Integrand size = 22, antiderivative size = 90 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^3 \, dx=\frac {(4 b c-a d) \sqrt {c+\frac {d}{x^2}} x^2}{8 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^4}{4 c}+\frac {d (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8 c^{3/2}} \]
1/4*a*(c+d/x^2)^(3/2)*x^4/c+1/8*d*(-a*d+4*b*c)*arctanh((c+d/x^2)^(1/2)/c^( 1/2))/c^(3/2)+1/8*(-a*d+4*b*c)*x^2*(c+d/x^2)^(1/2)/c
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.08 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^3 \, dx=\frac {\sqrt {c+\frac {d}{x^2}} x \left (\sqrt {c} x \sqrt {d+c x^2} \left (4 b c+a \left (d+2 c x^2\right )\right )+d (-4 b c+a d) \log \left (-\sqrt {c} x+\sqrt {d+c x^2}\right )\right )}{8 c^{3/2} \sqrt {d+c x^2}} \]
(Sqrt[c + d/x^2]*x*(Sqrt[c]*x*Sqrt[d + c*x^2]*(4*b*c + a*(d + 2*c*x^2)) + d*(-4*b*c + a*d)*Log[-(Sqrt[c]*x) + Sqrt[d + c*x^2]]))/(8*c^(3/2)*Sqrt[d + c*x^2])
Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {948, 87, 51, 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.
\(\displaystyle \int x^3 \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle -\frac {1}{2} \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^6d\frac {1}{x^2}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^4 \left (c+\frac {d}{x^2}\right )^{3/2}}{2 c}-\frac {(4 b c-a d) \int \sqrt {c+\frac {d}{x^2}} x^4d\frac {1}{x^2}}{4 c}\right )\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^4 \left (c+\frac {d}{x^2}\right )^{3/2}}{2 c}-\frac {(4 b c-a d) \left (\frac {1}{2} d \int \frac {x^2}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x^2}-x^2 \sqrt {c+\frac {d}{x^2}}\right )}{4 c}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^4 \left (c+\frac {d}{x^2}\right )^{3/2}}{2 c}-\frac {(4 b c-a d) \left (\int \frac {1}{\frac {1}{d x^4}-\frac {c}{d}}d\sqrt {c+\frac {d}{x^2}}-x^2 \sqrt {c+\frac {d}{x^2}}\right )}{4 c}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {a x^4 \left (c+\frac {d}{x^2}\right )^{3/2}}{2 c}-\frac {(4 b c-a d) \left (x^2 \left (-\sqrt {c+\frac {d}{x^2}}\right )-\frac {d \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{4 c}\right )\) |
((a*(c + d/x^2)^(3/2)*x^4)/(2*c) - ((4*b*c - a*d)*(-(Sqrt[c + d/x^2]*x^2) - (d*ArcTanh[Sqrt[c + d/x^2]/Sqrt[c]])/Sqrt[c]))/(4*c))/2
3.10.30.3.1 Defintions of rubi rules used
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] :> 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 = 91, normalized size of antiderivative = 1.01
method | result | size |
risch | \(\frac {x^{2} \left (2 a c \,x^{2}+a d +4 b c \right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{8 c}-\frac {d \left (a d -4 b c \right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}{8 c^{\frac {3}{2}} \sqrt {c \,x^{2}+d}}\) | \(91\) |
default | \(\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x \left (2 \left (c \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c}\, a x -\sqrt {c \,x^{2}+d}\, \sqrt {c}\, a d x +4 \sqrt {c \,x^{2}+d}\, c^{\frac {3}{2}} b x -\ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) a \,d^{2}+4 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) b c d \right )}{8 \sqrt {c \,x^{2}+d}\, c^{\frac {3}{2}}}\) | \(122\) |
1/8*x^2*(2*a*c*x^2+a*d+4*b*c)/c*((c*x^2+d)/x^2)^(1/2)-1/8*d*(a*d-4*b*c)/c^ (3/2)*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*((c*x^2+d)/x^2)^(1/2)*x/(c*x^2+d)^(1/2 )
Time = 0.43 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.12 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^3 \, dx=\left [-\frac {{\left (4 \, b c d - a d^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) - 2 \, {\left (2 \, a c^{2} x^{4} + {\left (4 \, b c^{2} + a c d\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{16 \, c^{2}}, -\frac {{\left (4 \, b c d - a d^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - {\left (2 \, a c^{2} x^{4} + {\left (4 \, b c^{2} + a c d\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{8 \, c^{2}}\right ] \]
[-1/16*((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 + (4*b*c^2 + a*c*d)*x^2)*sqrt((c*x^2 + d) /x^2))/c^2, -1/8*((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 + (4*b*c^2 + a*c*d)*x^2)*sqrt((c* x^2 + d)/x^2))/c^2]
Time = 17.85 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.60 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^3 \, dx=\frac {a c x^{5}}{4 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 a \sqrt {d} x^{3}}{8 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {a d^{\frac {3}{2}} x}{8 c \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {a d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{8 c^{\frac {3}{2}}} + \frac {b \sqrt {d} x \sqrt {\frac {c x^{2}}{d} + 1}}{2} + \frac {b d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2 \sqrt {c}} \]
a*c*x**5/(4*sqrt(d)*sqrt(c*x**2/d + 1)) + 3*a*sqrt(d)*x**3/(8*sqrt(c*x**2/ d + 1)) + a*d**(3/2)*x/(8*c*sqrt(c*x**2/d + 1)) - a*d**2*asinh(sqrt(c)*x/s qrt(d))/(8*c**(3/2)) + b*sqrt(d)*x*sqrt(c*x**2/d + 1)/2 + b*d*asinh(sqrt(c )*x/sqrt(d))/(2*sqrt(c))
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (74) = 148\).
Time = 0.30 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.77 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^3 \, dx=\frac {1}{16} \, {\left (\frac {d^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{2} + \sqrt {c + \frac {d}{x^{2}}} c d^{2}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} c - 2 \, {\left (c + \frac {d}{x^{2}}\right )} c^{2} + c^{3}}\right )} a + \frac {1}{4} \, {\left (2 \, \sqrt {c + \frac {d}{x^{2}}} x^{2} - \frac {d \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{\sqrt {c}}\right )} b \]
1/16*(d^2*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c)))/c^( 3/2) + 2*((c + d/x^2)^(3/2)*d^2 + sqrt(c + d/x^2)*c*d^2)/((c + d/x^2)^2*c - 2*(c + d/x^2)*c^2 + c^3))*a + 1/4*(2*sqrt(c + d/x^2)*x^2 - d*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c)))/sqrt(c))*b
Time = 0.28 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.17 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^3 \, dx=\frac {1}{8} \, {\left (2 \, a x^{2} \mathrm {sgn}\left (x\right ) + \frac {4 \, b c^{2} \mathrm {sgn}\left (x\right ) + a c d \mathrm {sgn}\left (x\right )}{c^{2}}\right )} \sqrt {c x^{2} + d} x - \frac {{\left (4 \, b c d \mathrm {sgn}\left (x\right ) - a d^{2} \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right )}{8 \, c^{\frac {3}{2}}} + \frac {{\left (4 \, b c d \log \left ({\left | d \right |}\right ) - a d^{2} \log \left ({\left | d \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{16 \, c^{\frac {3}{2}}} \]
1/8*(2*a*x^2*sgn(x) + (4*b*c^2*sgn(x) + a*c*d*sgn(x))/c^2)*sqrt(c*x^2 + d) *x - 1/8*(4*b*c*d*sgn(x) - a*d^2*sgn(x))*log(abs(-sqrt(c)*x + sqrt(c*x^2 + d)))/c^(3/2) + 1/16*(4*b*c*d*log(abs(d)) - a*d^2*log(abs(d)))*sgn(x)/c^(3 /2)
Time = 9.66 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.03 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^3 \, dx=\frac {a\,x^4\,\sqrt {c+\frac {d}{x^2}}}{8}+\frac {b\,x^2\,\sqrt {c+\frac {d}{x^2}}}{2}+\frac {a\,x^4\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{8\,c}+\frac {b\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2\,\sqrt {c}}-\frac {a\,d^2\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8\,c^{3/2}} \]