Optimal. Leaf size=76 \[ a c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )-\frac {1}{3} a \left (c+\frac {d}{x^2}\right )^{3/2}-a c \sqrt {c+\frac {d}{x^2}}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d} \]
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Rubi [A] time = 0.05, antiderivative size = 76, 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, 80, 50, 63, 208} \[ a c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )-\frac {1}{3} a \left (c+\frac {d}{x^2}\right )^{3/2}-a c \sqrt {c+\frac {d}{x^2}}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x) (c+d x)^{3/2}}{x} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d}-\frac {1}{2} a \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{3} a \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d}-\frac {1}{2} (a c) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=-a c \sqrt {c+\frac {d}{x^2}}-\frac {1}{3} a \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d}-\frac {1}{2} \left (a c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )\\ &=-a c \sqrt {c+\frac {d}{x^2}}-\frac {1}{3} a \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d}-\frac {\left (a c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+\frac {d}{x^2}}\right )}{d}\\ &=-a c \sqrt {c+\frac {d}{x^2}}-\frac {1}{3} a \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d}+a c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [C] time = 0.07, size = 90, normalized size = 1.18 \[ -\frac {\sqrt {c+\frac {d}{x^2}} \left (5 a d^2 x^2 \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {c x^2}{d}\right )+3 b \left (c x^2+d\right )^2 \sqrt {\frac {c x^2}{d}+1}\right )}{15 d x^4 \sqrt {\frac {c x^2}{d}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 213, normalized size = 2.80 \[ \left [\frac {15 \, a c^{\frac {3}{2}} d x^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) - 2 \, {\left ({\left (3 \, b c^{2} + 20 \, a c d\right )} x^{4} + 3 \, b d^{2} + {\left (6 \, b c d + 5 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{30 \, d x^{4}}, -\frac {15 \, a \sqrt {-c} c d x^{4} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left ({\left (3 \, b c^{2} + 20 \, a c d\right )} x^{4} + 3 \, b d^{2} + {\left (6 \, b c d + 5 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{15 \, d x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.07, size = 254, normalized size = 3.34 \[ -\frac {1}{2} \, a c^{\frac {3}{2}} \log \left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2}\right ) \mathrm {sgn}\relax (x) + \frac {2 \, {\left (15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {5}{2}} \mathrm {sgn}\relax (x) + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {3}{2}} d \mathrm {sgn}\relax (x) - 90 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {3}{2}} d^{2} \mathrm {sgn}\relax (x) + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {5}{2}} d^{2} \mathrm {sgn}\relax (x) + 110 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {3}{2}} d^{3} \mathrm {sgn}\relax (x) - 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {3}{2}} d^{4} \mathrm {sgn}\relax (x) + 3 \, b c^{\frac {5}{2}} d^{4} \mathrm {sgn}\relax (x) + 20 \, a c^{\frac {3}{2}} d^{5} \mathrm {sgn}\relax (x)\right )}}{15 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 153, normalized size = 2.01 \[ \frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (15 a \,c^{2} d^{2} x^{5} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+15 \sqrt {c \,x^{2}+d}\, a \,c^{\frac {5}{2}} d \,x^{6}+10 \left (c \,x^{2}+d \right )^{\frac {3}{2}} a \,c^{\frac {5}{2}} x^{6}-10 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a \,c^{\frac {3}{2}} x^{4}-5 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a \sqrt {c}\, d \,x^{2}-3 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \sqrt {c}\, d \right )}{15 \left (c \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c}\, d^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 80, normalized size = 1.05 \[ -\frac {b {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}}}{5 \, d} - \frac {1}{6} \, {\left (3 \, c^{\frac {3}{2}} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right ) + 2 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} + 6 \, \sqrt {c + \frac {d}{x^{2}}} c\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.83, size = 72, normalized size = 0.95 \[ a\,c^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )-\frac {a\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{3}-a\,c\,\sqrt {c+\frac {d}{x^2}}-\frac {b\,\sqrt {c+\frac {d}{x^2}}\,{\left (c\,x^2+d\right )}^2}{5\,d\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 54.14, size = 73, normalized size = 0.96 \[ - \frac {a c^{2} \operatorname {atan}{\left (\frac {\sqrt {c + \frac {d}{x^{2}}}}{\sqrt {- c}} \right )}}{\sqrt {- c}} - a c \sqrt {c + \frac {d}{x^{2}}} - \frac {a \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} - \frac {b \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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