Integrand size = 22, antiderivative size = 98 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^3 \, dx=-b d \sqrt {c+\frac {d}{x^2}}+\frac {1}{8} (4 b c+3 a d) \sqrt {c+\frac {d}{x^2}} x^2+\frac {1}{4} a \left (c+\frac {d}{x^2}\right )^{3/2} x^4+\frac {3 d (4 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8 \sqrt {c}} \] Output:
-b*d*(c+d/x^2)^(1/2)+1/8*(3*a*d+4*b*c)*(c+d/x^2)^(1/2)*x^2+1/4*a*(c+d/x^2) ^(3/2)*x^4+3/8*d*(a*d+4*b*c)*arctanh((c+d/x^2)^(1/2)/c^(1/2))/c^(1/2)
Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^3 \, dx=\frac {1}{8} \sqrt {c+\frac {d}{x^2}} \left (-8 b d+4 b c x^2+5 a d x^2+2 a c x^4+\frac {6 d (4 b c+a d) x \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {d}+\sqrt {d+c x^2}}\right )}{\sqrt {c} \sqrt {d+c x^2}}\right ) \] Input:
Integrate[(a + b/x^2)*(c + d/x^2)^(3/2)*x^3,x]
Output:
(Sqrt[c + d/x^2]*(-8*b*d + 4*b*c*x^2 + 5*a*d*x^2 + 2*a*c*x^4 + (6*d*(4*b*c + a*d)*x*ArcTanh[(Sqrt[c]*x)/(-Sqrt[d] + Sqrt[d + c*x^2])])/(Sqrt[c]*Sqrt [d + c*x^2])))/8
Time = 0.36 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {948, 87, 51, 60, 73, 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.
\(\displaystyle \int x^3 \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle -\frac {1}{2} \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^6d\frac {1}{x^2}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^4 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}-\frac {(a d+4 b c) \int \left (c+\frac {d}{x^2}\right )^{3/2} x^4d\frac {1}{x^2}}{4 c}\right )\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^4 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}-\frac {(a d+4 b c) \left (\frac {3}{2} d \int \sqrt {c+\frac {d}{x^2}} x^2d\frac {1}{x^2}-x^2 \left (c+\frac {d}{x^2}\right )^{3/2}\right )}{4 c}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^4 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}-\frac {(a d+4 b c) \left (\frac {3}{2} d \left (c \int \frac {x^2}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x^2}+2 \sqrt {c+\frac {d}{x^2}}\right )-x^2 \left (c+\frac {d}{x^2}\right )^{3/2}\right )}{4 c}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^4 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}-\frac {(a d+4 b c) \left (\frac {3}{2} d \left (\frac {2 c \int \frac {1}{\frac {1}{d x^4}-\frac {c}{d}}d\sqrt {c+\frac {d}{x^2}}}{d}+2 \sqrt {c+\frac {d}{x^2}}\right )-x^2 \left (c+\frac {d}{x^2}\right )^{3/2}\right )}{4 c}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^4 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c}-\frac {(a d+4 b c) \left (\frac {3}{2} d \left (2 \sqrt {c+\frac {d}{x^2}}-2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )\right )-x^2 \left (c+\frac {d}{x^2}\right )^{3/2}\right )}{4 c}\right )\) |
Input:
Int[(a + b/x^2)*(c + d/x^2)^(3/2)*x^3,x]
Output:
((a*(c + d/x^2)^(5/2)*x^4)/(2*c) - ((4*b*c + a*d)*(-((c + d/x^2)^(3/2)*x^2 ) + (3*d*(2*Sqrt[c + d/x^2] - 2*Sqrt[c]*ArcTanh[Sqrt[c + d/x^2]/Sqrt[c]])) /2))/(4*c))/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.98
method | result | size |
risch | \(\frac {\left (2 a c \,x^{4}+5 a d \,x^{2}+4 b c \,x^{2}-8 b d \right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{8}+\frac {3 d \left (a d +4 c b \right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}{8 \sqrt {c}\, \sqrt {c \,x^{2}+d}}\) | \(96\) |
default | \(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x^{2} \left (8 c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {3}{2}} b \,x^{2}+12 c^{\frac {3}{2}} \sqrt {c \,x^{2}+d}\, b d \,x^{2}+2 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {3}{2}} a d \,x^{2}-8 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {5}{2}} b +3 \sqrt {c}\, \sqrt {c \,x^{2}+d}\, a \,d^{2} x^{2}+3 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) a \,d^{3} x +12 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) b c \,d^{2} x \right )}{8 \left (c \,x^{2}+d \right )^{\frac {3}{2}} d \sqrt {c}}\) | \(174\) |
Input:
int((a+b/x^2)*(c+d/x^2)^(3/2)*x^3,x,method=_RETURNVERBOSE)
Output:
1/8*(2*a*c*x^4+5*a*d*x^2+4*b*c*x^2-8*b*d)*((c*x^2+d)/x^2)^(1/2)+3/8*d*(a*d +4*b*c)*ln(c^(1/2)*x+(c*x^2+d)^(1/2))/c^(1/2)*((c*x^2+d)/x^2)^(1/2)*x/(c*x ^2+d)^(1/2)
Time = 0.09 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.07 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^3 \, dx=\left [\frac {3 \, {\left (4 \, b c d + a d^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + 2 \, {\left (2 \, a c^{2} x^{4} - 8 \, b c d + {\left (4 \, b c^{2} + 5 \, a c d\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{16 \, c}, -\frac {3 \, {\left (4 \, b c d + a d^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - {\left (2 \, a c^{2} x^{4} - 8 \, b c d + {\left (4 \, b c^{2} + 5 \, a c d\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{8 \, c}\right ] \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^3,x, algorithm="fricas")
Output:
[1/16*(3*(4*b*c*d + a*d^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x^ 2 + d)/x^2) - d) + 2*(2*a*c^2*x^4 - 8*b*c*d + (4*b*c^2 + 5*a*c*d)*x^2)*sqr t((c*x^2 + d)/x^2))/c, -1/8*(3*(4*b*c*d + a*d^2)*sqrt(-c)*arctan(sqrt(-c)* x^2*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)) - (2*a*c^2*x^4 - 8*b*c*d + (4*b*c^2 + 5*a*c*d)*x^2)*sqrt((c*x^2 + d)/x^2))/c]
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (90) = 180\).
Time = 85.70 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.20 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^3 \, dx=\frac {a c^{2} x^{5}}{4 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 a c \sqrt {d} x^{3}}{8 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {a d^{\frac {3}{2}} x \sqrt {\frac {c x^{2}}{d} + 1}}{2} + \frac {a d^{\frac {3}{2}} x}{8 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 a d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{8 \sqrt {c}} + \frac {3 b \sqrt {c} d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2} + \frac {b c \sqrt {d} x \sqrt {\frac {c x^{2}}{d} + 1}}{2} - \frac {b c \sqrt {d} x}{\sqrt {\frac {c x^{2}}{d} + 1}} - \frac {b d^{\frac {3}{2}}}{x \sqrt {\frac {c x^{2}}{d} + 1}} \] Input:
integrate((a+b/x**2)*(c+d/x**2)**(3/2)*x**3,x)
Output:
a*c**2*x**5/(4*sqrt(d)*sqrt(c*x**2/d + 1)) + 3*a*c*sqrt(d)*x**3/(8*sqrt(c* x**2/d + 1)) + a*d**(3/2)*x*sqrt(c*x**2/d + 1)/2 + a*d**(3/2)*x/(8*sqrt(c* x**2/d + 1)) + 3*a*d**2*asinh(sqrt(c)*x/sqrt(d))/(8*sqrt(c)) + 3*b*sqrt(c) *d*asinh(sqrt(c)*x/sqrt(d))/2 + b*c*sqrt(d)*x*sqrt(c*x**2/d + 1)/2 - b*c*s qrt(d)*x/sqrt(c*x**2/d + 1) - b*d**(3/2)/(x*sqrt(c*x**2/d + 1))
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (80) = 160\).
Time = 0.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.74 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^3 \, dx=-\frac {1}{16} \, {\left (\frac {3 \, d^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{\sqrt {c}} - \frac {2 \, {\left (5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{2} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c d^{2}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} - 2 \, {\left (c + \frac {d}{x^{2}}\right )} c + c^{2}}\right )} a + \frac {1}{4} \, {\left (2 \, \sqrt {c + \frac {d}{x^{2}}} c x^{2} - 3 \, \sqrt {c} d \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right ) - 4 \, \sqrt {c + \frac {d}{x^{2}}} d\right )} b \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^3,x, algorithm="maxima")
Output:
-1/16*(3*d^2*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c)))/ sqrt(c) - 2*(5*(c + d/x^2)^(3/2)*d^2 - 3*sqrt(c + d/x^2)*c*d^2)/((c + d/x^ 2)^2 - 2*(c + d/x^2)*c + c^2))*a + 1/4*(2*sqrt(c + d/x^2)*c*x^2 - 3*sqrt(c )*d*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c))) - 4*sqrt( c + d/x^2)*d)*b
Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.23 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^3 \, dx=\frac {2 \, b \sqrt {c} d^{2} \mathrm {sgn}\left (x\right )}{{\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d} + \frac {1}{8} \, {\left (2 \, a c x^{2} \mathrm {sgn}\left (x\right ) + \frac {4 \, b c^{3} \mathrm {sgn}\left (x\right ) + 5 \, a c^{2} d \mathrm {sgn}\left (x\right )}{c^{2}}\right )} \sqrt {c x^{2} + d} x - \frac {3 \, {\left (4 \, b c d \mathrm {sgn}\left (x\right ) + a d^{2} \mathrm {sgn}\left (x\right )\right )} \log \left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2}\right )}{16 \, \sqrt {c}} \] Input:
integrate((a+b/x^2)*(c+d/x^2)^(3/2)*x^3,x, algorithm="giac")
Output:
2*b*sqrt(c)*d^2*sgn(x)/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d) + 1/8*(2*a*c* x^2*sgn(x) + (4*b*c^3*sgn(x) + 5*a*c^2*d*sgn(x))/c^2)*sqrt(c*x^2 + d)*x - 3/16*(4*b*c*d*sgn(x) + a*d^2*sgn(x))*log((sqrt(c)*x - sqrt(c*x^2 + d))^2)/ sqrt(c)
Time = 4.87 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.07 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^3 \, dx=\frac {5\,a\,x^4\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{8}-b\,d\,\sqrt {c+\frac {d}{x^2}}+\frac {3\,b\,\sqrt {c}\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2}+\frac {3\,a\,d^2\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8\,\sqrt {c}}-\frac {3\,a\,c\,x^4\,\sqrt {c+\frac {d}{x^2}}}{8}+\frac {b\,c\,x^2\,\sqrt {c+\frac {d}{x^2}}}{2} \] Input:
int(x^3*(a + b/x^2)*(c + d/x^2)^(3/2),x)
Output:
(5*a*x^4*(c + d/x^2)^(3/2))/8 - b*d*(c + d/x^2)^(1/2) + (3*b*c^(1/2)*d*ata nh((c + d/x^2)^(1/2)/c^(1/2)))/2 + (3*a*d^2*atanh((c + d/x^2)^(1/2)/c^(1/2 )))/(8*c^(1/2)) - (3*a*c*x^4*(c + d/x^2)^(1/2))/8 + (b*c*x^2*(c + d/x^2)^( 1/2))/2
Time = 0.20 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.47 \[ \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^3 \, dx=\frac {2 \sqrt {c \,x^{2}+d}\, a \,c^{2} x^{4}+5 \sqrt {c \,x^{2}+d}\, a c d \,x^{2}+4 \sqrt {c \,x^{2}+d}\, b \,c^{2} x^{2}-8 \sqrt {c \,x^{2}+d}\, b c d +3 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) a \,d^{2} x +12 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) b c d x -\sqrt {c}\, a \,d^{2} x -9 \sqrt {c}\, b c d x}{8 c x} \] Input:
int((a+b/x^2)*(c+d/x^2)^(3/2)*x^3,x)
Output:
(2*sqrt(c*x**2 + d)*a*c**2*x**4 + 5*sqrt(c*x**2 + d)*a*c*d*x**2 + 4*sqrt(c *x**2 + d)*b*c**2*x**2 - 8*sqrt(c*x**2 + d)*b*c*d + 3*sqrt(c)*log((sqrt(c* x**2 + d) + sqrt(c)*x)/sqrt(d))*a*d**2*x + 12*sqrt(c)*log((sqrt(c*x**2 + d ) + sqrt(c)*x)/sqrt(d))*b*c*d*x - sqrt(c)*a*d**2*x - 9*sqrt(c)*b*c*d*x)/(8 *c*x)