Optimal. Leaf size=110 \[ -\frac {\left (c+\frac {d}{x^2}\right )^{3/2} (3 a d+2 b c)}{6 c}-\frac {1}{2} \sqrt {c+\frac {d}{x^2}} (3 a d+2 b c)+\frac {1}{2} \sqrt {c} (3 a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )+\frac {a x^2 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c} \]
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Rubi [A] time = 0.07, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 78, 50, 63, 208} \begin {gather*} -\frac {\left (c+\frac {d}{x^2}\right )^{3/2} (3 a d+2 b c)}{6 c}-\frac {1}{2} \sqrt {c+\frac {d}{x^2}} (3 a d+2 b c)+\frac {1}{2} \sqrt {c} (3 a d+2 b c) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )+\frac {a x^2 \left (c+\frac {d}{x^2}\right )^{5/2}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
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 \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x) (c+d x)^{3/2}}{x^2} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^2}{2 c}-\frac {\left (b c+\frac {3 a d}{2}\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{x} \, dx,x,\frac {1}{x^2}\right )}{2 c}\\ &=-\frac {(2 b c+3 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{6 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^2}{2 c}-\frac {1}{4} (2 b c+3 a d) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{2} (2 b c+3 a d) \sqrt {c+\frac {d}{x^2}}-\frac {(2 b c+3 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{6 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^2}{2 c}-\frac {1}{4} (c (2 b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{2} (2 b c+3 a d) \sqrt {c+\frac {d}{x^2}}-\frac {(2 b c+3 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{6 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^2}{2 c}-\frac {(c (2 b c+3 a d)) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+\frac {d}{x^2}}\right )}{2 d}\\ &=-\frac {1}{2} (2 b c+3 a d) \sqrt {c+\frac {d}{x^2}}-\frac {(2 b c+3 a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{6 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^2}{2 c}+\frac {1}{2} \sqrt {c} (2 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [C] time = 0.10, size = 78, normalized size = 0.71 \begin {gather*} \frac {1}{3} \sqrt {c+\frac {d}{x^2}} \left (-\frac {(3 a d+2 b c) \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {c x^2}{d}\right )}{\sqrt {\frac {c x^2}{d}+1}}-\frac {b \left (c x^2+d\right )^2}{d x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 93, normalized size = 0.85 \begin {gather*} \frac {1}{2} \left (3 a \sqrt {c} d+2 b c^{3/2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {c x^2+d}{x^2}}}{\sqrt {c}}\right )+\frac {\sqrt {\frac {c x^2+d}{x^2}} \left (3 a c x^4-6 a d x^2-8 b c x^2-2 b d\right )}{6 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 195, normalized size = 1.77 \begin {gather*} \left [\frac {3 \, {\left (2 \, b c + 3 \, a d\right )} \sqrt {c} x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + 2 \, {\left (3 \, a c x^{4} - 2 \, {\left (4 \, b c + 3 \, a d\right )} x^{2} - 2 \, b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{12 \, x^{2}}, -\frac {3 \, {\left (2 \, b c + 3 \, a d\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - {\left (3 \, a c x^{4} - 2 \, {\left (4 \, b c + 3 \, a d\right )} x^{2} - 2 \, b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{6 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.54, size = 225, normalized size = 2.05 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2} + d} a c x \mathrm {sgn}\relax (x) - \frac {1}{4} \, {\left (2 \, b c^{\frac {3}{2}} \mathrm {sgn}\relax (x) + 3 \, a \sqrt {c} d \mathrm {sgn}\relax (x)\right )} \log \left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2}\right ) + \frac {2 \, {\left (6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {3}{2}} d \mathrm {sgn}\relax (x) + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a \sqrt {c} d^{2} \mathrm {sgn}\relax (x) - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {3}{2}} d^{2} \mathrm {sgn}\relax (x) - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a \sqrt {c} d^{3} \mathrm {sgn}\relax (x) + 4 \, b c^{\frac {3}{2}} d^{3} \mathrm {sgn}\relax (x) + 3 \, a \sqrt {c} d^{4} \mathrm {sgn}\relax (x)\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 216, normalized size = 1.96 \begin {gather*} \frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (9 a c \,d^{3} x^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+6 b \,c^{2} d^{2} x^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+9 \sqrt {c \,x^{2}+d}\, a \,c^{\frac {3}{2}} d^{2} x^{4}+6 \sqrt {c \,x^{2}+d}\, b \,c^{\frac {5}{2}} d \,x^{4}+6 \left (c \,x^{2}+d \right )^{\frac {3}{2}} a \,c^{\frac {3}{2}} d \,x^{4}+4 \left (c \,x^{2}+d \right )^{\frac {3}{2}} b \,c^{\frac {5}{2}} x^{4}-6 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a \sqrt {c}\, d \,x^{2}-4 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \,c^{\frac {3}{2}} x^{2}-2 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \sqrt {c}\, d \right )}{6 \left (c \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c}\, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 134, normalized size = 1.22 \begin {gather*} \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 )} a - \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 )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.65, size = 95, normalized size = 0.86 \begin {gather*} b\,c^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )-\frac {b\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{3}-a\,d\,\sqrt {c+\frac {d}{x^2}}-b\,c\,\sqrt {c+\frac {d}{x^2}}+\frac {3\,a\,\sqrt {c}\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2}+\frac {a\,c\,x^2\,\sqrt {c+\frac {d}{x^2}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 56.16, size = 187, normalized size = 1.70 \begin {gather*} \frac {3 a \sqrt {c} d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2} + \frac {a c \sqrt {d} x \sqrt {\frac {c x^{2}}{d} + 1}}{2} - \frac {a c \sqrt {d} x}{\sqrt {\frac {c x^{2}}{d} + 1}} - \frac {a d^{\frac {3}{2}}}{x \sqrt {\frac {c x^{2}}{d} + 1}} + b c^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )} - \frac {b c^{2} x}{\sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {b c \sqrt {d}}{x \sqrt {\frac {c x^{2}}{d} + 1}} + b d \left (\begin {cases} - \frac {\sqrt {c}}{2 x^{2}} & \text {for}\: d = 0 \\- \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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