Optimal. Leaf size=92 \[ \frac {(3 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{2 d^{5/2}}-\frac {3 b c-2 a d}{2 d^2 x \sqrt {c+\frac {d}{x^2}}}-\frac {b}{2 d x^3 \sqrt {c+\frac {d}{x^2}}} \]
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Rubi [A] time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {459, 335, 288, 217, 206} \begin {gather*} -\frac {3 b c-2 a d}{2 d^2 x \sqrt {c+\frac {d}{x^2}}}+\frac {(3 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{2 d^{5/2}}-\frac {b}{2 d x^3 \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 288
Rule 335
Rule 459
Rubi steps
\begin {align*} \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^4} \, dx &=-\frac {b}{2 d \sqrt {c+\frac {d}{x^2}} x^3}+\frac {(-3 b c+2 a d) \int \frac {1}{\left (c+\frac {d}{x^2}\right )^{3/2} x^4} \, dx}{2 d}\\ &=-\frac {b}{2 d \sqrt {c+\frac {d}{x^2}} x^3}-\frac {(-3 b c+2 a d) \operatorname {Subst}\left (\int \frac {x^2}{\left (c+d x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 d}\\ &=-\frac {b}{2 d \sqrt {c+\frac {d}{x^2}} x^3}-\frac {3 b c-2 a d}{2 d^2 \sqrt {c+\frac {d}{x^2}} x}+\frac {(3 b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{2 d^2}\\ &=-\frac {b}{2 d \sqrt {c+\frac {d}{x^2}} x^3}-\frac {3 b c-2 a d}{2 d^2 \sqrt {c+\frac {d}{x^2}} x}+\frac {(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 )}{2 d^2}\\ &=-\frac {b}{2 d \sqrt {c+\frac {d}{x^2}} x^3}-\frac {3 b c-2 a d}{2 d^2 \sqrt {c+\frac {d}{x^2}} x}+\frac {(3 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{2 d^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 57, normalized size = 0.62 \begin {gather*} \frac {x^2 (2 a d-3 b c) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c x^2}{d}+1\right )-b d}{2 d^2 x^3 \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 101, normalized size = 1.10 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (\frac {(3 b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {c x^2+d}}{\sqrt {d}}\right )}{2 d^{5/2}}+\frac {2 a d x^2-3 b c x^2-b d}{2 d^2 x^2 \sqrt {c x^2+d}}\right )}{\sqrt {c x^2+d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 248, normalized size = 2.70 \begin {gather*} \left [-\frac {{\left ({\left (3 \, b c^{2} - 2 \, a c d\right )} x^{3} + {\left (3 \, b c d - 2 \, a d^{2}\right )} x\right )} \sqrt {d} \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 (b d^{2} + {\left (3 \, b c d - 2 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{4 \, {\left (c d^{3} x^{3} + d^{4} x\right )}}, -\frac {{\left ({\left (3 \, b c^{2} - 2 \, a c d\right )} x^{3} + {\left (3 \, b c d - 2 \, a d^{2}\right )} x\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left (b d^{2} + {\left (3 \, b c d - 2 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{2 \, {\left (c d^{3} x^{3} + d^{4} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 132, normalized size = 1.43 \begin {gather*} \frac {\left (c \,x^{2}+d \right ) \left (-2 \sqrt {c \,x^{2}+d}\, a \,d^{2} x^{2} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )+3 \sqrt {c \,x^{2}+d}\, b c d \,x^{2} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )+2 a \,d^{\frac {5}{2}} x^{2}-3 b c \,d^{\frac {3}{2}} x^{2}-b \,d^{\frac {5}{2}}\right )}{2 \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} d^{\frac {7}{2}} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.22, size = 162, normalized size = 1.76 \begin {gather*} -\frac {1}{4} \, b {\left (\frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )} c x^{2} - 2 \, c d\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{2} x^{3} - \sqrt {c + \frac {d}{x^{2}}} d^{3} x} + \frac {3 \, c \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {5}{2}}}\right )} + \frac {1}{2} \, a {\left (\frac {\log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2}{\sqrt {c + \frac {d}{x^{2}}} d x}\right )} \end {gather*}
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 \begin {gather*} \int \frac {a+\frac {b}{x^2}}{x^4\,{\left (c+\frac {d}{x^2}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 18.44, size = 262, normalized size = 2.85 \begin {gather*} a \left (\frac {c d^{2} x^{2} \log {\left (\frac {c x^{2}}{d} \right )}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}} - \frac {2 c d^{2} x^{2} \log {\left (\sqrt {\frac {c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}} + \frac {2 d^{3} \sqrt {\frac {c x^{2}}{d} + 1}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}} + \frac {d^{3} \log {\left (\frac {c x^{2}}{d} \right )}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}} - \frac {2 d^{3} \log {\left (\sqrt {\frac {c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac {7}{2}} x^{2} + 2 d^{\frac {9}{2}}}\right ) + b \left (- \frac {3 \sqrt {c}}{2 d^{2} x \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {3 c \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{2 d^{\frac {5}{2}}} - \frac {1}{2 \sqrt {c} d x^{3} \sqrt {1 + \frac {d}{c x^{2}}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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