Integrand size = 24, antiderivative size = 150 \[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {x \sqrt {c+d x^2}}{b^2}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}-\frac {\sqrt {a} (3 b c-4 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 b^3 \sqrt {b c-a d}}+\frac {(b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 b^3 \sqrt {d}} \]
1/2*(-4*a*d+b*c)*arctanh(x*d^(1/2)/(d*x^2+c)^(1/2))/b^3/d^(1/2)-1/2*(-4*a* d+3*b*c)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))*a^(1/2)/b^3/(- a*d+b*c)^(1/2)+x*(d*x^2+c)^(1/2)/b^2-1/2*x^3*(d*x^2+c)^(1/2)/b/(b*x^2+a)
Time = 10.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.89 \[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {b x \left (2 a+b x^2\right ) \sqrt {c+d x^2}}{a+b x^2}+\frac {\sqrt {a} (-3 b c+4 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {b c-a d}}+\frac {(b c-4 a d) \log \left (d x+\sqrt {d} \sqrt {c+d x^2}\right )}{\sqrt {d}}}{2 b^3} \]
((b*x*(2*a + b*x^2)*Sqrt[c + d*x^2])/(a + b*x^2) + (Sqrt[a]*(-3*b*c + 4*a* d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[b*c - a*d] + ((b*c - 4*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(2*b^3)
Time = 0.34 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {369, 444, 27, 398, 224, 219, 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.
| \(\displaystyle \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle \frac {\int \frac {x^2 \left (4 d x^2+3 c\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 b}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {\frac {2 x \sqrt {c+d x^2}}{b}-\frac {\int \frac {2 d \left (2 a c-(b c-4 a d) x^2\right )}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 b d}}{2 b}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 x \sqrt {c+d x^2}}{b}-\frac {\int \frac {2 a c-(b c-4 a d) x^2}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}}{2 b}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {\frac {2 x \sqrt {c+d x^2}}{b}-\frac {\frac {a (3 b c-4 a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}-\frac {(b c-4 a d) \int \frac {1}{\sqrt {d x^2+c}}dx}{b}}{b}}{2 b}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {2 x \sqrt {c+d x^2}}{b}-\frac {\frac {a (3 b c-4 a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}-\frac {(b c-4 a d) \int \frac {1}{1-\frac {d x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}}{b}}{2 b}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {2 x \sqrt {c+d x^2}}{b}-\frac {\frac {a (3 b c-4 a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}-\frac {(b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}}{b}}{2 b}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {2 x \sqrt {c+d x^2}}{b}-\frac {\frac {a (3 b c-4 a d) \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}-\frac {(b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}}{b}}{2 b}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {2 x \sqrt {c+d x^2}}{b}-\frac {\frac {\sqrt {a} (3 b c-4 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b \sqrt {b c-a d}}-\frac {(b c-4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}}{b}}{2 b}-\frac {x^3 \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
-1/2*(x^3*Sqrt[c + d*x^2])/(b*(a + b*x^2)) + ((2*x*Sqrt[c + d*x^2])/b - (( Sqrt[a]*(3*b*c - 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2 ])])/(b*Sqrt[b*c - a*d]) - ((b*c - 4*a*d)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x ^2]])/(b*Sqrt[d]))/b)/(2*b)
3.8.31.3.1 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[((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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2* b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(m - 1) + d*(m + 2*q - 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 0 ] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[m, 1]
Time = 3.22 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97
| method | result | size |
| pseudoelliptic | \(-\frac {\frac {-\sqrt {d \,x^{2}+c}\, b x \sqrt {d}+4 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right ) a d -\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right ) b c}{\sqrt {d}}+a \left (-\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}-\frac {\left (4 a d -3 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2 b^{3}}\) | \(145\) |
| risch | \(\frac {x \sqrt {d \,x^{2}+c}}{2 b^{2}}-\frac {\frac {\left (4 a d -b c \right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b \sqrt {d}}-\frac {\left (a d -b c \right ) a \left (\frac {b \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{2 b^{2}}-\frac {\left (a d -b c \right ) a \left (\frac {b \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{2 b^{2}}+\frac {a \left (5 a d -3 b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {a \left (5 a d -3 b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{2 b^{2}}\) | \(907\) |
| default | \(\text {Expression too large to display}\) | \(2002\) |
-1/2/b^3*((-(d*x^2+c)^(1/2)*b*x*d^(1/2)+4*arctanh((d*x^2+c)^(1/2)/x/d^(1/2 ))*a*d-arctanh((d*x^2+c)^(1/2)/x/d^(1/2))*b*c)/d^(1/2)+a*(-b*(d*x^2+c)^(1/ 2)*x/(b*x^2+a)-(4*a*d-3*b*c)/((a*d-b*c)*a)^(1/2)*arctanh((d*x^2+c)^(1/2)/x *a/((a*d-b*c)*a)^(1/2))))
Time = 0.36 (sec) , antiderivative size = 1002, normalized size of antiderivative = 6.68 \[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\left [-\frac {2 \, {\left (a b c - 4 \, a^{2} d + {\left (b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + {\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, b^{2} c d - 4 \, a b d^{2}\right )} x^{2}\right )} \sqrt {-\frac {a}{b c - a 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^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (b^{2} d x^{3} + 2 \, a b d x\right )} \sqrt {d x^{2} + c}}{8 \, {\left (b^{4} d x^{2} + a b^{3} d\right )}}, -\frac {4 \, {\left (a b c - 4 \, a^{2} d + {\left (b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, b^{2} c d - 4 \, a b d^{2}\right )} x^{2}\right )} \sqrt {-\frac {a}{b c - a 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^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (b^{2} d x^{3} + 2 \, a b d x\right )} \sqrt {d x^{2} + c}}{8 \, {\left (b^{4} d x^{2} + a b^{3} d\right )}}, \frac {{\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, b^{2} c d - 4 \, a b d^{2}\right )} x^{2}\right )} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{3} + a c x\right )}}\right ) - {\left (a b c - 4 \, a^{2} d + {\left (b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (b^{2} d x^{3} + 2 \, a b d x\right )} \sqrt {d x^{2} + c}}{4 \, {\left (b^{4} d x^{2} + a b^{3} d\right )}}, -\frac {2 \, {\left (a b c - 4 \, a^{2} d + {\left (b^{2} c - 4 \, a b d\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (3 \, a b c d - 4 \, a^{2} d^{2} + {\left (3 \, b^{2} c d - 4 \, a b d^{2}\right )} x^{2}\right )} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{3} + a c x\right )}}\right ) - 2 \, {\left (b^{2} d x^{3} + 2 \, a b d x\right )} \sqrt {d x^{2} + c}}{4 \, {\left (b^{4} d x^{2} + a b^{3} d\right )}}\right ] \]
[-1/8*(2*(a*b*c - 4*a^2*d + (b^2*c - 4*a*b*d)*x^2)*sqrt(d)*log(-2*d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + (3*a*b*c*d - 4*a^2*d^2 + (3*b^2*c*d - 4 *a*b*d^2)*x^2)*sqrt(-a/(b*c - a*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^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^3 - (a*b*c^2 - a^2*c*d)*x)*sqrt(d*x^2 + c)*sqrt(-a/(b*c - a* d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(b^2*d*x^3 + 2*a*b*d*x)*sqrt(d*x^2 + c))/(b^4*d*x^2 + a*b^3*d), -1/8*(4*(a*b*c - 4*a^2*d + (b^2*c - 4*a*b*d)*x ^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + (3*a*b*c*d - 4*a^2*d^2 + (3*b^2*c*d - 4*a*b*d^2)*x^2)*sqrt(-a/(b*c - a*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^2*c^ 2 - 3*a*b*c*d + 2*a^2*d^2)*x^3 - (a*b*c^2 - a^2*c*d)*x)*sqrt(d*x^2 + c)*sq rt(-a/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(b^2*d*x^3 + 2*a*b*d* x)*sqrt(d*x^2 + c))/(b^4*d*x^2 + a*b^3*d), 1/4*((3*a*b*c*d - 4*a^2*d^2 + ( 3*b^2*c*d - 4*a*b*d^2)*x^2)*sqrt(a/(b*c - a*d))*arctan(-1/2*((b*c - 2*a*d) *x^2 - a*c)*sqrt(d*x^2 + c)*sqrt(a/(b*c - a*d))/(a*d*x^3 + a*c*x)) - (a*b* c - 4*a^2*d + (b^2*c - 4*a*b*d)*x^2)*sqrt(d)*log(-2*d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + 2*(b^2*d*x^3 + 2*a*b*d*x)*sqrt(d*x^2 + c))/(b^4*d*x^2 + a*b^3*d), -1/4*(2*(a*b*c - 4*a^2*d + (b^2*c - 4*a*b*d)*x^2)*sqrt(-d)*ar ctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - (3*a*b*c*d - 4*a^2*d^2 + (3*b^2*c*d - 4 *a*b*d^2)*x^2)*sqrt(a/(b*c - a*d))*arctan(-1/2*((b*c - 2*a*d)*x^2 - a*c...
\[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{4} \sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {d x^{2} + c} x^{4}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (124) = 248\).
Time = 0.31 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.89 \[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {d x^{2} + c} x}{2 \, b^{2}} + \frac {{\left (3 \, a b c \sqrt {d} - 4 \, a^{2} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} b^{3}} - \frac {{\left (b c - 4 \, a d\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{4 \, b^{3} \sqrt {d}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} d^{\frac {3}{2}} - a b c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} b^{3}} \]
1/2*sqrt(d*x^2 + c)*x/b^2 + 1/2*(3*a*b*c*sqrt(d) - 4*a^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))/(sqrt(a*b*c*d - a^2*d^2)*b^3) - 1/4*(b*c - 4*a*d)*log((sqrt(d)*x - sqr t(d*x^2 + c))^2)/(b^3*sqrt(d)) - ((sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c*sq rt(d) - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*d^(3/2) - a*b*c^2*sqrt(d))/( ((sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)*b^3)
Timed out. \[ \int \frac {x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^4\,\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^2} \,d x \]