Integrand size = 22, antiderivative size = 121 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=-\frac {d (3 b c+2 a d) \sqrt {c+\frac {d}{x^2}}}{2 c x}+\frac {(3 b c+2 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x}{3 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^3}{3 c}-\frac {1}{2} \sqrt {d} (3 b c+2 a d) \text {arctanh}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right ) \]
1/3*(2*a*d+3*b*c)*(c+d/x^2)^(3/2)*x/c+1/3*a*(c+d/x^2)^(5/2)*x^3/c-1/2*(2*a *d+3*b*c)*arctanh(d^(1/2)/x/(c+d/x^2)^(1/2))*d^(1/2)-1/2*d*(2*a*d+3*b*c)*( c+d/x^2)^(1/2)/c/x
Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.87 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=\frac {\sqrt {c+\frac {d}{x^2}} \left (\sqrt {d+c x^2} \left (-3 b d+6 b c x^2+8 a d x^2+2 a c x^4\right )-3 \sqrt {d} (3 b c+2 a d) x^2 \text {arctanh}\left (\frac {\sqrt {d+c x^2}}{\sqrt {d}}\right )\right )}{6 x \sqrt {d+c x^2}} \]
(Sqrt[c + d/x^2]*(Sqrt[d + c*x^2]*(-3*b*d + 6*b*c*x^2 + 8*a*d*x^2 + 2*a*c* x^4) - 3*Sqrt[d]*(3*b*c + 2*a*d)*x^2*ArcTanh[Sqrt[d + c*x^2]/Sqrt[d]]))/(6 *x*Sqrt[d + c*x^2])
Time = 0.22 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {955, 773, 247, 211, 224, 219}
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.
| \(\displaystyle \int x^2 \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 955 |
| \(\displaystyle \frac {(2 a d+3 b c) \int \left (c+\frac {d}{x^2}\right )^{3/2}dx}{3 c}+\frac {a x^3 \left (c+\frac {d}{x^2}\right )^{5/2}}{3 c}\) |
| \(\Big \downarrow \) 773 |
| \(\displaystyle \frac {a x^3 \left (c+\frac {d}{x^2}\right )^{5/2}}{3 c}-\frac {(2 a d+3 b c) \int \left (c+\frac {d}{x^2}\right )^{3/2} x^2d\frac {1}{x}}{3 c}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle \frac {a x^3 \left (c+\frac {d}{x^2}\right )^{5/2}}{3 c}-\frac {(2 a d+3 b c) \left (3 d \int \sqrt {c+\frac {d}{x^2}}d\frac {1}{x}-x \left (c+\frac {d}{x^2}\right )^{3/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {a x^3 \left (c+\frac {d}{x^2}\right )^{5/2}}{3 c}-\frac {(2 a d+3 b c) \left (3 d \left (\frac {1}{2} c \int \frac {1}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}+\frac {\sqrt {c+\frac {d}{x^2}}}{2 x}\right )-x \left (c+\frac {d}{x^2}\right )^{3/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {a x^3 \left (c+\frac {d}{x^2}\right )^{5/2}}{3 c}-\frac {(2 a d+3 b c) \left (3 d \left (\frac {1}{2} c \int \frac {1}{1-\frac {d}{x^2}}d\frac {1}{\sqrt {c+\frac {d}{x^2}} x}+\frac {\sqrt {c+\frac {d}{x^2}}}{2 x}\right )-x \left (c+\frac {d}{x^2}\right )^{3/2}\right )}{3 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {a x^3 \left (c+\frac {d}{x^2}\right )^{5/2}}{3 c}-\frac {(2 a d+3 b c) \left (3 d \left (\frac {c \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{2 \sqrt {d}}+\frac {\sqrt {c+\frac {d}{x^2}}}{2 x}\right )-x \left (c+\frac {d}{x^2}\right )^{3/2}\right )}{3 c}\) |
(a*(c + d/x^2)^(5/2)*x^3)/(3*c) - ((3*b*c + 2*a*d)*(-((c + d/x^2)^(3/2)*x) + 3*d*(Sqrt[c + d/x^2]/(2*x) + (c*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/( 2*Sqrt[d]))))/(3*c)
3.10.58.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^ 2, x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && !IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Time = 0.11 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.21
| method | result | size |
| risch | \(-\frac {b d \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{2 x}+\frac {\left (c^{2} a \left (\frac {x^{2} \sqrt {c \,x^{2}+d}}{3 c}-\frac {2 d \sqrt {c \,x^{2}+d}}{3 c^{2}}\right )+\sqrt {c \,x^{2}+d}\, b c +2 a d \sqrt {c \,x^{2}+d}-\frac {\sqrt {d}\, \left (2 a d +3 b c \right ) \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right )}{2}\right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}{\sqrt {c \,x^{2}+d}}\) | \(147\) |
| default | \(-\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x \left (6 d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) a \,x^{2}+9 d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) b c \,x^{2}-2 \left (c \,x^{2}+d \right )^{\frac {3}{2}} a d \,x^{2}-3 \left (c \,x^{2}+d \right )^{\frac {3}{2}} b c \,x^{2}+3 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b -6 \sqrt {c \,x^{2}+d}\, a \,d^{2} x^{2}-9 \sqrt {c \,x^{2}+d}\, b c d \,x^{2}\right )}{6 \left (c \,x^{2}+d \right )^{\frac {3}{2}} d}\) | \(170\) |
-1/2*b*d/x*((c*x^2+d)/x^2)^(1/2)+(c^2*a*(1/3*x^2/c*(c*x^2+d)^(1/2)-2/3*d/c ^2*(c*x^2+d)^(1/2))+(c*x^2+d)^(1/2)*b*c+2*a*d*(c*x^2+d)^(1/2)-1/2*d^(1/2)* (2*a*d+3*b*c)*ln((2*d+2*d^(1/2)*(c*x^2+d)^(1/2))/x))*((c*x^2+d)/x^2)^(1/2) *x/(c*x^2+d)^(1/2)
Time = 0.34 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.57 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=\left [\frac {3 \, {\left (3 \, b c + 2 \, a d\right )} \sqrt {d} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (2 \, a c x^{4} + 2 \, {\left (3 \, b c + 4 \, a d\right )} x^{2} - 3 \, b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{12 \, x}, \frac {3 \, {\left (3 \, b c + 2 \, a d\right )} \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left (2 \, a c x^{4} + 2 \, {\left (3 \, b c + 4 \, a d\right )} x^{2} - 3 \, b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{6 \, x}\right ] \]
[1/12*(3*(3*b*c + 2*a*d)*sqrt(d)*x*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^2 + d)/x^2) + 2*d)/x^2) + 2*(2*a*c*x^4 + 2*(3*b*c + 4*a*d)*x^2 - 3*b*d)*sqrt( (c*x^2 + d)/x^2))/x, 1/6*(3*(3*b*c + 2*a*d)*sqrt(-d)*x*arctan(sqrt(-d)*x*s qrt((c*x^2 + d)/x^2)/(c*x^2 + d)) + (2*a*c*x^4 + 2*(3*b*c + 4*a*d)*x^2 - 3 *b*d)*sqrt((c*x^2 + d)/x^2))/x]
Time = 3.24 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.67 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=\frac {a \sqrt {c} d x}{\sqrt {1 + \frac {d}{c x^{2}}}} + \frac {a c \sqrt {d} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{3} + \frac {a d^{\frac {3}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{3} - a d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )} + \frac {a d^{2}}{\sqrt {c} x \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {b c^{\frac {3}{2}} x}{\sqrt {1 + \frac {d}{c x^{2}}}} - \frac {b \sqrt {c} d \sqrt {1 + \frac {d}{c x^{2}}}}{2 x} + \frac {b \sqrt {c} d}{x \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 b c \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{2} \]
a*sqrt(c)*d*x/sqrt(1 + d/(c*x**2)) + a*c*sqrt(d)*x**2*sqrt(c*x**2/d + 1)/3 + a*d**(3/2)*sqrt(c*x**2/d + 1)/3 - a*d**(3/2)*asinh(sqrt(d)/(sqrt(c)*x)) + a*d**2/(sqrt(c)*x*sqrt(1 + d/(c*x**2))) + b*c**(3/2)*x/sqrt(1 + d/(c*x* *2)) - b*sqrt(c)*d*sqrt(1 + d/(c*x**2))/(2*x) + b*sqrt(c)*d/(x*sqrt(1 + d/ (c*x**2))) - 3*b*c*sqrt(d)*asinh(sqrt(d)/(sqrt(c)*x))/2
Time = 0.31 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.35 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=\frac {1}{6} \, {\left (2 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{3} + 6 \, \sqrt {c + \frac {d}{x^{2}}} d x + 3 \, d^{\frac {3}{2}} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )\right )} a + \frac {1}{4} \, {\left (4 \, \sqrt {c + \frac {d}{x^{2}}} c x - \frac {2 \, \sqrt {c + \frac {d}{x^{2}}} c d x}{{\left (c + \frac {d}{x^{2}}\right )} x^{2} - d} + 3 \, c \sqrt {d} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )\right )} b \]
1/6*(2*(c + d/x^2)^(3/2)*x^3 + 6*sqrt(c + d/x^2)*d*x + 3*d^(3/2)*log((sqrt (c + d/x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d))))*a + 1/4*(4*sqrt(c + d/x^2)*c*x - 2*sqrt(c + d/x^2)*c*d*x/((c + d/x^2)*x^2 - d) + 3*c*sqrt(d )*log((sqrt(c + d/x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d))))*b
Time = 0.29 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.95 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=\frac {2 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a c \mathrm {sgn}\left (x\right ) + 6 \, \sqrt {c x^{2} + d} b c^{2} \mathrm {sgn}\left (x\right ) + 6 \, \sqrt {c x^{2} + d} a c d \mathrm {sgn}\left (x\right ) - \frac {3 \, \sqrt {c x^{2} + d} b c d \mathrm {sgn}\left (x\right )}{x^{2}} + \frac {3 \, {\left (3 \, b c^{2} d \mathrm {sgn}\left (x\right ) + 2 \, a c d^{2} \mathrm {sgn}\left (x\right )\right )} \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {-d}}\right )}{\sqrt {-d}}}{6 \, c} \]
1/6*(2*(c*x^2 + d)^(3/2)*a*c*sgn(x) + 6*sqrt(c*x^2 + d)*b*c^2*sgn(x) + 6*s qrt(c*x^2 + d)*a*c*d*sgn(x) - 3*sqrt(c*x^2 + d)*b*c*d*sgn(x)/x^2 + 3*(3*b* c^2*d*sgn(x) + 2*a*c*d^2*sgn(x))*arctan(sqrt(c*x^2 + d)/sqrt(-d))/sqrt(-d) )/c
Timed out. \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx=\int x^2\,\left (a+\frac {b}{x^2}\right )\,{\left (c+\frac {d}{x^2}\right )}^{3/2} \,d x \]