Integrand size = 24, antiderivative size = 170 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {2 b c+3 a d}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {2 b c+3 a d}{2 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\sqrt {b} (2 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 (b c-a d)^{7/2}} \]
1/6*(3*a*d+2*b*c)/b/(-a*d+b*c)^2/(d*x^2+c)^(3/2)+1/2*a/b/(-a*d+b*c)/(b*x^2 +a)/(d*x^2+c)^(3/2)-1/2*(3*a*d+2*b*c)*arctanh(b^(1/2)*(d*x^2+c)^(1/2)/(-a* d+b*c)^(1/2))*b^(1/2)/(-a*d+b*c)^(7/2)+1/2*(3*a*d+2*b*c)/(-a*d+b*c)^3/(d*x ^2+c)^(1/2)
Time = 0.46 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {1}{6} \left (\frac {2 a^2 d \left (2 c+3 d x^2\right )+2 b^2 c x^2 \left (4 c+3 d x^2\right )+a b \left (11 c^2+16 c d x^2+9 d^2 x^4\right )}{(b c-a d)^3 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {3 \sqrt {b} (2 b c+3 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{7/2}}\right ) \]
((2*a^2*d*(2*c + 3*d*x^2) + 2*b^2*c*x^2*(4*c + 3*d*x^2) + a*b*(11*c^2 + 16 *c*d*x^2 + 9*d^2*x^4))/((b*c - a*d)^3*(a + b*x^2)*(c + d*x^2)^(3/2)) - (3* Sqrt[b]*(2*b*c + 3*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[-(b*c) + a*d ]])/(-(b*c) + a*d)^(7/2))/6
Time = 0.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {354, 87, 61, 61, 73, 221}
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.
| \(\displaystyle \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2}{\left (b x^2+a\right )^2 \left (d x^2+c\right )^{5/2}}dx^2\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{2} \left (\frac {(3 a d+2 b c) \int \frac {1}{\left (b x^2+a\right ) \left (d x^2+c\right )^{5/2}}dx^2}{2 b (b c-a d)}+\frac {a}{b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{2} \left (\frac {(3 a d+2 b c) \left (\frac {b \int \frac {1}{\left (b x^2+a\right ) \left (d x^2+c\right )^{3/2}}dx^2}{b c-a d}+\frac {2}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{2 b (b c-a d)}+\frac {a}{b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{2} \left (\frac {(3 a d+2 b c) \left (\frac {b \left (\frac {b \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx^2}{b c-a d}+\frac {2}{\sqrt {c+d x^2} (b c-a d)}\right )}{b c-a d}+\frac {2}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{2 b (b c-a d)}+\frac {a}{b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {(3 a d+2 b c) \left (\frac {b \left (\frac {2 b \int \frac {1}{\frac {b x^4}{d}+a-\frac {b c}{d}}d\sqrt {d x^2+c}}{d (b c-a d)}+\frac {2}{\sqrt {c+d x^2} (b c-a d)}\right )}{b c-a d}+\frac {2}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{2 b (b c-a d)}+\frac {a}{b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {(3 a d+2 b c) \left (\frac {b \left (\frac {2}{\sqrt {c+d x^2} (b c-a d)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}\right )}{b c-a d}+\frac {2}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{2 b (b c-a d)}+\frac {a}{b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}\right )\) |
(a/(b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) + ((2*b*c + 3*a*d)*(2/(3* (b*c - a*d)*(c + d*x^2)^(3/2)) + (b*(2/((b*c - a*d)*Sqrt[c + d*x^2]) - (2* Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3 /2)))/(b*c - a*d)))/(2*b*(b*c - a*d)))/2
3.8.77.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 3.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.95
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\frac {9 \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )^{\frac {3}{2}} b \left (a d +\frac {2 b c}{3}\right ) \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{4}+\left (2 x^{2} \left (\frac {3 d \,x^{2}}{4}+c \right ) c \,b^{2}+\frac {11 \left (\frac {9}{11} d^{2} x^{4}+\frac {16}{11} c d \,x^{2}+c^{2}\right ) a b}{4}+d \,a^{2} \left (\frac {3 d \,x^{2}}{2}+c \right )\right ) \sqrt {\left (a d -b c \right ) b}\right )}{3 \left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\left (a d -b c \right ) b}\, \left (b \,x^{2}+a \right ) \left (a d -b c \right )^{3}}\) | \(162\) |
| default | \(\text {Expression too large to display}\) | \(3483\) |
-2/3*(9/4*(b*x^2+a)*(d*x^2+c)^(3/2)*b*(a*d+2/3*b*c)*arctan(b*(d*x^2+c)^(1/ 2)/((a*d-b*c)*b)^(1/2))+(2*x^2*(3/4*d*x^2+c)*c*b^2+11/4*(9/11*d^2*x^4+16/1 1*c*d*x^2+c^2)*a*b+d*a^2*(3/2*d*x^2+c))*((a*d-b*c)*b)^(1/2))/(d*x^2+c)^(3/ 2)/((a*d-b*c)*b)^(1/2)/(b*x^2+a)/(a*d-b*c)^3
Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (146) = 292\).
Time = 0.36 (sec) , antiderivative size = 993, normalized size of antiderivative = 5.84 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (2 \, b^{2} c d^{2} + 3 \, a b d^{3}\right )} x^{6} + 2 \, a b c^{3} + 3 \, a^{2} c^{2} d + {\left (4 \, b^{2} c^{2} d + 8 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{4} + {\left (2 \, b^{2} c^{3} + 7 \, a b c^{2} d + 6 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (3 \, {\left (2 \, b^{2} c d + 3 \, a b d^{2}\right )} x^{4} + 11 \, a b c^{2} + 4 \, a^{2} c d + 2 \, {\left (4 \, b^{2} c^{2} + 8 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}, \frac {3 \, {\left ({\left (2 \, b^{2} c d^{2} + 3 \, a b d^{3}\right )} x^{6} + 2 \, a b c^{3} + 3 \, a^{2} c^{2} d + {\left (4 \, b^{2} c^{2} d + 8 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{4} + {\left (2 \, b^{2} c^{3} + 7 \, a b c^{2} d + 6 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{b c - a d}}}{2 \, {\left (b d x^{2} + b c\right )}}\right ) + 2 \, {\left (3 \, {\left (2 \, b^{2} c d + 3 \, a b d^{2}\right )} x^{4} + 11 \, a b c^{2} + 4 \, a^{2} c d + 2 \, {\left (4 \, b^{2} c^{2} + 8 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}\right ] \]
[-1/24*(3*((2*b^2*c*d^2 + 3*a*b*d^3)*x^6 + 2*a*b*c^3 + 3*a^2*c^2*d + (4*b^ 2*c^2*d + 8*a*b*c*d^2 + 3*a^2*d^3)*x^4 + (2*b^2*c^3 + 7*a*b*c^2*d + 6*a^2* c*d^2)*x^2)*sqrt(b/(b*c - a*d))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 + 4*(2*b^2*c^2 - 3*a*b*c*d + a^2* d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*c - a*d)))/(b^2*x ^4 + 2*a*b*x^2 + a^2)) - 4*(3*(2*b^2*c*d + 3*a*b*d^2)*x^4 + 11*a*b*c^2 + 4 *a^2*c*d + 2*(4*b^2*c^2 + 8*a*b*c*d + 3*a^2*d^2)*x^2)*sqrt(d*x^2 + c))/(a* b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b ^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a *b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*x^2), 1/12 *(3*((2*b^2*c*d^2 + 3*a*b*d^3)*x^6 + 2*a*b*c^3 + 3*a^2*c^2*d + (4*b^2*c^2* d + 8*a*b*c*d^2 + 3*a^2*d^3)*x^4 + (2*b^2*c^3 + 7*a*b*c^2*d + 6*a^2*c*d^2) *x^2)*sqrt(-b/(b*c - a*d))*arctan(1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-b/(b*c - a*d))/(b*d*x^2 + b*c)) + 2*(3*(2*b^2*c*d + 3*a*b*d^2)*x ^4 + 11*a*b*c^2 + 4*a^2*c*d + 2*(4*b^2*c^2 + 8*a*b*c*d + 3*a^2*d^2)*x^2)*s qrt(d*x^2 + c))/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d ^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + ( 2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5) *x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2...
\[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {x^{3}}{\left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Time = 0.29 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.53 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {\frac {3 \, \sqrt {d x^{2} + c} a b d^{2}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}} + \frac {3 \, {\left (2 \, b^{2} c d + 3 \, a b d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, {\left (3 \, {\left (d x^{2} + c\right )} b c d + b c^{2} d + 3 \, {\left (d x^{2} + c\right )} a d^{2} - a c d^{2}\right )}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}}{6 \, d} \]
1/6*(3*sqrt(d*x^2 + c)*a*b*d^2/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*((d*x^2 + c)*b - b*c + a*d)) + 3*(2*b^2*c*d + 3*a*b*d^2)*arctan( sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2* b*c*d^2 - a^3*d^3)*sqrt(-b^2*c + a*b*d)) + 2*(3*(d*x^2 + c)*b*c*d + b*c^2* d + 3*(d*x^2 + c)*a*d^2 - a*c*d^2)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d ^2 - a^3*d^3)*(d*x^2 + c)^(3/2)))/d
Time = 5.66 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.14 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=-\frac {\frac {\left (d\,x^2+c\right )\,\left (3\,a\,d+2\,b\,c\right )}{3\,{\left (a\,d-b\,c\right )}^2}-\frac {c}{3\,\left (a\,d-b\,c\right )}+\frac {b\,{\left (d\,x^2+c\right )}^2\,\left (3\,a\,d+2\,b\,c\right )}{2\,{\left (a\,d-b\,c\right )}^3}}{b\,{\left (d\,x^2+c\right )}^{5/2}+{\left (d\,x^2+c\right )}^{3/2}\,\left (a\,d-b\,c\right )}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^2+c}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^{7/2}}\right )\,\left (3\,a\,d+2\,b\,c\right )}{2\,{\left (a\,d-b\,c\right )}^{7/2}} \]
- (((c + d*x^2)*(3*a*d + 2*b*c))/(3*(a*d - b*c)^2) - c/(3*(a*d - b*c)) + ( b*(c + d*x^2)^2*(3*a*d + 2*b*c))/(2*(a*d - b*c)^3))/(b*(c + d*x^2)^(5/2) + (c + d*x^2)^(3/2)*(a*d - b*c)) - (b^(1/2)*atan((b^(1/2)*(c + d*x^2)^(1/2) *(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*d - b*c)^(7/2))*( 3*a*d + 2*b*c))/(2*(a*d - b*c)^(7/2))