Integrand size = 24, antiderivative size = 739 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {d (2 b c+a d) x^{3/2}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+21 a b c d-5 a^2 d^2\right ) x^{3/2}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {b^{9/4} (b c-13 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^4}+\frac {b^{9/4} (b c-13 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^4}-\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^4}+\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^4}+\frac {b^{9/4} (b c-13 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^4}-\frac {b^{9/4} (b c-13 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^4}+\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^4}-\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^4} \]
1/4*d*(a*d+2*b*c)*x^(3/2)/a/c/(-a*d+b*c)^2/(d*x^2+c)^2+1/2*b*x^(3/2)/a/(-a *d+b*c)/(b*x^2+a)/(d*x^2+c)^2+1/16*d*(-5*a^2*d^2+21*a*b*c*d+8*b^2*c^2)*x^( 3/2)/a/c^2/(-a*d+b*c)^3/(d*x^2+c)-1/8*b^(9/4)*(-13*a*d+b*c)*arctan(1-b^(1/ 4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(5/4)/(-a*d+b*c)^4*2^(1/2)+1/8*b^(9/4)*(-13* a*d+b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(5/4)/(-a*d+b*c)^4*2^ (1/2)-1/64*d^(5/4)*(5*a^2*d^2-26*a*b*c*d+117*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)^4*2^(1/2)+1/64*d^(5/4)*(5*a^2*d^2 -26*a*b*c*d+117*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)^4*2^(1/2)+1/16*b^(9/4)*(-13*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1 /4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(5/4)/(-a*d+b*c)^4*2^(1/2)-1/16*b^(9/4)*(-1 3*a*d+b*c)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(5/4)/( -a*d+b*c)^4*2^(1/2)+1/128*d^(5/4)*(5*a^2*d^2-26*a*b*c*d+117*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)^4*2^(1 /2)-1/128*d^(5/4)*(5*a^2*d^2-26*a*b*c*d+117*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)^4*2^(1/2)
Time = 1.71 (sec) , antiderivative size = 451, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {1}{64} \left (-\frac {4 x^{3/2} \left (8 b^3 c^2 \left (c+d x^2\right )^2-a^3 d^3 \left (9 c+5 d x^2\right )+a b^2 c d^2 x^2 \left (25 c+21 d x^2\right )+a^2 b d^2 \left (25 c^2+12 c d x^2-5 d^2 x^4\right )\right )}{a c^2 (-b c+a d)^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {8 \sqrt {2} b^{9/4} (-b c+13 a d) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{5/4} (b c-a d)^4}-\frac {\sqrt {2} d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{9/4} (b c-a d)^4}+\frac {8 \sqrt {2} b^{9/4} (-b c+13 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{5/4} (b c-a d)^4}-\frac {\sqrt {2} d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{9/4} (b c-a d)^4}\right ) \]
((-4*x^(3/2)*(8*b^3*c^2*(c + d*x^2)^2 - a^3*d^3*(9*c + 5*d*x^2) + a*b^2*c* d^2*x^2*(25*c + 21*d*x^2) + a^2*b*d^2*(25*c^2 + 12*c*d*x^2 - 5*d^2*x^4)))/ (a*c^2*(-(b*c) + a*d)^3*(a + b*x^2)*(c + d*x^2)^2) + (8*Sqrt[2]*b^(9/4)*(- (b*c) + 13*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt [x])])/(a^(5/4)*(b*c - a*d)^4) - (Sqrt[2]*d^(5/4)*(117*b^2*c^2 - 26*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)^4) + (8*Sqrt[2]*b^(9/4)*(-(b*c) + 13*a*d)*ArcTa nh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(5/4)*(b*c - a*d)^4) - (Sqrt[2]*d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*ArcTa nh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c^(9/4)*(b*c - a*d)^4))/64
Time = 1.05 (sec) , antiderivative size = 796, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {368, 972, 25, 1049, 27, 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.
| \(\displaystyle \int \frac {\sqrt {x}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {x}{\left (b x^2+a\right )^2 \left (d x^2+c\right )^3}d\sqrt {x}\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle 2 \left (\frac {b x^{3/2}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac {\int -\frac {x \left (9 b d x^2+b c-4 a d\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^3}d\sqrt {x}}{4 a (b c-a d)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {\int \frac {x \left (9 b d x^2+b c-4 a d\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^3}d\sqrt {x}}{4 a (b c-a d)}+\frac {b x^{3/2}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {4 x \left (2 b^2 c^2-16 a b d c+5 a^2 d^2+5 b d (2 b c+a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{8 c (b c-a d)}+\frac {d x^{3/2} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b x^{3/2}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {\frac {\int \frac {x \left (2 b^2 c^2-16 a b d c+5 a^2 d^2+5 b d (2 b c+a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}}{2 c (b c-a d)}+\frac {d x^{3/2} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b x^{3/2}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\int \frac {x \left (8 b^3 c^3-96 a b^2 d c^2+21 a^2 b d^2 c-5 a^3 d^3+b d \left (8 b^2 c^2+21 a b d c-5 a^2 d^2\right ) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 c (b c-a d)}+\frac {d x^{3/2} \left (-5 a^2 d^2+21 a b c d+8 b^2 c^2\right )}{4 c \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d x^{3/2} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b x^{3/2}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 \left (\frac {\frac {\frac {\int \left (\frac {8 b^3 c^2 (b c-13 a d) x}{(b c-a d) \left (b x^2+a\right )}-\frac {a d^2 \left (117 b^2 c^2-26 a b d c+5 a^2 d^2\right ) x}{(a d-b c) \left (d x^2+c\right )}\right )d\sqrt {x}}{4 c (b c-a d)}+\frac {d x^{3/2} \left (-5 a^2 d^2+21 a b c d+8 b^2 c^2\right )}{4 c \left (c+d x^2\right ) (b c-a d)}}{2 c (b c-a d)}+\frac {d x^{3/2} (a d+2 b c)}{2 c \left (c+d x^2\right )^2 (b c-a d)}}{4 a (b c-a d)}+\frac {b x^{3/2}}{4 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {b x^{3/2}}{4 a (b c-a d) \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac {\frac {d (2 b c+a d) x^{3/2}}{2 c (b c-a d) \left (d x^2+c\right )^2}+\frac {\frac {d \left (8 b^2 c^2+21 a b d c-5 a^2 d^2\right ) x^{3/2}}{4 c (b c-a d) \left (d x^2+c\right )}+\frac {-\frac {2 \sqrt {2} c^2 (b c-13 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) b^{9/4}}{\sqrt [4]{a} (b c-a d)}+\frac {2 \sqrt {2} c^2 (b c-13 a d) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) b^{9/4}}{\sqrt [4]{a} (b c-a d)}+\frac {\sqrt {2} c^2 (b c-13 a d) \log \left (\sqrt {b} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}\right ) b^{9/4}}{\sqrt [4]{a} (b c-a d)}-\frac {\sqrt {2} c^2 (b c-13 a d) \log \left (\sqrt {b} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}\right ) b^{9/4}}{\sqrt [4]{a} (b c-a d)}-\frac {a d^{5/4} \left (117 b^2 c^2-26 a b d c+5 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {a d^{5/4} \left (117 b^2 c^2-26 a b d c+5 a^2 d^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {a d^{5/4} \left (117 b^2 c^2-26 a b d c+5 a^2 d^2\right ) \log \left (\sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {a d^{5/4} \left (117 b^2 c^2-26 a b d c+5 a^2 d^2\right ) \log \left (\sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)}}{4 c (b c-a d)}}{2 c (b c-a d)}}{4 a (b c-a d)}\right )\) |
2*((b*x^(3/2))/(4*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + ((d*(2*b*c + a*d)*x^(3/2))/(2*c*(b*c - a*d)*(c + d*x^2)^2) + ((d*(8*b^2*c^2 + 21*a*b*c* d - 5*a^2*d^2)*x^(3/2))/(4*c*(b*c - a*d)*(c + d*x^2)) + ((-2*Sqrt[2]*b^(9/ 4)*c^2*(b*c - 13*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1 /4)*(b*c - a*d)) + (2*Sqrt[2]*b^(9/4)*c^2*(b*c - 13*a*d)*ArcTan[1 + (Sqrt[ 2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(b*c - a*d)) - (a*d^(5/4)*(117*b^2* c^2 - 26*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)) + (a*d^(5/4)*(117*b^2*c^2 - 26*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)) + (Sqrt[2]*b^(9/4)*c^2*(b*c - 13*a*d)*Log[Sqrt[a] - Sqrt [2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*(b*c - a*d)) - (Sqrt[2] *b^(9/4)*c^2*(b*c - 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*(b*c - a*d)) + (a*d^(5/4)*(117*b^2*c^2 - 26*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)) - (a*d^(5/4)*(117*b^2*c^2 - 26*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)))/(4*c*(b*c - a*d)))/(2*c*(b*c - a*d)))/(4*a*(b *c - a*d)))
3.5.99.3.1 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[(-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 = 381, normalized size of antiderivative = 0.52
| method | result | size |
| derivativedivides | \(-\frac {2 b^{3} \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 a \left (b \,x^{2}+a \right )}+\frac {\left (13 a d -b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}+\frac {2 d^{2} \left (\frac {\frac {d \left (5 a^{2} d^{2}-26 a b c d +21 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{32 c^{2}}+\frac {\left (9 a^{2} d^{2}-34 a b c d +25 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}-26 a b c d +117 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{2} d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}\) | \(381\) |
| default | \(-\frac {2 b^{3} \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 a \left (b \,x^{2}+a \right )}+\frac {\left (13 a d -b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}+\frac {2 d^{2} \left (\frac {\frac {d \left (5 a^{2} d^{2}-26 a b c d +21 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{32 c^{2}}+\frac {\left (9 a^{2} d^{2}-34 a b c d +25 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}-26 a b c d +117 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{2} d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}\) | \(381\) |
-2*b^3/(a*d-b*c)^4*(1/4*(a*d-b*c)/a*x^(3/2)/(b*x^2+a)+1/32*(13*a*d-b*c)/a/ 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*d^2/(a*d-b*c)^4*((1/32* d*(5*a^2*d^2-26*a*b*c*d+21*b^2*c^2)/c^2*x^(7/2)+1/32*(9*a^2*d^2-34*a*b*c*d +25*b^2*c^2)/c*x^(3/2))/(d*x^2+c)^2+1/256*(5*a^2*d^2-26*a*b*c*d+117*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*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 = 246.76 (sec) , antiderivative size = 9098, normalized size of antiderivative = 12.31 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.33 (sec) , antiderivative size = 845, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\frac {{\left (b^{4} c - 13 \, a b^{3} d\right )} {\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 )}}{16 \, {\left (a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d + 6 \, a^{3} b^{2} c^{2} d^{2} - 4 \, a^{4} b c d^{3} + a^{5} d^{4}\right )}} + \frac {{\left (117 \, b^{2} c^{2} d^{2} - 26 \, a b c d^{3} + 5 \, a^{2} d^{4}\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^{4} c^{6} - 4 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} - 4 \, a^{3} b c^{3} d^{3} + a^{4} c^{2} d^{4}\right )}} + \frac {{\left (8 \, b^{3} c^{2} d^{2} + 21 \, a b^{2} c d^{3} - 5 \, a^{2} b d^{4}\right )} x^{\frac {11}{2}} + {\left (16 \, b^{3} c^{3} d + 25 \, a b^{2} c^{2} d^{2} + 12 \, a^{2} b c d^{3} - 5 \, a^{3} d^{4}\right )} x^{\frac {7}{2}} + {\left (8 \, b^{3} c^{4} + 25 \, a^{2} b c^{2} d^{2} - 9 \, a^{3} c d^{3}\right )} x^{\frac {3}{2}}}{16 \, {\left (a^{2} b^{3} c^{7} - 3 \, a^{3} b^{2} c^{6} d + 3 \, a^{4} b c^{5} d^{2} - a^{5} c^{4} d^{3} + {\left (a b^{4} c^{5} d^{2} - 3 \, a^{2} b^{3} c^{4} d^{3} + 3 \, a^{3} b^{2} c^{3} d^{4} - a^{4} b c^{2} d^{5}\right )} x^{6} + {\left (2 \, a b^{4} c^{6} d - 5 \, a^{2} b^{3} c^{5} d^{2} + 3 \, a^{3} b^{2} c^{4} d^{3} + a^{4} b c^{3} d^{4} - a^{5} c^{2} d^{5}\right )} x^{4} + {\left (a b^{4} c^{7} - a^{2} b^{3} c^{6} d - 3 \, a^{3} b^{2} c^{5} d^{2} + 5 \, a^{4} b c^{4} d^{3} - 2 \, a^{5} c^{3} d^{4}\right )} x^{2}\right )}} \]
1/16*(b^4*c - 13*a*b^3*d)*(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)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a ))/(a^(1/4)*b^(3/4)))/(a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4 *a^4*b*c*d^3 + a^5*d^4) + 1/128*(117*b^2*c^2*d^2 - 26*a*b*c*d^3 + 5*a^2*d^ 4)*(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)*ar ctan(-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(-s qrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)))/( b^4*c^6 - 4*a*b^3*c^5*d + 6*a^2*b^2*c^4*d^2 - 4*a^3*b*c^3*d^3 + a^4*c^2*d^ 4) + 1/16*((8*b^3*c^2*d^2 + 21*a*b^2*c*d^3 - 5*a^2*b*d^4)*x^(11/2) + (16*b ^3*c^3*d + 25*a*b^2*c^2*d^2 + 12*a^2*b*c*d^3 - 5*a^3*d^4)*x^(7/2) + (8*b^3 *c^4 + 25*a^2*b*c^2*d^2 - 9*a^3*c*d^3)*x^(3/2))/(a^2*b^3*c^7 - 3*a^3*b^2*c ^6*d + 3*a^4*b*c^5*d^2 - a^5*c^4*d^3 + (a*b^4*c^5*d^2 - 3*a^2*b^3*c^4*d^3 + 3*a^3*b^2*c^3*d^4 - a^4*b*c^2*d^5)*x^6 + (2*a*b^4*c^6*d - 5*a^2*b^3*c...
Leaf count of result is larger than twice the leaf count of optimal. 1233 vs. \(2 (583) = 1166\).
Time = 0.57 (sec) , antiderivative size = 1233, normalized size of antiderivative = 1.67 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
1/2*b^3*x^(3/2)/((a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*( b*x^2 + a)) + 1/4*((a*b^3)^(3/4)*b*c - 13*(a*b^3)^(3/4)*a*d)*arctan(1/2*sq rt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^2*b^4*c^4 - 4*sqrt(2)*a^3*b^3*c^3*d + 6*sqrt(2)*a^4*b^2*c^2*d^2 - 4*sqrt(2)*a^5*b*c* d^3 + sqrt(2)*a^6*d^4) + 1/4*((a*b^3)^(3/4)*b*c - 13*(a*b^3)^(3/4)*a*d)*ar ctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)* a^2*b^4*c^4 - 4*sqrt(2)*a^3*b^3*c^3*d + 6*sqrt(2)*a^4*b^2*c^2*d^2 - 4*sqrt (2)*a^5*b*c*d^3 + sqrt(2)*a^6*d^4) + 1/32*(117*(c*d^3)^(3/4)*b^2*c^2 - 26* (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^4*c^7*d - 4*sqrt(2)*a* b^3*c^6*d^2 + 6*sqrt(2)*a^2*b^2*c^5*d^3 - 4*sqrt(2)*a^3*b*c^4*d^4 + sqrt(2 )*a^4*c^3*d^5) + 1/32*(117*(c*d^3)^(3/4)*b^2*c^2 - 26*(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^4*c^7*d - 4*sqrt(2)*a*b^3*c^6*d^2 + 6*sqr t(2)*a^2*b^2*c^5*d^3 - 4*sqrt(2)*a^3*b*c^4*d^4 + sqrt(2)*a^4*c^3*d^5) - 1/ 8*((a*b^3)^(3/4)*b*c - 13*(a*b^3)^(3/4)*a*d)*log(sqrt(2)*sqrt(x)*(a/b)^(1/ 4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^4*c^4 - 4*sqrt(2)*a^3*b^3*c^3*d + 6*sqr t(2)*a^4*b^2*c^2*d^2 - 4*sqrt(2)*a^5*b*c*d^3 + sqrt(2)*a^6*d^4) + 1/8*((a* b^3)^(3/4)*b*c - 13*(a*b^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^2*b^4*c^4 - 4*sqrt(2)*a^3*b^3*c^3*d + 6*sqrt(...
Time = 14.69 (sec) , antiderivative size = 45858, normalized size of antiderivative = 62.05 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx=\text {Too large to display} \]
((x^(7/2)*(16*b^3*c^3*d - 5*a^3*d^4 + 25*a*b^2*c^2*d^2 + 12*a^2*b*c*d^3))/ (16*a*c*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)) - (x^(3/2 )*(8*b^3*c^3 - 9*a^3*d^3 + 25*a^2*b*c*d^2))/(16*a*c*(a^3*d^3 - b^3*c^3 + 3 *a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (b*d^2*x^(11/2)*(8*b^2*c^2 - 5*a^2*d^2 + 21*a*b*c*d))/(16*a*c*(b^3*c^4 - a^3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3* d)))/(a*c^2 + x^2*(b*c^2 + 2*a*c*d) + x^4*(a*d^2 + 2*b*c*d) + b*d^2*x^6) - atan(((-(625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 166567752*a*b^7*c^7*d^6 + 87554844*a^2*b^6*c^6*d^7 - 29580408*a^3*b^5*c^5*d^8 + 7255846*a^4*b^4*c^4 *d^9 - 1264120*a^5*b^3*c^3*d^10 + 159900*a^6*b^2*c^2*d^11 - 13000*a^7*b*c* d^12)/(16777216*b^16*c^25 + 16777216*a^16*c^9*d^16 - 268435456*a^15*b*c^10 *d^15 + 2013265920*a^2*b^14*c^23*d^2 - 9395240960*a^3*b^13*c^22*d^3 + 3053 4533120*a^4*b^12*c^21*d^4 - 73282879488*a^5*b^11*c^20*d^5 + 134351945728*a ^6*b^10*c^19*d^6 - 191931351040*a^7*b^9*c^18*d^7 + 215922769920*a^8*b^8*c^ 17*d^8 - 191931351040*a^9*b^7*c^16*d^9 + 134351945728*a^10*b^6*c^15*d^10 - 73282879488*a^11*b^5*c^14*d^11 + 30534533120*a^12*b^4*c^13*d^12 - 9395240 960*a^13*b^3*c^12*d^13 + 2013265920*a^14*b^2*c^11*d^14 - 268435456*a*b^15* c^24*d))^(1/4)*((-(625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 166567752*a*b^7* c^7*d^6 + 87554844*a^2*b^6*c^6*d^7 - 29580408*a^3*b^5*c^5*d^8 + 7255846*a^ 4*b^4*c^4*d^9 - 1264120*a^5*b^3*c^3*d^10 + 159900*a^6*b^2*c^2*d^11 - 13000 *a^7*b*c*d^12)/(16777216*b^16*c^25 + 16777216*a^16*c^9*d^16 - 268435456...