Optimal. Leaf size=112 \[ -\frac {\left (c+\frac {d}{x^2}\right )^{3/2} (4 a d+b c)}{4 c x}-\frac {3 \sqrt {c+\frac {d}{x^2}} (4 a d+b c)}{8 x}-\frac {3 c (4 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{8 \sqrt {d}}+\frac {a x \left (c+\frac {d}{x^2}\right )^{5/2}}{c} \]
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Rubi [A] time = 0.06, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {375, 453, 195, 217, 206} \begin {gather*} -\frac {\left (c+\frac {d}{x^2}\right )^{3/2} (4 a d+b c)}{4 c x}-\frac {3 \sqrt {c+\frac {d}{x^2}} (4 a d+b c)}{8 x}-\frac {3 c (4 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{8 \sqrt {d}}+\frac {a x \left (c+\frac {d}{x^2}\right )^{5/2}}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 375
Rule 453
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}{x^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x}{c}+\frac {(-b c-4 a d) \operatorname {Subst}\left (\int \left (c+d x^2\right )^{3/2} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {(b c+4 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{4 c x}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x}{c}-\frac {1}{4} (3 (b c+4 a d)) \operatorname {Subst}\left (\int \sqrt {c+d x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3 (b c+4 a d) \sqrt {c+\frac {d}{x^2}}}{8 x}-\frac {(b c+4 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{4 c x}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x}{c}-\frac {1}{8} (3 c (b c+4 a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3 (b c+4 a d) \sqrt {c+\frac {d}{x^2}}}{8 x}-\frac {(b c+4 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{4 c x}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x}{c}-\frac {1}{8} (3 c (b c+4 a d)) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {1}{\sqrt {c+\frac {d}{x^2}} x}\right )\\ &=-\frac {3 (b c+4 a d) \sqrt {c+\frac {d}{x^2}}}{8 x}-\frac {(b c+4 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{4 c x}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x}{c}-\frac {3 c (b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{8 \sqrt {d}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 68, normalized size = 0.61 \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (c x^4 (4 a d+b c) \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {c x^2}{d}+1\right )-5 b d^2\right )}{20 d^3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 107, normalized size = 0.96 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (\frac {\sqrt {c x^2+d} \left (8 a c x^4-4 a d x^2-5 b c x^2-2 b d\right )}{8 x^4}-\frac {3 \left (4 a c d+b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c x^2+d}}{\sqrt {d}}\right )}{8 \sqrt {d}}\right )}{\sqrt {c x^2+d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 216, normalized size = 1.93 \begin {gather*} \left [\frac {3 \, {\left (b c^{2} + 4 \, a c d\right )} \sqrt {d} x^{3} \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 (8 \, a c d x^{4} - 2 \, b d^{2} - {\left (5 \, b c d + 4 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{16 \, d x^{3}}, \frac {3 \, {\left (b c^{2} + 4 \, a c d\right )} \sqrt {-d} x^{3} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left (8 \, a c d x^{4} - 2 \, b d^{2} - {\left (5 \, b c d + 4 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{8 \, d x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 145, normalized size = 1.29 \begin {gather*} \frac {8 \, \sqrt {c x^{2} + d} a c^{2} \mathrm {sgn}\relax (x) + \frac {3 \, {\left (b c^{3} \mathrm {sgn}\relax (x) + 4 \, a c^{2} d \mathrm {sgn}\relax (x)\right )} \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {-d}}\right )}{\sqrt {-d}} - \frac {5 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c^{3} \mathrm {sgn}\relax (x) + 4 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a c^{2} d \mathrm {sgn}\relax (x) - 3 \, \sqrt {c x^{2} + d} b c^{3} d \mathrm {sgn}\relax (x) - 4 \, \sqrt {c x^{2} + d} a c^{2} d^{2} \mathrm {sgn}\relax (x)}{c^{2} x^{4}}}{8 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 213, normalized size = 1.90 \begin {gather*} -\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (12 a c \,d^{\frac {5}{2}} x^{4} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )+3 b \,c^{2} d^{\frac {3}{2}} x^{4} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )-12 \sqrt {c \,x^{2}+d}\, a c \,d^{2} x^{4}-3 \sqrt {c \,x^{2}+d}\, b \,c^{2} d \,x^{4}-4 \left (c \,x^{2}+d \right )^{\frac {3}{2}} a c d \,x^{4}-\left (c \,x^{2}+d \right )^{\frac {3}{2}} b \,c^{2} x^{4}+4 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a d \,x^{2}+\left (c \,x^{2}+d \right )^{\frac {5}{2}} b c \,x^{2}+2 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b d \right )}{8 \left (c \,x^{2}+d \right )^{\frac {3}{2}} d^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.42, size = 207, normalized size = 1.85 \begin {gather*} \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 )} a + \frac {1}{16} \, {\left (\frac {3 \, c^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{\sqrt {d}} - \frac {2 \, {\left (5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{2} x^{3} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c^{2} d x\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} x^{4} - 2 \, {\left (c + \frac {d}{x^{2}}\right )} d x^{2} + d^{2}}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.86, size = 78, normalized size = 0.70 \begin {gather*} \frac {a\,x\,{\left (c\,x^2+d\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {d}{c\,x^2}\right )}{{\left (\frac {d}{c}+x^2\right )}^{3/2}}-\frac {b\,{\left (c\,x^2+d\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {d}{c\,x^2}\right )}{x\,{\left (\frac {d}{c}+x^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 11.78, size = 216, normalized size = 1.93 \begin {gather*} \frac {a c^{\frac {3}{2}} x}{\sqrt {1 + \frac {d}{c x^{2}}}} - \frac {a \sqrt {c} d \sqrt {1 + \frac {d}{c x^{2}}}}{2 x} + \frac {a \sqrt {c} d}{x \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 a c \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{2} - \frac {b c^{\frac {3}{2}} \sqrt {1 + \frac {d}{c x^{2}}}}{2 x} - \frac {b c^{\frac {3}{2}}}{8 x \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 b \sqrt {c} d}{8 x^{3} \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 b c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{8 \sqrt {d}} - \frac {b d^{2}}{4 \sqrt {c} x^{5} \sqrt {1 + \frac {d}{c x^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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