Integrand size = 21, antiderivative size = 199 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=\frac {c (5 b c-6 a d) x \sqrt {c+d x^2}}{16 a^3 (b c-a d) \left (a+b x^2\right )}+\frac {(5 b c-6 a d) x \left (c+d x^2\right )^{3/2}}{24 a^2 (b c-a d) \left (a+b x^2\right )^2}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a (b c-a d) \left (a+b x^2\right )^3}+\frac {c^2 (5 b c-6 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{16 a^{7/2} (b c-a d)^{3/2}} \]
1/24*(-6*a*d+5*b*c)*x*(d*x^2+c)^(3/2)/a^2/(-a*d+b*c)/(b*x^2+a)^2+1/6*b*x*( d*x^2+c)^(5/2)/a/(-a*d+b*c)/(b*x^2+a)^3+1/16*c^2*(-6*a*d+5*b*c)*arctan(x*( -a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/a^(7/2)/(-a*d+b*c)^(3/2)+1/16*c*( -6*a*d+5*b*c)*x*(d*x^2+c)^(1/2)/a^3/(-a*d+b*c)/(b*x^2+a)
Time = 15.18 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=\frac {-\frac {\sqrt {a} x \sqrt {c+d x^2} \left (15 b^3 c^2 x^4+8 a b^2 c x^2 \left (5 c-d x^2\right )-6 a^3 d \left (5 c+2 d x^2\right )+a^2 b \left (33 c^2-22 c d x^2-4 d^2 x^4\right )\right )}{(-b c+a d) \left (a+b x^2\right )^3}+\frac {3 c^2 (5 b c-6 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{3/2}}}{48 a^{7/2}} \]
(-((Sqrt[a]*x*Sqrt[c + d*x^2]*(15*b^3*c^2*x^4 + 8*a*b^2*c*x^2*(5*c - d*x^2 ) - 6*a^3*d*(5*c + 2*d*x^2) + a^2*b*(33*c^2 - 22*c*d*x^2 - 4*d^2*x^4)))/(( -(b*c) + a*d)*(a + b*x^2)^3)) + (3*c^2*(5*b*c - 6*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(b*c - a*d)^(3/2))/(48*a^(7/2))
Time = 0.28 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {296, 292, 292, 291, 218}
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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 296 |
| \(\displaystyle \frac {(5 b c-6 a d) \int \frac {\left (d x^2+c\right )^{3/2}}{\left (b x^2+a\right )^3}dx}{6 a (b c-a d)}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a \left (a+b x^2\right )^3 (b c-a d)}\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle \frac {(5 b c-6 a d) \left (\frac {3 c \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right )^2}dx}{4 a}+\frac {x \left (c+d x^2\right )^{3/2}}{4 a \left (a+b x^2\right )^2}\right )}{6 a (b c-a d)}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a \left (a+b x^2\right )^3 (b c-a d)}\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle \frac {(5 b c-6 a d) \left (\frac {3 c \left (\frac {c \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 a}+\frac {x \sqrt {c+d x^2}}{2 a \left (a+b x^2\right )}\right )}{4 a}+\frac {x \left (c+d x^2\right )^{3/2}}{4 a \left (a+b x^2\right )^2}\right )}{6 a (b c-a d)}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a \left (a+b x^2\right )^3 (b c-a d)}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {(5 b c-6 a d) \left (\frac {3 c \left (\frac {c \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{2 a}+\frac {x \sqrt {c+d x^2}}{2 a \left (a+b x^2\right )}\right )}{4 a}+\frac {x \left (c+d x^2\right )^{3/2}}{4 a \left (a+b x^2\right )^2}\right )}{6 a (b c-a d)}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a \left (a+b x^2\right )^3 (b c-a d)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(5 b c-6 a d) \left (\frac {3 c \left (\frac {c \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} \sqrt {b c-a d}}+\frac {x \sqrt {c+d x^2}}{2 a \left (a+b x^2\right )}\right )}{4 a}+\frac {x \left (c+d x^2\right )^{3/2}}{4 a \left (a+b x^2\right )^2}\right )}{6 a (b c-a d)}+\frac {b x \left (c+d x^2\right )^{5/2}}{6 a \left (a+b x^2\right )^3 (b c-a d)}\) |
(b*x*(c + d*x^2)^(5/2))/(6*a*(b*c - a*d)*(a + b*x^2)^3) + ((5*b*c - 6*a*d) *((x*(c + d*x^2)^(3/2))/(4*a*(a + b*x^2)^2) + (3*c*((x*Sqrt[c + d*x^2])/(2 *a*(a + b*x^2)) + (c*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])] )/(2*a^(3/2)*Sqrt[b*c - a*d])))/(4*a)))/(6*a*(b*c - a*d))
3.2.4.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Si mp[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] - Simp[c*(q/( a*(p + 1))) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1), x], x] /; FreeQ[ {a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0] && Gt Q[q, 0] && NeQ[p, -1]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d)) Int[ (a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1 ]) && NeQ[p, -1]
Time = 2.48 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(\frac {5 \left (d \left (\frac {2 d \,x^{2}}{5}+c \right ) a^{3}-\frac {11 \left (-\frac {4}{33} d^{2} x^{4}-\frac {2}{3} c d \,x^{2}+c^{2}\right ) b \,a^{2}}{10}-\frac {4 \left (-\frac {d \,x^{2}}{5}+c \right ) x^{2} b^{2} c a}{3}-\frac {c^{2} b^{3} x^{4}}{2}\right ) x \sqrt {d \,x^{2}+c}\, \sqrt {\left (a d -b c \right ) a}+3 \left (b \,x^{2}+a \right )^{3} \left (a d -\frac {5 b c}{6}\right ) c^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{8 \sqrt {\left (a d -b c \right ) a}\, \left (a d -b c \right ) a^{3} \left (b \,x^{2}+a \right )^{3}}\) | \(179\) |
| default | \(\text {Expression too large to display}\) | \(12958\) |
1/8*(5*(d*(2/5*d*x^2+c)*a^3-11/10*(-4/33*d^2*x^4-2/3*c*d*x^2+c^2)*b*a^2-4/ 3*(-1/5*d*x^2+c)*x^2*b^2*c*a-1/2*c^2*b^3*x^4)*x*(d*x^2+c)^(1/2)*((a*d-b*c) *a)^(1/2)+3*(b*x^2+a)^3*(a*d-5/6*b*c)*c^2*arctanh((d*x^2+c)^(1/2)/x*a/((a* d-b*c)*a)^(1/2)))/((a*d-b*c)*a)^(1/2)/(a*d-b*c)/a^3/(b*x^2+a)^3
Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (175) = 350\).
Time = 0.48 (sec) , antiderivative size = 972, normalized size of antiderivative = 4.88 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=\left [-\frac {3 \, {\left (5 \, a^{3} b c^{3} - 6 \, a^{4} c^{2} d + {\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{6} + 3 \, {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{4} + 3 \, {\left (5 \, a^{2} b^{2} c^{3} - 6 \, a^{3} b c^{2} d\right )} x^{2}\right )} \sqrt {-a b c + a^{2} d} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left ({\left (15 \, a b^{4} c^{3} - 23 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} + 4 \, a^{4} b d^{3}\right )} x^{5} + 2 \, {\left (20 \, a^{2} b^{3} c^{3} - 31 \, a^{3} b^{2} c^{2} d + 5 \, a^{4} b c d^{2} + 6 \, a^{5} d^{3}\right )} x^{3} + 3 \, {\left (11 \, a^{3} b^{2} c^{3} - 21 \, a^{4} b c^{2} d + 10 \, a^{5} c d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{192 \, {\left (a^{7} b^{2} c^{2} - 2 \, a^{8} b c d + a^{9} d^{2} + {\left (a^{4} b^{5} c^{2} - 2 \, a^{5} b^{4} c d + a^{6} b^{3} d^{2}\right )} x^{6} + 3 \, {\left (a^{5} b^{4} c^{2} - 2 \, a^{6} b^{3} c d + a^{7} b^{2} d^{2}\right )} x^{4} + 3 \, {\left (a^{6} b^{3} c^{2} - 2 \, a^{7} b^{2} c d + a^{8} b d^{2}\right )} x^{2}\right )}}, \frac {3 \, {\left (5 \, a^{3} b c^{3} - 6 \, a^{4} c^{2} d + {\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{6} + 3 \, {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{4} + 3 \, {\left (5 \, a^{2} b^{2} c^{3} - 6 \, a^{3} b c^{2} d\right )} x^{2}\right )} \sqrt {a b c - a^{2} d} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left ({\left (15 \, a b^{4} c^{3} - 23 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} + 4 \, a^{4} b d^{3}\right )} x^{5} + 2 \, {\left (20 \, a^{2} b^{3} c^{3} - 31 \, a^{3} b^{2} c^{2} d + 5 \, a^{4} b c d^{2} + 6 \, a^{5} d^{3}\right )} x^{3} + 3 \, {\left (11 \, a^{3} b^{2} c^{3} - 21 \, a^{4} b c^{2} d + 10 \, a^{5} c d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{96 \, {\left (a^{7} b^{2} c^{2} - 2 \, a^{8} b c d + a^{9} d^{2} + {\left (a^{4} b^{5} c^{2} - 2 \, a^{5} b^{4} c d + a^{6} b^{3} d^{2}\right )} x^{6} + 3 \, {\left (a^{5} b^{4} c^{2} - 2 \, a^{6} b^{3} c d + a^{7} b^{2} d^{2}\right )} x^{4} + 3 \, {\left (a^{6} b^{3} c^{2} - 2 \, a^{7} b^{2} c d + a^{8} b d^{2}\right )} x^{2}\right )}}\right ] \]
[-1/192*(3*(5*a^3*b*c^3 - 6*a^4*c^2*d + (5*b^4*c^3 - 6*a*b^3*c^2*d)*x^6 + 3*(5*a*b^3*c^3 - 6*a^2*b^2*c^2*d)*x^4 + 3*(5*a^2*b^2*c^3 - 6*a^3*b*c^2*d)* x^2)*sqrt(-a*b*c + a^2*d)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^2 *c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 - 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt( -a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*((15*a*b ^4*c^3 - 23*a^2*b^3*c^2*d + 4*a^3*b^2*c*d^2 + 4*a^4*b*d^3)*x^5 + 2*(20*a^2 *b^3*c^3 - 31*a^3*b^2*c^2*d + 5*a^4*b*c*d^2 + 6*a^5*d^3)*x^3 + 3*(11*a^3*b ^2*c^3 - 21*a^4*b*c^2*d + 10*a^5*c*d^2)*x)*sqrt(d*x^2 + c))/(a^7*b^2*c^2 - 2*a^8*b*c*d + a^9*d^2 + (a^4*b^5*c^2 - 2*a^5*b^4*c*d + a^6*b^3*d^2)*x^6 + 3*(a^5*b^4*c^2 - 2*a^6*b^3*c*d + a^7*b^2*d^2)*x^4 + 3*(a^6*b^3*c^2 - 2*a^ 7*b^2*c*d + a^8*b*d^2)*x^2), 1/96*(3*(5*a^3*b*c^3 - 6*a^4*c^2*d + (5*b^4*c ^3 - 6*a*b^3*c^2*d)*x^6 + 3*(5*a*b^3*c^3 - 6*a^2*b^2*c^2*d)*x^4 + 3*(5*a^2 *b^2*c^3 - 6*a^3*b*c^2*d)*x^2)*sqrt(a*b*c - a^2*d)*arctan(1/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) + 2*((15*a*b^4*c^3 - 23*a^2*b^3*c^2*d + 4*a^3*b ^2*c*d^2 + 4*a^4*b*d^3)*x^5 + 2*(20*a^2*b^3*c^3 - 31*a^3*b^2*c^2*d + 5*a^4 *b*c*d^2 + 6*a^5*d^3)*x^3 + 3*(11*a^3*b^2*c^3 - 21*a^4*b*c^2*d + 10*a^5*c* d^2)*x)*sqrt(d*x^2 + c))/(a^7*b^2*c^2 - 2*a^8*b*c*d + a^9*d^2 + (a^4*b^5*c ^2 - 2*a^5*b^4*c*d + a^6*b^3*d^2)*x^6 + 3*(a^5*b^4*c^2 - 2*a^6*b^3*c*d + a ^7*b^2*d^2)*x^4 + 3*(a^6*b^3*c^2 - 2*a^7*b^2*c*d + a^8*b*d^2)*x^2)]
Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=\text {Timed out} \]
\[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{4}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 919 vs. \(2 (175) = 350\).
Time = 1.26 (sec) , antiderivative size = 919, normalized size of antiderivative = 4.62 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=-\frac {{\left (5 \, b c^{3} \sqrt {d} - 6 \, a c^{2} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{16 \, {\left (a^{3} b c - a^{4} d\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} b^{5} c^{3} \sqrt {d} - 18 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{10} a b^{4} c^{2} d^{\frac {3}{2}} - 75 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{5} c^{4} \sqrt {d} + 240 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b^{4} c^{3} d^{\frac {3}{2}} - 180 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} b^{3} c^{2} d^{\frac {5}{2}} - 96 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{3} b^{2} c d^{\frac {7}{2}} + 96 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{4} b d^{\frac {9}{2}} + 150 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{5} c^{5} \sqrt {d} - 620 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b^{4} c^{4} d^{\frac {3}{2}} + 968 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} b^{3} c^{3} d^{\frac {5}{2}} - 720 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{3} b^{2} c^{2} d^{\frac {7}{2}} + 64 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{4} b c d^{\frac {9}{2}} + 128 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{5} d^{\frac {11}{2}} - 150 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{5} c^{6} \sqrt {d} + 600 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b^{4} c^{5} d^{\frac {3}{2}} - 864 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} b^{3} c^{4} d^{\frac {5}{2}} + 288 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{3} b^{2} c^{3} d^{\frac {7}{2}} + 96 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{4} b c^{2} d^{\frac {9}{2}} + 75 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{5} c^{7} \sqrt {d} - 210 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{4} c^{6} d^{\frac {3}{2}} + 72 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} b^{3} c^{5} d^{\frac {5}{2}} + 48 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{3} b^{2} c^{4} d^{\frac {7}{2}} - 15 \, b^{5} c^{8} \sqrt {d} + 8 \, a b^{4} c^{7} d^{\frac {3}{2}} + 4 \, a^{2} b^{3} c^{6} d^{\frac {5}{2}}}{24 \, {\left (a^{3} b^{3} c - a^{4} b^{2} d\right )} {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}^{3}} \]
-1/16*(5*b*c^3*sqrt(d) - 6*a*c^2*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d* x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/((a^3*b*c - a^4*d)*s qrt(a*b*c*d - a^2*d^2)) - 1/24*(15*(sqrt(d)*x - sqrt(d*x^2 + c))^10*b^5*c^ 3*sqrt(d) - 18*(sqrt(d)*x - sqrt(d*x^2 + c))^10*a*b^4*c^2*d^(3/2) - 75*(sq rt(d)*x - sqrt(d*x^2 + c))^8*b^5*c^4*sqrt(d) + 240*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a*b^4*c^3*d^(3/2) - 180*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^2*b^3*c ^2*d^(5/2) - 96*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^3*b^2*c*d^(7/2) + 96*(sq rt(d)*x - sqrt(d*x^2 + c))^8*a^4*b*d^(9/2) + 150*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^5*c^5*sqrt(d) - 620*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a*b^4*c^4*d^( 3/2) + 968*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^2*b^3*c^3*d^(5/2) - 720*(sqrt (d)*x - sqrt(d*x^2 + c))^6*a^3*b^2*c^2*d^(7/2) + 64*(sqrt(d)*x - sqrt(d*x^ 2 + c))^6*a^4*b*c*d^(9/2) + 128*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^5*d^(11/ 2) - 150*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^5*c^6*sqrt(d) + 600*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b^4*c^5*d^(3/2) - 864*(sqrt(d)*x - sqrt(d*x^2 + c)) ^4*a^2*b^3*c^4*d^(5/2) + 288*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^3*b^2*c^3*d ^(7/2) + 96*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^4*b*c^2*d^(9/2) + 75*(sqrt(d )*x - sqrt(d*x^2 + c))^2*b^5*c^7*sqrt(d) - 210*(sqrt(d)*x - sqrt(d*x^2 + c ))^2*a*b^4*c^6*d^(3/2) + 72*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*b^3*c^5*d^ (5/2) + 48*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^3*b^2*c^4*d^(7/2) - 15*b^5*c^ 8*sqrt(d) + 8*a*b^4*c^7*d^(3/2) + 4*a^2*b^3*c^6*d^(5/2))/((a^3*b^3*c - ...
Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^4} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}}{{\left (b\,x^2+a\right )}^4} \,d x \]