Integrand size = 24, antiderivative size = 618 \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}-\frac {b^{13/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {b^{13/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {b^{13/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {b^{13/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2} \]
1/10*(9*a*d-4*b*c)/a/c^2/(-a*d+b*c)/x^(5/2)-1/2*d/c/(-a*d+b*c)/x^(5/2)/(d* x^2+c)-1/2*b^(13/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)/(-a* d+b*c)^2*2^(1/2)+1/2*b^(13/4)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^ (9/4)/(-a*d+b*c)^2*2^(1/2)+1/8*d^(9/4)*(-9*a*d+13*b*c)*arctan(1-d^(1/4)*2^ (1/2)*x^(1/2)/c^(1/4))/c^(13/4)/(-a*d+b*c)^2*2^(1/2)-1/8*d^(9/4)*(-9*a*d+1 3*b*c)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(13/4)/(-a*d+b*c)^2*2^( 1/2)+1/4*b^(13/4)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^ (9/4)/(-a*d+b*c)^2*2^(1/2)-1/4*b^(13/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/ 4)*2^(1/2)*x^(1/2))/a^(9/4)/(-a*d+b*c)^2*2^(1/2)-1/16*d^(9/4)*(-9*a*d+13*b *c)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(13/4)/(-a*d+b *c)^2*2^(1/2)+1/16*d^(9/4)*(-9*a*d+13*b*c)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^ (1/4)*2^(1/2)*x^(1/2))/c^(13/4)/(-a*d+b*c)^2*2^(1/2)+1/2*(-9*a^2*d^2+4*a*b *c*d+4*b^2*c^2)/a^2/c^3/(-a*d+b*c)/x^(1/2)
Time = 1.23 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {1}{40} \left (-\frac {4 \left (20 b^2 c^2 x^2 \left (c+d x^2\right )+a^2 d \left (4 c^2-36 c d x^2-45 d^2 x^4\right )-4 a b c \left (c^2-4 c d x^2-5 d^2 x^4\right )\right )}{a^2 c^3 (-b c+a d) x^{5/2} \left (c+d x^2\right )}-\frac {20 \sqrt {2} b^{13/4} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{9/4} (b c-a d)^2}+\frac {5 \sqrt {2} d^{9/4} (13 b c-9 a d) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{13/4} (b c-a d)^2}-\frac {20 \sqrt {2} b^{13/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{9/4} (b c-a d)^2}+\frac {5 \sqrt {2} d^{9/4} (13 b c-9 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{13/4} (b c-a d)^2}\right ) \]
((-4*(20*b^2*c^2*x^2*(c + d*x^2) + a^2*d*(4*c^2 - 36*c*d*x^2 - 45*d^2*x^4) - 4*a*b*c*(c^2 - 4*c*d*x^2 - 5*d^2*x^4)))/(a^2*c^3*(-(b*c) + a*d)*x^(5/2) *(c + d*x^2)) - (20*Sqrt[2]*b^(13/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2] *a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(9/4)*(b*c - a*d)^2) + (5*Sqrt[2]*d^(9/4)*( 13*b*c - 9*a*d)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt [x])])/(c^(13/4)*(b*c - a*d)^2) - (20*Sqrt[2]*b^(13/4)*ArcTanh[(Sqrt[2]*a^ (1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(9/4)*(b*c - a*d)^2) + ( 5*Sqrt[2]*d^(9/4)*(13*b*c - 9*a*d)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x ])/(Sqrt[c] + Sqrt[d]*x)])/(c^(13/4)*(b*c - a*d)^2))/40
Time = 0.90 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {368, 972, 1053, 27, 1053, 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.
| \(\displaystyle \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle 2 \int \frac {1}{x^3 \left (b x^2+a\right ) \left (d x^2+c\right )^2}d\sqrt {x}\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle 2 \left (\frac {\int \frac {-9 b d x^2+4 b c-9 a d}{x^3 \left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{4 c (b c-a d)}-\frac {d}{4 c x^{5/2} \left (c+d x^2\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle 2 \left (\frac {-\frac {\int \frac {5 \left (4 b^2 c^2+4 a b d c-9 a^2 d^2+b d (4 b c-9 a d) x^2\right )}{x \left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{5 a c}-\frac {4 b c-9 a d}{5 a c x^{5/2}}}{4 c (b c-a d)}-\frac {d}{4 c x^{5/2} \left (c+d x^2\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {-\frac {\int \frac {4 b^2 c^2+4 a b d c-9 a^2 d^2+b d (4 b c-9 a d) x^2}{x \left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{a c}-\frac {4 b c-9 a d}{5 a c x^{5/2}}}{4 c (b c-a d)}-\frac {d}{4 c x^{5/2} \left (c+d x^2\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle 2 \left (\frac {-\frac {-\frac {\int \frac {x \left (4 b^3 c^3+4 a b^2 d c^2+4 a^2 b d^2 c-9 a^3 d^3+b d \left (4 b^2 c^2+4 a b d c-9 a^2 d^2\right ) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )}d\sqrt {x}}{a c}-\frac {\frac {4 b^2 c}{a}-\frac {9 a d^2}{c}+4 b d}{\sqrt {x}}}{a c}-\frac {4 b c-9 a d}{5 a c x^{5/2}}}{4 c (b c-a d)}-\frac {d}{4 c x^{5/2} \left (c+d x^2\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 \left (\frac {-\frac {-\frac {\int \left (\frac {4 b^4 c^3 x}{(b c-a d) \left (b x^2+a\right )}-\frac {a^2 d^3 (9 a d-13 b c) x}{(a d-b c) \left (d x^2+c\right )}\right )d\sqrt {x}}{a c}-\frac {\frac {4 b^2 c}{a}-\frac {9 a d^2}{c}+4 b d}{\sqrt {x}}}{a c}-\frac {4 b c-9 a d}{5 a c x^{5/2}}}{4 c (b c-a d)}-\frac {d}{4 c x^{5/2} \left (c+d x^2\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {-\frac {-\frac {\frac {a^2 d^{9/4} (13 b c-9 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {a^2 d^{9/4} (13 b c-9 a d) \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {a^2 d^{9/4} (13 b c-9 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {a^2 d^{9/4} (13 b c-9 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{4 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {\sqrt {2} b^{13/4} c^3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} (b c-a d)}+\frac {\sqrt {2} b^{13/4} c^3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a} (b c-a d)}+\frac {b^{13/4} c^3 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)}-\frac {b^{13/4} c^3 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)}}{a c}-\frac {\frac {4 b^2 c}{a}-\frac {9 a d^2}{c}+4 b d}{\sqrt {x}}}{a c}-\frac {4 b c-9 a d}{5 a c x^{5/2}}}{4 c (b c-a d)}-\frac {d}{4 c x^{5/2} \left (c+d x^2\right ) (b c-a d)}\right )\) |
2*(-1/4*d/(c*(b*c - a*d)*x^(5/2)*(c + d*x^2)) + (-1/5*(4*b*c - 9*a*d)/(a*c *x^(5/2)) - (-(((4*b^2*c)/a + 4*b*d - (9*a*d^2)/c)/Sqrt[x]) - (-((Sqrt[2]* b^(13/4)*c^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(b*c - a*d))) + (Sqrt[2]*b^(13/4)*c^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1 /4)])/(a^(1/4)*(b*c - a*d)) + (a^2*d^(9/4)*(13*b*c - 9*a*d)*ArcTan[1 - (Sq rt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(2*Sqrt[2]*c^(1/4)*(b*c - a*d)) - (a^2*d^ (9/4)*(13*b*c - 9*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(2*S qrt[2]*c^(1/4)*(b*c - a*d)) + (b^(13/4)*c^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)* b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(Sqrt[2]*a^(1/4)*(b*c - a*d)) - (b^(13/4)*c^ 3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(Sqrt[2]*a^( 1/4)*(b*c - a*d)) - (a^2*d^(9/4)*(13*b*c - 9*a*d)*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^2 *d^(9/4)*(13*b*c - 9*a*d)*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*c))/(a*c))/(4*c*(b*c - a*d )))
3.5.79.3.1 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[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -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 = 2.90 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.50
| method | result | size |
| derivativedivides | \(\frac {2 d^{3} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {9 a d}{4}-\frac {13 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{c^{3} \left (a d -b c \right )^{2}}-\frac {2}{5 a \,c^{2} x^{\frac {5}{2}}}-\frac {2 \left (-2 a d -b c \right )}{a^{2} c^{3} \sqrt {x}}+\frac {b^{3} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(306\) |
| default | \(\frac {2 d^{3} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {9 a d}{4}-\frac {13 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{c^{3} \left (a d -b c \right )^{2}}-\frac {2}{5 a \,c^{2} x^{\frac {5}{2}}}-\frac {2 \left (-2 a d -b c \right )}{a^{2} c^{3} \sqrt {x}}+\frac {b^{3} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(306\) |
| risch | \(-\frac {2 \left (-10 a d \,x^{2}-5 c b \,x^{2}+a c \right )}{5 a^{2} c^{3} x^{\frac {5}{2}}}+\frac {\frac {b^{3} c^{3} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {2 a^{2} d^{3} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {9 a d}{4}-\frac {13 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{2}}}{a^{2} c^{3}}\) | \(312\) |
2*d^3/c^3/(a*d-b*c)^2*((1/4*a*d-1/4*b*c)*x^(3/2)/(d*x^2+c)+1/8*(9/4*a*d-13 /4*b*c)/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)))-2/5/a/c^2/x^(5/2) -2*(-2*a*d-b*c)/a^2/c^3/x^(1/2)+1/4*b^3/a^2/(a*d-b*c)^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*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 = 42.38 (sec) , antiderivative size = 3783, normalized size of antiderivative = 6.12 \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
1/40*(20*(-b^13/(a^9*b^8*c^8 - 8*a^10*b^7*c^7*d + 28*a^11*b^6*c^6*d^2 - 56 *a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*d^5 + 28*a^15*b^ 2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))^(1/4)*((a^2*b*c^4*d - a^3*c^3*d^2) *x^5 + (a^2*b*c^5 - a^3*c^4*d)*x^3)*log(b^10*sqrt(x) + (a^7*b^6*c^6 - 6*a^ 8*b^5*c^5*d + 15*a^9*b^4*c^4*d^2 - 20*a^10*b^3*c^3*d^3 + 15*a^11*b^2*c^2*d ^4 - 6*a^12*b*c*d^5 + a^13*d^6)*(-b^13/(a^9*b^8*c^8 - 8*a^10*b^7*c^7*d + 2 8*a^11*b^6*c^6*d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b ^3*c^3*d^5 + 28*a^15*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))^(3/4)) - 20 *(-b^13/(a^9*b^8*c^8 - 8*a^10*b^7*c^7*d + 28*a^11*b^6*c^6*d^2 - 56*a^12*b^ 5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*d^5 + 28*a^15*b^2*c^2*d^ 6 - 8*a^16*b*c*d^7 + a^17*d^8))^(1/4)*((a^2*b*c^4*d - a^3*c^3*d^2)*x^5 + ( a^2*b*c^5 - a^3*c^4*d)*x^3)*log(b^10*sqrt(x) - (a^7*b^6*c^6 - 6*a^8*b^5*c^ 5*d + 15*a^9*b^4*c^4*d^2 - 20*a^10*b^3*c^3*d^3 + 15*a^11*b^2*c^2*d^4 - 6*a ^12*b*c*d^5 + a^13*d^6)*(-b^13/(a^9*b^8*c^8 - 8*a^10*b^7*c^7*d + 28*a^11*b ^6*c^6*d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*d ^5 + 28*a^15*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))^(3/4)) + 20*(-b^13/ (a^9*b^8*c^8 - 8*a^10*b^7*c^7*d + 28*a^11*b^6*c^6*d^2 - 56*a^12*b^5*c^5*d^ 3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*d^5 + 28*a^15*b^2*c^2*d^6 - 8*a^ 16*b*c*d^7 + a^17*d^8))^(1/4)*(I*(a^2*b*c^4*d - a^3*c^3*d^2)*x^5 + I*(a^2* b*c^5 - a^3*c^4*d)*x^3)*log(b^10*sqrt(x) - (I*a^7*b^6*c^6 - 6*I*a^8*b^5...
Timed out. \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 551, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b^{4} {\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 (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} - \frac {{\left (13 \, b c d^{3} - 9 \, a 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 )}}{16 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}} - \frac {4 \, a b c^{3} - 4 \, a^{2} c^{2} d - 5 \, {\left (4 \, b^{2} c^{2} d + 4 \, a b c d^{2} - 9 \, a^{2} d^{3}\right )} x^{4} - 4 \, {\left (5 \, b^{2} c^{3} + 4 \, a b c^{2} d - 9 \, a^{2} c d^{2}\right )} x^{2}}{10 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{\frac {9}{2}} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{\frac {5}{2}}\right )}} \]
1/4*b^4*(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 )))/(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2) - 1/16*(13*b*c*d^3 - 9*a*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)*arctan( -1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sq rt(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^2*c ^5 - 2*a*b*c^4*d + a^2*c^3*d^2) - 1/10*(4*a*b*c^3 - 4*a^2*c^2*d - 5*(4*b^2 *c^2*d + 4*a*b*c*d^2 - 9*a^2*d^3)*x^4 - 4*(5*b^2*c^3 + 4*a*b*c^2*d - 9*a^2 *c*d^2)*x^2)/((a^2*b*c^4*d - a^3*c^3*d^2)*x^(9/2) + (a^2*b*c^5 - a^3*c^4*d )*x^(5/2))
Time = 0.44 (sec) , antiderivative size = 715, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=-\frac {d^{3} x^{\frac {3}{2}}}{2 \, {\left (b c^{4} - a c^{3} d\right )} {\left (d x^{2} + c\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} b \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} a^{3} b^{2} c^{2} - 2 \, \sqrt {2} a^{4} b c d + \sqrt {2} a^{5} d^{2}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} b \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} a^{3} b^{2} c^{2} - 2 \, \sqrt {2} a^{4} b c d + \sqrt {2} a^{5} d^{2}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} b \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{3} b^{2} c^{2} - 2 \, \sqrt {2} a^{4} b c d + \sqrt {2} a^{5} d^{2}\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} b \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{3} b^{2} c^{2} - 2 \, \sqrt {2} a^{4} b c d + \sqrt {2} a^{5} d^{2}\right )}} - \frac {{\left (13 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 9 \, \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^{6} - 2 \, \sqrt {2} a b c^{5} d + \sqrt {2} a^{2} c^{4} d^{2}\right )}} - \frac {{\left (13 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 9 \, \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^{6} - 2 \, \sqrt {2} a b c^{5} d + \sqrt {2} a^{2} c^{4} d^{2}\right )}} + \frac {{\left (13 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 9 \, \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^{6} - 2 \, \sqrt {2} a b c^{5} d + \sqrt {2} a^{2} c^{4} d^{2}\right )}} - \frac {{\left (13 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 9 \, \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^{6} - 2 \, \sqrt {2} a b c^{5} d + \sqrt {2} a^{2} c^{4} d^{2}\right )}} + \frac {2 \, {\left (5 \, b c x^{2} + 10 \, a d x^{2} - a c\right )}}{5 \, a^{2} c^{3} x^{\frac {5}{2}}} \]
-1/2*d^3*x^(3/2)/((b*c^4 - a*c^3*d)*(d*x^2 + c)) + (a*b^3)^(3/4)*b*arctan( 1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^3*b^ 2*c^2 - 2*sqrt(2)*a^4*b*c*d + sqrt(2)*a^5*d^2) + (a*b^3)^(3/4)*b*arctan(-1 /2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^3*b^2 *c^2 - 2*sqrt(2)*a^4*b*c*d + sqrt(2)*a^5*d^2) - 1/2*(a*b^3)^(3/4)*b*log(sq rt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^2*c^2 - 2*sqrt(2 )*a^4*b*c*d + sqrt(2)*a^5*d^2) + 1/2*(a*b^3)^(3/4)*b*log(-sqrt(2)*sqrt(x)* (a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^2*c^2 - 2*sqrt(2)*a^4*b*c*d + sqrt(2)*a^5*d^2) - 1/4*(13*(c*d^3)^(3/4)*b*c - 9*(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 ^6 - 2*sqrt(2)*a*b*c^5*d + sqrt(2)*a^2*c^4*d^2) - 1/4*(13*(c*d^3)^(3/4)*b* c - 9*(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^6 - 2*sqrt(2)*a*b*c^5*d + sqrt(2)*a^2*c^4 *d^2) + 1/8*(13*(c*d^3)^(3/4)*b*c - 9*(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^6 - 2*sqrt(2)*a*b*c^5*d + s qrt(2)*a^2*c^4*d^2) - 1/8*(13*(c*d^3)^(3/4)*b*c - 9*(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^6 - 2*sqrt(2 )*a*b*c^5*d + sqrt(2)*a^2*c^4*d^2) + 2/5*(5*b*c*x^2 + 10*a*d*x^2 - a*c)/(a ^2*c^3*x^(5/2))
Time = 10.52 (sec) , antiderivative size = 17850, normalized size of antiderivative = 28.88 \[ \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \]
- (2/(5*a*c) - (2*x^2*(9*a*d + 5*b*c))/(5*a^2*c^2) + (d*x^4*(4*b^2*c^2 - 9 *a^2*d^2 + 4*a*b*c*d))/(2*a^2*c^3*(a*d - b*c)))/(c*x^(5/2) + d*x^(9/2)) - 2*atan((524288*a^3*b^16*c^32*x^(1/2)*(-(6561*a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12)/(409 6*b^8*c^21 + 4096*a^8*c^13*d^8 - 32768*a^7*b*c^14*d^7 + 114688*a^2*b^6*c^1 9*d^2 - 229376*a^3*b^5*c^18*d^3 + 286720*a^4*b^4*c^17*d^4 - 229376*a^5*b^3 *c^16*d^5 + 114688*a^6*b^2*c^15*d^6 - 32768*a*b^7*c^20*d))^(5/4) + 2654208 *a^19*c^16*d^16*x^(1/2)*(-(6561*a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a*b^3 *c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12)/(4096*b^8*c^21 + 4096*a^8*c^13*d^8 - 32768*a^7*b*c^14*d^7 + 114688*a^2*b^6*c^19*d^2 - 22937 6*a^3*b^5*c^18*d^3 + 286720*a^4*b^4*c^17*d^4 - 229376*a^5*b^3*c^16*d^5 + 1 14688*a^6*b^2*c^15*d^6 - 32768*a*b^7*c^20*d))^(5/4) + 346112*b^15*c^18*d^6 *x^(1/2)*(-(6561*a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 821 34*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12)/(4096*b^8*c^21 + 4096*a^8*c^13*d ^8 - 32768*a^7*b*c^14*d^7 + 114688*a^2*b^6*c^19*d^2 - 229376*a^3*b^5*c^18* d^3 + 286720*a^4*b^4*c^17*d^4 - 229376*a^5*b^3*c^16*d^5 + 114688*a^6*b^2*c ^15*d^6 - 32768*a*b^7*c^20*d))^(1/4) - 479232*a*b^14*c^17*d^7*x^(1/2)*(-(6 561*a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^ 2*d^11 - 37908*a^3*b*c*d^12)/(4096*b^8*c^21 + 4096*a^8*c^13*d^8 - 32768*a^ 7*b*c^14*d^7 + 114688*a^2*b^6*c^19*d^2 - 229376*a^3*b^5*c^18*d^3 + 2867...