Integrand size = 24, antiderivative size = 120 \[ \int \frac {x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=-\frac {x \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}+\frac {(b c-2 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 \sqrt {a} b^2 \sqrt {b c-a d}}+\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b^2} \]
arctanh(x*d^(1/2)/(d*x^2+c)^(1/2))*d^(1/2)/b^2+1/2*(-2*a*d+b*c)*arctan(x*( -a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/b^2/a^(1/2)/(-a*d+b*c)^(1/2)-1/2* x*(d*x^2+c)^(1/2)/b/(b*x^2+a)
Time = 0.44 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.15 \[ \int \frac {x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {-\frac {b x \sqrt {c+d x^2}}{a+b x^2}+\frac {(-b c+2 a d) \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {a} \sqrt {b c-a d}}-2 \sqrt {d} \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{2 b^2} \]
(-((b*x*Sqrt[c + d*x^2])/(a + b*x^2)) + ((-(b*c) + 2*a*d)*ArcTan[(a*Sqrt[d ] + b*x*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt[b*c - a*d])])/(Sqrt[a ]*Sqrt[b*c - a*d]) - 2*Sqrt[d]*Log[-(Sqrt[d]*x) + Sqrt[c + d*x^2]])/(2*b^2 )
Time = 0.25 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {369, 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.
| \(\displaystyle \int \frac {x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle \frac {\int \frac {2 d x^2+c}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{2 b}-\frac {x \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {\frac {(b c-2 a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}+\frac {2 d \int \frac {1}{\sqrt {d x^2+c}}dx}{b}}{2 b}-\frac {x \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {(b c-2 a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}+\frac {2 d \int \frac {1}{1-\frac {d x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}}{2 b}-\frac {x \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {(b c-2 a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}+\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b}}{2 b}-\frac {x \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {(b c-2 a d) \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}+\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b}}{2 b}-\frac {x \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {(b c-2 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} b \sqrt {b c-a d}}+\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b}}{2 b}-\frac {x \sqrt {c+d x^2}}{2 b \left (a+b x^2\right )}\) |
-1/2*(x*Sqrt[c + d*x^2])/(b*(a + b*x^2)) + (((b*c - 2*a*d)*ArcTan[(Sqrt[b* c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(Sqrt[a]*b*Sqrt[b*c - a*d]) + (2*S qrt[d]*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/b)/(2*b)
3.8.33.3.1 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[((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]
Time = 3.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.82
| method | result | size |
| pseudoelliptic | \(-\frac {-2 \sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )+\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}+\frac {\left (2 a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{2 b^{2}}\) | \(99\) |
| default | \(\text {Expression too large to display}\) | \(1959\) |
-1/2/b^2*(-2*d^(1/2)*arctanh((d*x^2+c)^(1/2)/x/d^(1/2))+b*(d*x^2+c)^(1/2)* x/(b*x^2+a)+(2*a*d-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)))
Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (98) = 196\).
Time = 0.33 (sec) , antiderivative size = 1069, normalized size of antiderivative = 8.91 \[ \int \frac {x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\left [-\frac {4 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c} x - 4 \, {\left (a^{2} b c - a^{3} d + {\left (a b^{2} c - a^{2} 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 (a b c - 2 \, a^{2} d + {\left (b^{2} c - 2 \, a b 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 )}{8 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + {\left (a b^{4} c - a^{2} b^{3} d\right )} x^{2}\right )}}, -\frac {4 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c} x + 8 \, {\left (a^{2} b c - a^{3} d + {\left (a b^{2} c - a^{2} b d\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (a b c - 2 \, a^{2} d + {\left (b^{2} c - 2 \, a b 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 )}{8 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + {\left (a b^{4} c - a^{2} b^{3} d\right )} x^{2}\right )}}, -\frac {2 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c} x - \sqrt {a b c - a^{2} d} {\left (a b c - 2 \, a^{2} d + {\left (b^{2} c - 2 \, a b d\right )} x^{2}\right )} \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 (a^{2} b c - a^{3} d + {\left (a b^{2} c - a^{2} 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 )}{4 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + {\left (a b^{4} c - a^{2} b^{3} d\right )} x^{2}\right )}}, -\frac {2 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c} x - \sqrt {a b c - a^{2} d} {\left (a b c - 2 \, a^{2} d + {\left (b^{2} c - 2 \, a b d\right )} x^{2}\right )} \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 ) + 4 \, {\left (a^{2} b c - a^{3} d + {\left (a b^{2} c - a^{2} b d\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right )}{4 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + {\left (a b^{4} c - a^{2} b^{3} d\right )} x^{2}\right )}}\right ] \]
[-1/8*(4*(a*b^2*c - a^2*b*d)*sqrt(d*x^2 + c)*x - 4*(a^2*b*c - a^3*d + (a*b ^2*c - a^2*b*d)*x^2)*sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - (a*b*c - 2*a^2*d + (b^2*c - 2*a*b*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)))/(a^2*b^3*c - a^3*b^2*d + (a*b^4*c - a^2*b^3 *d)*x^2), -1/8*(4*(a*b^2*c - a^2*b*d)*sqrt(d*x^2 + c)*x + 8*(a^2*b*c - a^3 *d + (a*b^2*c - a^2*b*d)*x^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - (a*b*c - 2*a^2*d + (b^2*c - 2*a*b*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)))/(a^2*b^3*c - a^3*b^2*d + (a*b^4*c - a^2*b^3*d) *x^2), -1/4*(2*(a*b^2*c - a^2*b*d)*sqrt(d*x^2 + c)*x - sqrt(a*b*c - a^2*d) *(a*b*c - 2*a^2*d + (b^2*c - 2*a*b*d)*x^2)*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*(a^2*b*c - a^3*d + (a*b^2*c - a^2*b*d)*x^2)*sqrt(d) *log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c))/(a^2*b^3*c - a^3*b^2*d + (a*b^4*c - a^2*b^3*d)*x^2), -1/4*(2*(a*b^2*c - a^2*b*d)*sqrt(d*x^2 + c)*x - sqrt(a*b*c - a^2*d)*(a*b*c - 2*a^2*d + (b^2*c - 2*a*b*d)*x^2)*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...
\[ \int \frac {x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^{2} \sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {d x^{2} + c} x^{2}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (98) = 196\).
Time = 0.32 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.09 \[ \int \frac {x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=-\frac {{\left (b c \sqrt {d} - 2 \, a d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt {a b c d - a^{2} d^{2}} b^{2}} - \frac {\sqrt {d} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{2 \, b^{2}} + \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d^{\frac {3}{2}} - b c^{2} \sqrt {d}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} b^{2}} \]
-1/2*(b*c*sqrt(d) - 2*a*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^2) - 1/2*sqrt(d)*log((sqrt(d)*x - sqrt(d*x^2 + c))^2)/b^2 + ((sqrt(d)*x - sq rt(d*x^2 + c))^2*b*c*sqrt(d) - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d^(3/2) - b*c^2*sqrt(d))/(((sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqr t(d*x^2 + c))^2*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)*b^2)
Timed out. \[ \int \frac {x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x^2\,\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^2} \,d x \]