | size |
| derivativedivides | \(\frac {2 d \left (\frac {\frac {d \left (5 a^{2} d^{2}-18 a b c d +13 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{32 c^{2}}+\frac {\left (9 a^{2} d^{2}-26 a b c d +17 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}-18 a b c d +45 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{2} d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}-\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(336\) |
| default | \(\frac {2 d \left (\frac {\frac {d \left (5 a^{2} d^{2}-18 a b c d +13 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{32 c^{2}}+\frac {\left (9 a^{2} d^{2}-26 a b c d +17 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}-18 a b c d +45 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{2} d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{3}}-\frac {b^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(336\) |
2*d/(a*d-b*c)^3*((1/32*d*(5*a^2*d^2-18*a*b*c*d+13*b^2*c^2)/c^2*x^(7/2)+1/3 2*(9*a^2*d^2-26*a*b*c*d+17*b^2*c^2)/c*x^(3/2))/(d*x^2+c)^2+1/256*(5*a^2*d^ 2-18*a*b*c*d+45*b^2*c^2)/c^2/d/(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*ar ctan(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*b^2/(a*d-b*c)^3/(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))
Result contains complex when optimal does not.
Time = 43.26 (sec) , antiderivative size = 5966, normalized size of antiderivative = 9.42 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 594, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\frac {b^{3} {\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^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac {{\left (45 \, b^{2} c^{2} d - 18 \, a b c d^{2} + 5 \, a^{2} d^{3}\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 )}}{128 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )}} - \frac {{\left (13 \, b c d^{2} - 5 \, a d^{3}\right )} x^{\frac {7}{2}} + {\left (17 \, b c^{2} d - 9 \, a c d^{2}\right )} x^{\frac {3}{2}}}{16 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} + {\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x^{2}\right )}} \]
1/4*b^3*(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^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - 1/128*(45*b^2*c^ 2*d - 18*a*b*c*d^2 + 5*a^2*d^3)*(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)*sqr t(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 + sqrt(c))/(c^(1/4)*d^(3/4)))/(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a ^3*c^2*d^3) - 1/16*((13*b*c*d^2 - 5*a*d^3)*x^(7/2) + (17*b*c^2*d - 9*a*c*d ^2)*x^(3/2))/(b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c ^3*d^3 + a^2*c^2*d^4)*x^4 + 2*(b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*x^ 2)
Time = 0.45 (sec) , antiderivative size = 968, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
-1/32*(45*(c*d^3)^(3/4)*b^2*c^2 - 18*(c*d^3)^(3/4)*a*b*c*d + 5*(c*d^3)^(3/ 4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/ 4))/(sqrt(2)*b^3*c^6*d^2 - 3*sqrt(2)*a*b^2*c^5*d^3 + 3*sqrt(2)*a^2*b*c^4*d ^4 - sqrt(2)*a^3*c^3*d^5) - 1/32*(45*(c*d^3)^(3/4)*b^2*c^2 - 18*(c*d^3)^(3 /4)*a*b*c*d + 5*(c*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^ (1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^6*d^2 - 3*sqrt(2)*a*b^2*c^5 *d^3 + 3*sqrt(2)*a^2*b*c^4*d^4 - sqrt(2)*a^3*c^3*d^5) + 1/64*(45*(c*d^3)^( 3/4)*b^2*c^2 - 18*(c*d^3)^(3/4)*a*b*c*d + 5*(c*d^3)^(3/4)*a^2*d^2)*log(sqr t(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^6*d^2 - 3*sqrt(2) *a*b^2*c^5*d^3 + 3*sqrt(2)*a^2*b*c^4*d^4 - sqrt(2)*a^3*c^3*d^5) - 1/64*(45 *(c*d^3)^(3/4)*b^2*c^2 - 18*(c*d^3)^(3/4)*a*b*c*d + 5*(c*d^3)^(3/4)*a^2*d^ 2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^6*d^2 - 3*sqrt(2)*a*b^2*c^5*d^3 + 3*sqrt(2)*a^2*b*c^4*d^4 - sqrt(2)*a^3*c^3*d^5) + (a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b )^(1/4))/(sqrt(2)*a*b^3*c^3 - 3*sqrt(2)*a^2*b^2*c^2*d + 3*sqrt(2)*a^3*b*c* d^2 - sqrt(2)*a^4*d^3) + (a*b^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^ (1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b^3*c^3 - 3*sqrt(2)*a^2*b^2*c^2 *d + 3*sqrt(2)*a^3*b*c*d^2 - sqrt(2)*a^4*d^3) - 1/2*(a*b^3)^(3/4)*log(sqrt (2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b^3*c^3 - 3*sqrt(2)*a^ 2*b^2*c^2*d + 3*sqrt(2)*a^3*b*c*d^2 - sqrt(2)*a^4*d^3) + 1/2*(a*b^3)^(3...
Time = 9.12 (sec) , antiderivative size = 32735, normalized size of antiderivative = 51.71 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
2*atan((((((2048*a*b^23*c^20*d^4 + (125*a^20*b^4*c*d^23)/16 - 22528*a^2*b^ 22*c^19*d^5 + (1711115*a^3*b^21*c^18*d^6)/16 - (4294995*a^4*b^20*c^17*d^7) /16 + (565575*a^5*b^19*c^16*d^8)/2 + (844557*a^6*b^18*c^15*d^9)/2 - (93477 99*a^7*b^17*c^14*d^10)/4 + (20337495*a^8*b^16*c^13*d^11)/4 - (14638795*a^9 *b^15*c^12*d^12)/2 + (15550975*a^10*b^14*c^11*d^13)/2 - (50934983*a^11*b^1 3*c^10*d^14)/8 + (32835743*a^12*b^12*c^9*d^15)/8 - (4207335*a^13*b^11*c^8* d^16)/2 + (1717635*a^14*b^10*c^7*d^17)/2 - (1110975*a^15*b^9*c^6*d^18)/4 + (280623*a^16*b^8*c^5*d^19)/4 - (26949*a^17*b^7*c^4*d^20)/2 + (3745*a^18*b ^6*c^3*d^21)/2 - (2725*a^19*b^5*c^2*d^22)/16)*1i)/(b^14*c^20 + a^14*c^6*d^ 14 - 14*a^13*b*c^7*d^13 + 91*a^2*b^12*c^18*d^2 - 364*a^3*b^11*c^17*d^3 + 1 001*a^4*b^10*c^16*d^4 - 2002*a^5*b^9*c^15*d^5 + 3003*a^6*b^8*c^14*d^6 - 34 32*a^7*b^7*c^13*d^7 + 3003*a^8*b^6*c^12*d^8 - 2002*a^9*b^5*c^11*d^9 + 1001 *a^10*b^4*c^10*d^10 - 364*a^11*b^3*c^9*d^11 + 91*a^12*b^2*c^8*d^12 - 14*a* b^13*c^19*d) - (x^(1/2)*(-b^9/(16*a^13*d^12 + 16*a*b^12*c^12 - 192*a^2*b^1 1*c^11*d + 1056*a^3*b^10*c^10*d^2 - 3520*a^4*b^9*c^9*d^3 + 7920*a^5*b^8*c^ 8*d^4 - 12672*a^6*b^7*c^7*d^5 + 14784*a^7*b^6*c^6*d^6 - 12672*a^8*b^5*c^5* d^7 + 7920*a^9*b^4*c^4*d^8 - 3520*a^10*b^3*c^3*d^9 + 1056*a^11*b^2*c^2*d^1 0 - 192*a^12*b*c*d^11))^(1/4)*(16777216*a*b^22*c^21*d^4 - 201326592*a^2*b^ 21*c^20*d^5 + 1140473856*a^3*b^20*c^19*d^6 - 4115660800*a^4*b^19*c^18*d^7 + 10825629696*a^5*b^18*c^17*d^8 - 22493528064*a^6*b^17*c^16*d^9 + 38637...