Optimal. Leaf size=121 \[ \frac {x \left (c+\frac {d}{x^2}\right )^{3/2} (2 a d+3 b c)}{3 c}-\frac {d \sqrt {c+\frac {d}{x^2}} (2 a d+3 b c)}{2 c x}-\frac {1}{2} \sqrt {d} (2 a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )+\frac {a x^3 \left (c+\frac {d}{x^2}\right )^{5/2}}{3 c} \]
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Rubi [A] time = 0.06, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {453, 242, 277, 195, 217, 206} \[ \frac {x \left (c+\frac {d}{x^2}\right )^{3/2} (2 a d+3 b c)}{3 c}-\frac {d \sqrt {c+\frac {d}{x^2}} (2 a d+3 b c)}{2 c x}-\frac {1}{2} \sqrt {d} (2 a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )+\frac {a x^3 \left (c+\frac {d}{x^2}\right )^{5/2}}{3 c} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 242
Rule 277
Rule 453
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^2 \, dx &=\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^3}{3 c}+\frac {(3 b c+2 a d) \int \left (c+\frac {d}{x^2}\right )^{3/2} \, dx}{3 c}\\ &=\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^3}{3 c}-\frac {(3 b c+2 a d) \operatorname {Subst}\left (\int \frac {\left (c+d x^2\right )^{3/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=\frac {(3 b c+2 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x}{3 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^3}{3 c}-\frac {(d (3 b c+2 a d)) \operatorname {Subst}\left (\int \sqrt {c+d x^2} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {d (3 b c+2 a d) \sqrt {c+\frac {d}{x^2}}}{2 c x}+\frac {(3 b c+2 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x}{3 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^3}{3 c}-\frac {1}{2} (d (3 b c+2 a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {d (3 b c+2 a d) \sqrt {c+\frac {d}{x^2}}}{2 c x}+\frac {(3 b c+2 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x}{3 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^3}{3 c}-\frac {1}{2} (d (3 b c+2 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 {d (3 b c+2 a d) \sqrt {c+\frac {d}{x^2}}}{2 c x}+\frac {(3 b c+2 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x}{3 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^3}{3 c}-\frac {1}{2} \sqrt {d} (3 b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 105, normalized size = 0.87 \[ \frac {\sqrt {c+\frac {d}{x^2}} \left (\sqrt {c x^2+d} \left (2 a c x^4+8 a d x^2+6 b c x^2-3 b d\right )-3 \sqrt {d} x^2 (2 a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {c x^2+d}}{\sqrt {d}}\right )\right )}{6 x \sqrt {c x^2+d}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 190, normalized size = 1.57 \[ \left [\frac {3 \, {\left (3 \, b c + 2 \, a d\right )} \sqrt {d} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (2 \, a c x^{4} + 2 \, {\left (3 \, b c + 4 \, a d\right )} x^{2} - 3 \, b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{12 \, x}, \frac {3 \, {\left (3 \, b c + 2 \, a d\right )} \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left (2 \, a c x^{4} + 2 \, {\left (3 \, b c + 4 \, a d\right )} x^{2} - 3 \, b d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{6 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 115, normalized size = 0.95 \[ \frac {2 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a c \mathrm {sgn}\relax (x) + 6 \, \sqrt {c x^{2} + d} b c^{2} \mathrm {sgn}\relax (x) + 6 \, \sqrt {c x^{2} + d} a c d \mathrm {sgn}\relax (x) - \frac {3 \, \sqrt {c x^{2} + d} b c d \mathrm {sgn}\relax (x)}{x^{2}} + \frac {3 \, {\left (3 \, b c^{2} d \mathrm {sgn}\relax (x) + 2 \, a c d^{2} \mathrm {sgn}\relax (x)\right )} \arctan \left (\frac {\sqrt {c x^{2} + d}}{\sqrt {-d}}\right )}{\sqrt {-d}}}{6 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 170, normalized size = 1.40 \[ -\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (6 a \,d^{\frac {5}{2}} x^{2} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )+9 b c \,d^{\frac {3}{2}} x^{2} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )-6 \sqrt {c \,x^{2}+d}\, a \,d^{2} x^{2}-9 \sqrt {c \,x^{2}+d}\, b c d \,x^{2}-2 \left (c \,x^{2}+d \right )^{\frac {3}{2}} a d \,x^{2}-3 \left (c \,x^{2}+d \right )^{\frac {3}{2}} b c \,x^{2}+3 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \right ) x}{6 \left (c \,x^{2}+d \right )^{\frac {3}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 163, normalized size = 1.35 \[ \frac {1}{6} \, {\left (2 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{3} + 6 \, \sqrt {c + \frac {d}{x^{2}}} d x + 3 \, d^{\frac {3}{2}} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )\right )} a + \frac {1}{4} \, {\left (4 \, \sqrt {c + \frac {d}{x^{2}}} c x - \frac {2 \, \sqrt {c + \frac {d}{x^{2}}} c d x}{{\left (c + \frac {d}{x^{2}}\right )} x^{2} - d} + 3 \, c \sqrt {d} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\left (a+\frac {b}{x^2}\right )\,{\left (c+\frac {d}{x^2}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.65, size = 202, normalized size = 1.67 \[ \frac {a \sqrt {c} d x}{\sqrt {1 + \frac {d}{c x^{2}}}} + \frac {a c \sqrt {d} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{3} + \frac {a d^{\frac {3}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{3} - a d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )} + \frac {a d^{2}}{\sqrt {c} x \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {b c^{\frac {3}{2}} x}{\sqrt {1 + \frac {d}{c x^{2}}}} - \frac {b \sqrt {c} d \sqrt {1 + \frac {d}{c x^{2}}}}{2 x} + \frac {b \sqrt {c} d}{x \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 b c \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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