Integrand size = 31, antiderivative size = 118 \[ \int \frac {x^2 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {\left (3 b c^2+4 a d^2\right ) x \sqrt {-c+d x} \sqrt {c+d x}}{8 d^4}+\frac {b x^3 \sqrt {-c+d x} \sqrt {c+d x}}{4 d^2}+\frac {c^2 \left (3 b c^2+4 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{4 d^5} \]
1/4*c^2*(4*a*d^2+3*b*c^2)*arctanh((d*x-c)^(1/2)/(d*x+c)^(1/2))/d^5+1/8*(4* a*d^2+3*b*c^2)*x*(d*x-c)^(1/2)*(d*x+c)^(1/2)/d^4+1/4*b*x^3*(d*x-c)^(1/2)*( d*x+c)^(1/2)/d^2
Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.78 \[ \int \frac {x^2 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {d x \sqrt {-c+d x} \sqrt {c+d x} \left (3 b c^2+4 a d^2+2 b d^2 x^2\right )+\left (6 b c^4+8 a c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{8 d^5} \]
(d*x*Sqrt[-c + d*x]*Sqrt[c + d*x]*(3*b*c^2 + 4*a*d^2 + 2*b*d^2*x^2) + (6*b *c^4 + 8*a*c^2*d^2)*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x]])/(8*d^5)
Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {960, 101, 27, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b x^2\right )}{\sqrt {d x-c} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 960 |
| \(\displaystyle \frac {1}{4} \left (4 a+\frac {3 b c^2}{d^2}\right ) \int \frac {x^2}{\sqrt {d x-c} \sqrt {c+d x}}dx+\frac {b x^3 \sqrt {d x-c} \sqrt {c+d x}}{4 d^2}\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {1}{4} \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {\int \frac {c^2}{\sqrt {d x-c} \sqrt {c+d x}}dx}{2 d^2}+\frac {x \sqrt {d x-c} \sqrt {c+d x}}{2 d^2}\right )+\frac {b x^3 \sqrt {d x-c} \sqrt {c+d x}}{4 d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {c^2 \int \frac {1}{\sqrt {d x-c} \sqrt {c+d x}}dx}{2 d^2}+\frac {x \sqrt {d x-c} \sqrt {c+d x}}{2 d^2}\right )+\frac {b x^3 \sqrt {d x-c} \sqrt {c+d x}}{4 d^2}\) |
| \(\Big \downarrow \) 45 |
| \(\displaystyle \frac {1}{4} \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {c^2 \int \frac {1}{d-\frac {d (d x-c)}{c+d x}}d\frac {\sqrt {d x-c}}{\sqrt {c+d x}}}{d^2}+\frac {x \sqrt {d x-c} \sqrt {c+d x}}{2 d^2}\right )+\frac {b x^3 \sqrt {d x-c} \sqrt {c+d x}}{4 d^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{4} \left (4 a+\frac {3 b c^2}{d^2}\right ) \left (\frac {c^2 \text {arctanh}\left (\frac {\sqrt {d x-c}}{\sqrt {c+d x}}\right )}{d^3}+\frac {x \sqrt {d x-c} \sqrt {c+d x}}{2 d^2}\right )+\frac {b x^3 \sqrt {d x-c} \sqrt {c+d x}}{4 d^2}\) |
(b*x^3*Sqrt[-c + d*x]*Sqrt[c + d*x])/(4*d^2) + ((4*a + (3*b*c^2)/d^2)*((x* Sqrt[-c + d*x]*Sqrt[c + d*x])/(2*d^2) + (c^2*ArcTanh[Sqrt[-c + d*x]/Sqrt[c + d*x]])/d^3))/4
3.4.60.3.1 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_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
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 = 4.21 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(-\frac {x \left (2 b \,d^{2} x^{2}+4 a \,d^{2}+3 b \,c^{2}\right ) \left (-d x +c \right ) \sqrt {d x +c}}{8 d^{4} \sqrt {d x -c}}+\frac {c^{2} \left (4 a \,d^{2}+3 b \,c^{2}\right ) \ln \left (\frac {x \,d^{2}}{\sqrt {d^{2}}}+\sqrt {d^{2} x^{2}-c^{2}}\right ) \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{8 d^{4} \sqrt {d^{2}}\, \sqrt {d x -c}\, \sqrt {d x +c}}\) | \(136\) |
| default | \(\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (2 \,\operatorname {csgn}\left (d \right ) b \,d^{3} x^{3} \sqrt {d^{2} x^{2}-c^{2}}+4 \sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right ) d^{3} a x +3 \sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right ) d b \,c^{2} x +4 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) a \,c^{2} d^{2}+3 \ln \left (\left (\sqrt {d^{2} x^{2}-c^{2}}\, \operatorname {csgn}\left (d \right )+d x \right ) \operatorname {csgn}\left (d \right )\right ) b \,c^{4}\right ) \operatorname {csgn}\left (d \right )}{8 d^{5} \sqrt {d^{2} x^{2}-c^{2}}}\) | \(182\) |
-1/8*x*(2*b*d^2*x^2+4*a*d^2+3*b*c^2)*(-d*x+c)*(d*x+c)^(1/2)/d^4/(d*x-c)^(1 /2)+1/8*c^2*(4*a*d^2+3*b*c^2)/d^4*ln(x*d^2/(d^2)^(1/2)+(d^2*x^2-c^2)^(1/2) )/(d^2)^(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 = 90, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {{\left (2 \, b d^{3} x^{3} + {\left (3 \, b c^{2} d + 4 \, a d^{3}\right )} x\right )} \sqrt {d x + c} \sqrt {d x - c} - {\left (3 \, b c^{4} + 4 \, a c^{2} d^{2}\right )} \log \left (-d x + \sqrt {d x + c} \sqrt {d x - c}\right )}{8 \, d^{5}} \]
1/8*((2*b*d^3*x^3 + (3*b*c^2*d + 4*a*d^3)*x)*sqrt(d*x + c)*sqrt(d*x - c) - (3*b*c^4 + 4*a*c^2*d^2)*log(-d*x + sqrt(d*x + c)*sqrt(d*x - c)))/d^5
Timed out. \[ \int \frac {x^2 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.20 \[ \int \frac {x^2 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {\sqrt {d^{2} x^{2} - c^{2}} b x^{3}}{4 \, d^{2}} + \frac {3 \, b c^{4} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{8 \, d^{5}} + \frac {a c^{2} \log \left (2 \, d^{2} x + 2 \, \sqrt {d^{2} x^{2} - c^{2}} d\right )}{2 \, d^{3}} + \frac {3 \, \sqrt {d^{2} x^{2} - c^{2}} b c^{2} x}{8 \, d^{4}} + \frac {\sqrt {d^{2} x^{2} - c^{2}} a x}{2 \, d^{2}} \]
1/4*sqrt(d^2*x^2 - c^2)*b*x^3/d^2 + 3/8*b*c^4*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)*d)/d^5 + 1/2*a*c^2*log(2*d^2*x + 2*sqrt(d^2*x^2 - c^2)*d)/d^3 + 3/ 8*sqrt(d^2*x^2 - c^2)*b*c^2*x/d^4 + 1/2*sqrt(d^2*x^2 - c^2)*a*x/d^2
Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.19 \[ \int \frac {x^2 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {{\left ({\left (d x + c\right )} {\left (2 \, {\left (d x + c\right )} {\left (\frac {{\left (d x + c\right )} b}{d^{4}} - \frac {3 \, b c}{d^{4}}\right )} + \frac {9 \, b c^{2} d^{16} + 4 \, a d^{18}}{d^{20}}\right )} - \frac {5 \, b c^{3} d^{16} + 4 \, a c d^{18}}{d^{20}}\right )} \sqrt {d x + c} \sqrt {d x - c} - \frac {2 \, {\left (3 \, b c^{4} + 4 \, a c^{2} d^{2}\right )} \log \left ({\left | -\sqrt {d x + c} + \sqrt {d x - c} \right |}\right )}{d^{4}}}{8 \, d} \]
1/8*(((d*x + c)*(2*(d*x + c)*((d*x + c)*b/d^4 - 3*b*c/d^4) + (9*b*c^2*d^16 + 4*a*d^18)/d^20) - (5*b*c^3*d^16 + 4*a*c*d^18)/d^20)*sqrt(d*x + c)*sqrt( d*x - c) - 2*(3*b*c^4 + 4*a*c^2*d^2)*log(abs(-sqrt(d*x + c) + sqrt(d*x - c )))/d^4)/d
Time = 33.91 (sec) , antiderivative size = 1048, normalized size of antiderivative = 8.88 \[ \int \frac {x^2 \left (a+b x^2\right )}{\sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {\frac {2\,a\,c^2\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}{\sqrt {-c}-\sqrt {d\,x-c}}+\frac {14\,a\,c^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^3}+\frac {14\,a\,c^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^5}+\frac {2\,a\,c^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^7}}{d^3-\frac {4\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+\frac {6\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}-\frac {4\,d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^6}+\frac {d^3\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^8}}-\frac {\frac {23\,b\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^3}-\frac {3\,b\,c^4\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}{2\,\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}+\frac {333\,b\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^5}+\frac {671\,b\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^7}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^7}+\frac {671\,b\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^9}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^9}+\frac {333\,b\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{11}}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{11}}+\frac {23\,b\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{13}}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{13}}-\frac {3\,b\,c^4\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{15}}{2\,{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{15}}}{d^5-\frac {8\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^2}+\frac {28\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^4}-\frac {56\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^6}+\frac {70\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^8}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^8}-\frac {56\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{10}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{10}}+\frac {28\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{12}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{12}}-\frac {8\,d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{14}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{14}}+\frac {d^5\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^{16}}{{\left (\sqrt {-c}-\sqrt {d\,x-c}\right )}^{16}}}-\frac {2\,a\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}-\sqrt {c}}{\sqrt {-c}-\sqrt {d\,x-c}}\right )}{d^3}-\frac {3\,b\,c^4\,\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}-\sqrt {c}}{\sqrt {-c}-\sqrt {d\,x-c}}\right )}{2\,d^5} \]
((2*a*c^2*((c + d*x)^(1/2) - c^(1/2)))/((-c)^(1/2) - (d*x - c)^(1/2)) + (1 4*a*c^2*((c + d*x)^(1/2) - c^(1/2))^3)/((-c)^(1/2) - (d*x - c)^(1/2))^3 + (14*a*c^2*((c + d*x)^(1/2) - c^(1/2))^5)/((-c)^(1/2) - (d*x - c)^(1/2))^5 + (2*a*c^2*((c + d*x)^(1/2) - c^(1/2))^7)/((-c)^(1/2) - (d*x - c)^(1/2))^7 )/(d^3 - (4*d^3*((c + d*x)^(1/2) - c^(1/2))^2)/((-c)^(1/2) - (d*x - c)^(1/ 2))^2 + (6*d^3*((c + d*x)^(1/2) - c^(1/2))^4)/((-c)^(1/2) - (d*x - c)^(1/2 ))^4 - (4*d^3*((c + d*x)^(1/2) - c^(1/2))^6)/((-c)^(1/2) - (d*x - c)^(1/2) )^6 + (d^3*((c + d*x)^(1/2) - c^(1/2))^8)/((-c)^(1/2) - (d*x - c)^(1/2))^8 ) - ((23*b*c^4*((c + d*x)^(1/2) - c^(1/2))^3)/(2*((-c)^(1/2) - (d*x - c)^( 1/2))^3) - (3*b*c^4*((c + d*x)^(1/2) - c^(1/2)))/(2*((-c)^(1/2) - (d*x - c )^(1/2))) + (333*b*c^4*((c + d*x)^(1/2) - c^(1/2))^5)/(2*((-c)^(1/2) - (d* x - c)^(1/2))^5) + (671*b*c^4*((c + d*x)^(1/2) - c^(1/2))^7)/(2*((-c)^(1/2 ) - (d*x - c)^(1/2))^7) + (671*b*c^4*((c + d*x)^(1/2) - c^(1/2))^9)/(2*((- c)^(1/2) - (d*x - c)^(1/2))^9) + (333*b*c^4*((c + d*x)^(1/2) - c^(1/2))^11 )/(2*((-c)^(1/2) - (d*x - c)^(1/2))^11) + (23*b*c^4*((c + d*x)^(1/2) - c^( 1/2))^13)/(2*((-c)^(1/2) - (d*x - c)^(1/2))^13) - (3*b*c^4*((c + d*x)^(1/2 ) - c^(1/2))^15)/(2*((-c)^(1/2) - (d*x - c)^(1/2))^15))/(d^5 - (8*d^5*((c + d*x)^(1/2) - c^(1/2))^2)/((-c)^(1/2) - (d*x - c)^(1/2))^2 + (28*d^5*((c + d*x)^(1/2) - c^(1/2))^4)/((-c)^(1/2) - (d*x - c)^(1/2))^4 - (56*d^5*((c + d*x)^(1/2) - c^(1/2))^6)/((-c)^(1/2) - (d*x - c)^(1/2))^6 + (70*d^5*(...