Optimal. Leaf size=123 \[ \frac {d^2 (6 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{16 c^{3/2}}+\frac {d x^2 \sqrt {c+\frac {d}{x^2}} (6 b c-a d)}{16 c}+\frac {x^4 \left (c+\frac {d}{x^2}\right )^{3/2} (6 b c-a d)}{24 c}+\frac {a x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{6 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 78, 47, 63, 208} \[ \frac {d^2 (6 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{16 c^{3/2}}+\frac {x^4 \left (c+\frac {d}{x^2}\right )^{3/2} (6 b c-a d)}{24 c}+\frac {d x^2 \sqrt {c+\frac {d}{x^2}} (6 b c-a d)}{16 c}+\frac {a x^6 \left (c+\frac {d}{x^2}\right )^{5/2}}{6 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^5 \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x) (c+d x)^{3/2}}{x^4} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^6}{6 c}-\frac {\left (3 b c-\frac {a d}{2}\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{x^3} \, dx,x,\frac {1}{x^2}\right )}{6 c}\\ &=\frac {(6 b c-a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^4}{24 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^6}{6 c}-\frac {(d (6 b c-a d)) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x^2} \, dx,x,\frac {1}{x^2}\right )}{16 c}\\ &=\frac {d (6 b c-a d) \sqrt {c+\frac {d}{x^2}} x^2}{16 c}+\frac {(6 b c-a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^4}{24 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^6}{6 c}-\frac {\left (d^2 (6 b c-a d)\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )}{32 c}\\ &=\frac {d (6 b c-a d) \sqrt {c+\frac {d}{x^2}} x^2}{16 c}+\frac {(6 b c-a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^4}{24 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^6}{6 c}-\frac {(d (6 b c-a d)) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+\frac {d}{x^2}}\right )}{16 c}\\ &=\frac {d (6 b c-a d) \sqrt {c+\frac {d}{x^2}} x^2}{16 c}+\frac {(6 b c-a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^4}{24 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^6}{6 c}+\frac {d^2 (6 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{16 c^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 123, normalized size = 1.00 \[ \frac {x \sqrt {c+\frac {d}{x^2}} \left (\sqrt {c} x \sqrt {\frac {c x^2}{d}+1} \left (a \left (8 c^2 x^4+14 c d x^2+3 d^2\right )+6 b c \left (2 c x^2+5 d\right )\right )-3 d^{3/2} (a d-6 b c) \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )\right )}{48 c^{3/2} \sqrt {\frac {c x^2}{d}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.67, size = 243, normalized size = 1.98 \[ \left [-\frac {3 \, {\left (6 \, b c d^{2} - a d^{3}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) - 2 \, {\left (8 \, a c^{3} x^{6} + 2 \, {\left (6 \, b c^{3} + 7 \, a c^{2} d\right )} x^{4} + 3 \, {\left (10 \, b c^{2} d + a c d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{96 \, c^{2}}, -\frac {3 \, {\left (6 \, b c d^{2} - a d^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - {\left (8 \, a c^{3} x^{6} + 2 \, {\left (6 \, b c^{3} + 7 \, a c^{2} d\right )} x^{4} + 3 \, {\left (10 \, b c^{2} d + a c d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{48 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 144, normalized size = 1.17 \[ \frac {1}{48} \, {\left (2 \, {\left (4 \, a c x^{2} \mathrm {sgn}\relax (x) + \frac {6 \, b c^{5} \mathrm {sgn}\relax (x) + 7 \, a c^{4} d \mathrm {sgn}\relax (x)}{c^{4}}\right )} x^{2} + \frac {3 \, {\left (10 \, b c^{4} d \mathrm {sgn}\relax (x) + a c^{3} d^{2} \mathrm {sgn}\relax (x)\right )}}{c^{4}}\right )} \sqrt {c x^{2} + d} x - \frac {{\left (6 \, b c d^{2} \mathrm {sgn}\relax (x) - a d^{3} \mathrm {sgn}\relax (x)\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right )}{16 \, c^{\frac {3}{2}}} + \frac {{\left (6 \, b c d^{2} \log \left ({\left | d \right |}\right ) - a d^{3} \log \left ({\left | d \right |}\right )\right )} \mathrm {sgn}\relax (x)}{32 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 162, normalized size = 1.32 \[ \frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (-3 a \,d^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+18 b c \,d^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )-3 \sqrt {c \,x^{2}+d}\, a \sqrt {c}\, d^{2} x +18 \sqrt {c \,x^{2}+d}\, b \,c^{\frac {3}{2}} d x -2 \left (c \,x^{2}+d \right )^{\frac {3}{2}} a \sqrt {c}\, d x +12 \left (c \,x^{2}+d \right )^{\frac {3}{2}} b \,c^{\frac {3}{2}} x +8 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a \sqrt {c}\, x \right ) x^{3}}{48 \left (c \,x^{2}+d \right )^{\frac {3}{2}} c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.30, size = 240, normalized size = 1.95 \[ \frac {1}{96} \, {\left (\frac {3 \, d^{3} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{3} + 8 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c d^{3} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c^{2} d^{3}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{3} c - 3 \, {\left (c + \frac {d}{x^{2}}\right )}^{2} c^{2} + 3 \, {\left (c + \frac {d}{x^{2}}\right )} c^{3} - c^{4}}\right )} a - \frac {1}{16} \, {\left (\frac {3 \, d^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{\sqrt {c}} - \frac {2 \, {\left (5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{2} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c d^{2}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} - 2 \, {\left (c + \frac {d}{x^{2}}\right )} c + c^{2}}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.78, size = 130, normalized size = 1.06 \[ \frac {a\,x^6\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{6}+\frac {5\,b\,x^4\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{8}+\frac {a\,x^6\,{\left (c+\frac {d}{x^2}\right )}^{5/2}}{16\,c}+\frac {3\,b\,d^2\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8\,\sqrt {c}}-\frac {a\,c\,x^6\,\sqrt {c+\frac {d}{x^2}}}{16}-\frac {3\,b\,c\,x^4\,\sqrt {c+\frac {d}{x^2}}}{8}+\frac {a\,d^3\,\mathrm {atan}\left (\frac {\sqrt {c+\frac {d}{x^2}}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{16\,c^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 100.22, size = 253, normalized size = 2.06 \[ \frac {a c^{2} x^{7}}{6 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {11 a c \sqrt {d} x^{5}}{24 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {17 a d^{\frac {3}{2}} x^{3}}{48 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {a d^{\frac {5}{2}} x}{16 c \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {a d^{3} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{16 c^{\frac {3}{2}}} + \frac {b c^{2} x^{5}}{4 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 b c \sqrt {d} x^{3}}{8 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {b d^{\frac {3}{2}} x \sqrt {\frac {c x^{2}}{d} + 1}}{2} + \frac {b d^{\frac {3}{2}} x}{8 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 b d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{8 \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________