Integrand size = 31, antiderivative size = 152 \[ \int \frac {x^2 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {c \left (3 b c^2+2 a d^2\right )}{2 d^5 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {b x^3}{2 d^2 \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (3 b c^2+2 a d^2\right ) \sqrt {-c+d x}}{2 d^5 \sqrt {c+d x}}+\frac {\left (3 b c^2+2 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{d^5} \]
(2*a*d^2+3*b*c^2)*arctanh((d*x-c)^(1/2)/(d*x+c)^(1/2))/d^5-1/2*c*(2*a*d^2+ 3*b*c^2)/d^5/(d*x-c)^(1/2)/(d*x+c)^(1/2)+1/2*b*x^3/d^2/(d*x-c)^(1/2)/(d*x+ c)^(1/2)-1/2*(2*a*d^2+3*b*c^2)*(d*x-c)^(1/2)/d^5/(d*x+c)^(1/2)
Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.71 \[ \int \frac {x^2 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {-3 b c^2 d x-2 a d^3 x+b d^3 x^3+2 \left (3 b c^2+2 a d^2\right ) \sqrt {-c+d x} \sqrt {c+d x} \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {-c+d x}}\right )}{2 d^5 \sqrt {-c+d x} \sqrt {c+d x}} \]
(-3*b*c^2*d*x - 2*a*d^3*x + b*d^3*x^3 + 2*(3*b*c^2 + 2*a*d^2)*Sqrt[-c + d* x]*Sqrt[c + d*x]*ArcTanh[Sqrt[c + d*x]/Sqrt[-c + d*x]])/(2*d^5*Sqrt[-c + d *x]*Sqrt[c + d*x])
Time = 0.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {960, 100, 27, 87, 45, 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.
\(\displaystyle \int \frac {x^2 \left (a+b x^2\right )}{(d x-c)^{3/2} (c+d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 960 |
\(\displaystyle \frac {1}{2} \left (2 a+\frac {3 b c^2}{d^2}\right ) \int \frac {x^2}{(d x-c)^{3/2} (c+d x)^{3/2}}dx+\frac {b x^3}{2 d^2 \sqrt {d x-c} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 100 |
\(\displaystyle \frac {1}{2} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {\int \frac {c d^2 x}{\sqrt {d x-c} (c+d x)^{3/2}}dx}{c d^3}-\frac {c}{d^3 \sqrt {d x-c} \sqrt {c+d x}}\right )+\frac {b x^3}{2 d^2 \sqrt {d x-c} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {\int \frac {x}{\sqrt {d x-c} (c+d x)^{3/2}}dx}{d}-\frac {c}{d^3 \sqrt {d x-c} \sqrt {c+d x}}\right )+\frac {b x^3}{2 d^2 \sqrt {d x-c} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{2} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {\frac {\int \frac {1}{\sqrt {d x-c} \sqrt {c+d x}}dx}{d}-\frac {\sqrt {d x-c}}{d^2 \sqrt {c+d x}}}{d}-\frac {c}{d^3 \sqrt {d x-c} \sqrt {c+d x}}\right )+\frac {b x^3}{2 d^2 \sqrt {d x-c} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 45 |
\(\displaystyle \frac {1}{2} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {\frac {2 \int \frac {1}{d-\frac {d (d x-c)}{c+d x}}d\frac {\sqrt {d x-c}}{\sqrt {c+d x}}}{d}-\frac {\sqrt {d x-c}}{d^2 \sqrt {c+d x}}}{d}-\frac {c}{d^3 \sqrt {d x-c} \sqrt {c+d x}}\right )+\frac {b x^3}{2 d^2 \sqrt {d x-c} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (2 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{d^2}-\frac {\sqrt {d x-c}}{d^2 \sqrt {c+d x}}}{d}-\frac {c}{d^3 \sqrt {d x-c} \sqrt {c+d x}}\right )+\frac {b x^3}{2 d^2 \sqrt {d x-c} \sqrt {c+d x}}\) |
(b*x^3)/(2*d^2*Sqrt[-c + d*x]*Sqrt[c + d*x]) + ((2*a + (3*b*c^2)/d^2)*(-(c /(d^3*Sqrt[-c + d*x]*Sqrt[c + d*x])) + (-(Sqrt[-c + d*x]/(d^2*Sqrt[c + d*x ])) + (2*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x]])/d^2)/d))/2
3.4.70.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && !GtQ[c, 0]
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*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) *(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^( m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(m + n *(p + 1) + 1))), x] - Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/ (b1*b2*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n /2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]
Time = 4.26 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.65
method | result | size |
risch | \(-\frac {b x \left (-d x +c \right ) \sqrt {d x +c}}{2 d^{4} \sqrt {d x -c}}+\frac {\left (\frac {2 a \,d^{2} \ln \left (\frac {x \,d^{2}}{\sqrt {d^{2}}}+\sqrt {d^{2} x^{2}-c^{2}}\right )}{\sqrt {d^{2}}}+\frac {3 b \,c^{2} \ln \left (\frac {x \,d^{2}}{\sqrt {d^{2}}}+\sqrt {d^{2} x^{2}-c^{2}}\right )}{\sqrt {d^{2}}}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d^{2} \left (x -\frac {c}{d}\right )^{2}+2 c d \left (x -\frac {c}{d}\right )}}{d^{2} \left (x -\frac {c}{d}\right )}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d^{2} \left (x +\frac {c}{d}\right )^{2}-2 c d \left (x +\frac {c}{d}\right )}}{d^{2} \left (x +\frac {c}{d}\right )}\right ) \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{2 d^{4} \sqrt {d x -c}\, \sqrt {d x +c}}\) | \(251\) |
default | \(\frac {\sqrt {d x -c}\, \left (-\operatorname {csgn}\left (d \right ) b \,d^{3} x^{3} \sqrt {d^{2} x^{2}-c^{2}}-2 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) a \,d^{4} x^{2}-3 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) b \,c^{2} d^{2} x^{2}+2 \sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right ) d^{3} a x +3 \sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right ) d b \,c^{2} x +2 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) a \,c^{2} d^{2}+3 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) b \,c^{4}\right ) \operatorname {csgn}\left (d \right )}{2 \left (-d x +c \right ) \sqrt {d^{2} x^{2}-c^{2}}\, d^{5} \sqrt {d x +c}}\) | \(263\) |
-1/2*b*x*(-d*x+c)*(d*x+c)^(1/2)/d^4/(d*x-c)^(1/2)+1/2/d^4*(2*a*d^2*ln(x*d^ 2/(d^2)^(1/2)+(d^2*x^2-c^2)^(1/2))/(d^2)^(1/2)+3*b*c^2*ln(x*d^2/(d^2)^(1/2 )+(d^2*x^2-c^2)^(1/2))/(d^2)^(1/2)-(a*d^2+b*c^2)/d^2/(x-c/d)*(d^2*(x-c/d)^ 2+2*c*d*(x-c/d))^(1/2)-(a*d^2+b*c^2)/d^2/(x+c/d)*(d^2*(x+c/d)^2-2*c*d*(x+c /d))^(1/2))*((d*x-c)*(d*x+c))^(1/2)/(d*x-c)^(1/2)/(d*x+c)^(1/2)
Time = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.05 \[ \int \frac {x^2 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 \, b c^{4} + 2 \, a c^{2} d^{2} - 2 \, {\left (b c^{2} d^{2} + a d^{4}\right )} x^{2} + {\left (b d^{3} x^{3} - {\left (3 \, b c^{2} d + 2 \, a d^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {d x - c} + {\left (3 \, b c^{4} + 2 \, a c^{2} d^{2} - {\left (3 \, b c^{2} d^{2} + 2 \, a d^{4}\right )} x^{2}\right )} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right )}{2 \, {\left (d^{7} x^{2} - c^{2} d^{5}\right )}} \]
1/2*(2*b*c^4 + 2*a*c^2*d^2 - 2*(b*c^2*d^2 + a*d^4)*x^2 + (b*d^3*x^3 - (3*b *c^2*d + 2*a*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c) + (3*b*c^4 + 2*a*c^2*d^2 - (3*b*c^2*d^2 + 2*a*d^4)*x^2)*log(-d*x + sqrt(d*x + c)*sqrt(d*x - c)))/(d ^7*x^2 - c^2*d^5)
Timed out. \[ \int \frac {x^2 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {b x^{3}}{2 \, \sqrt {d^{2} x^{2} - c^{2}} d^{2}} - \frac {3 \, b c^{2} x}{2 \, \sqrt {d^{2} x^{2} - c^{2}} d^{4}} - \frac {a x}{\sqrt {d^{2} x^{2} - c^{2}} d^{2}} + \frac {3 \, b c^{2} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{2 \, d^{5}} + \frac {a \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{d^{3}} \]
1/2*b*x^3/(sqrt(d^2*x^2 - c^2)*d^2) - 3/2*b*c^2*x/(sqrt(d^2*x^2 - c^2)*d^4 ) - a*x/(sqrt(d^2*x^2 - c^2)*d^2) + 3/2*b*c^2*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)*d)/d^5 + a*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)*d)/d^3
Time = 0.32 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.97 \[ \int \frac {x^2 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {\sqrt {d x + c} {\left ({\left (d x + c\right )} {\left (\frac {{\left (d x + c\right )} b}{d^{5}} - \frac {3 \, b c}{d^{5}}\right )} + \frac {b c^{2} d^{15} - a d^{17}}{d^{20}}\right )}}{2 \, \sqrt {d x - c}} - \frac {{\left (3 \, b c^{2} + 2 \, a d^{2}\right )} \log \left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}\right )}{2 \, d^{5}} - \frac {2 \, {\left (b c^{3} + a c d^{2}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 2 \, c\right )} d^{5}} \]
1/2*sqrt(d*x + c)*((d*x + c)*((d*x + c)*b/d^5 - 3*b*c/d^5) + (b*c^2*d^15 - a*d^17)/d^20)/sqrt(d*x - c) - 1/2*(3*b*c^2 + 2*a*d^2)*log((sqrt(d*x + c) - sqrt(d*x - c))^2)/d^5 - 2*(b*c^3 + a*c*d^2)/(((sqrt(d*x + c) - sqrt(d*x - c))^2 + 2*c)*d^5)
Timed out. \[ \int \frac {x^2 \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {x^2\,\left (b\,x^2+a\right )}{{\left (c+d\,x\right )}^{3/2}\,{\left (d\,x-c\right )}^{3/2}} \,d x \]