Optimal. Leaf size=93 \[ -\frac {c (3 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{8 d^{5/2}}+\frac {\sqrt {c+\frac {d}{x^2}} (3 b c-4 a d)}{8 d^2 x}-\frac {b \sqrt {c+\frac {d}{x^2}}}{4 d x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 93, 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, 321, 217, 206} \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}} (3 b c-4 a d)}{8 d^2 x}-\frac {c (3 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{8 d^{5/2}}-\frac {b \sqrt {c+\frac {d}{x^2}}}{4 d x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 321
Rule 335
Rule 459
Rubi steps
\begin {align*} \int \frac {a+\frac {b}{x^2}}{\sqrt {c+\frac {d}{x^2}} x^4} \, dx &=-\frac {b \sqrt {c+\frac {d}{x^2}}}{4 d x^3}+\frac {(-3 b c+4 a d) \int \frac {1}{\sqrt {c+\frac {d}{x^2}} x^4} \, dx}{4 d}\\ &=-\frac {b \sqrt {c+\frac {d}{x^2}}}{4 d x^3}-\frac {(-3 b c+4 a d) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{4 d}\\ &=-\frac {b \sqrt {c+\frac {d}{x^2}}}{4 d x^3}+\frac {(3 b c-4 a d) \sqrt {c+\frac {d}{x^2}}}{8 d^2 x}-\frac {(c (3 b c-4 a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{8 d^2}\\ &=-\frac {b \sqrt {c+\frac {d}{x^2}}}{4 d x^3}+\frac {(3 b c-4 a d) \sqrt {c+\frac {d}{x^2}}}{8 d^2 x}-\frac {(c (3 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 )}{8 d^2}\\ &=-\frac {b \sqrt {c+\frac {d}{x^2}}}{4 d x^3}+\frac {(3 b c-4 a d) \sqrt {c+\frac {d}{x^2}}}{8 d^2 x}-\frac {c (3 b c-4 a d) \tanh ^{-1}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{8 d^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 107, normalized size = 1.15 \begin {gather*} -\frac {\left (c x^2+d\right ) \left (d \sqrt {\frac {c x^2}{d}+1} \left (4 a d x^2-3 b c x^2+2 b d\right )+c x^4 (3 b c-4 a d) \tanh ^{-1}\left (\sqrt {\frac {c x^2}{d}+1}\right )\right )}{8 d^3 x^5 \sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c x^2}{d}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 104, normalized size = 1.12 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (\frac {\left (4 a c d-3 b c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c x^2+d}}{\sqrt {d}}\right )}{8 d^{5/2}}+\frac {\sqrt {c x^2+d} \left (-4 a d x^2+3 b c x^2-2 b d\right )}{8 d^2 x^4}\right )}{\sqrt {c x^2+d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 201, normalized size = 2.16 \begin {gather*} \left [-\frac {{\left (3 \, 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 (2 \, b d^{2} - {\left (3 \, b c d - 4 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{16 \, d^{3} x^{3}}, \frac {{\left (3 \, 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 (2 \, b d^{2} - {\left (3 \, b c d - 4 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{8 \, d^{3} x^{3}}\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 = 146, normalized size = 1.57 \begin {gather*} -\frac {\sqrt {c \,x^{2}+d}\, \left (-4 a c \,d^{2} x^{4} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )+3 b \,c^{2} d \,x^{4} \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )+4 \sqrt {c \,x^{2}+d}\, a \,d^{\frac {5}{2}} x^{2}-3 \sqrt {c \,x^{2}+d}\, b c \,d^{\frac {3}{2}} x^{2}+2 \sqrt {c \,x^{2}+d}\, b \,d^{\frac {5}{2}}\right )}{8 \sqrt {\frac {c \,x^{2}+d}{x^{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 = 1.23, size = 200, normalized size = 2.15 \begin {gather*} -\frac {1}{4} \, {\left (\frac {2 \, \sqrt {c + \frac {d}{x^{2}}} c x}{{\left (c + \frac {d}{x^{2}}\right )} d x^{2} - d^{2}} + \frac {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 {3}{2}}}\right )} a + \frac {1}{16} \, b {\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 )}{d^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{2} x^{3} - 5 \, \sqrt {c + \frac {d}{x^{2}}} c^{2} d x\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} d^{2} x^{4} - 2 \, {\left (c + \frac {d}{x^{2}}\right )} d^{3} x^{2} + d^{4}}\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\,\sqrt {c+\frac {d}{x^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.46, size = 150, normalized size = 1.61 \begin {gather*} - \frac {a \sqrt {c} \sqrt {1 + \frac {d}{c x^{2}}}}{2 d x} + \frac {a c \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{2 d^{\frac {3}{2}}} + \frac {3 b c^{\frac {3}{2}}}{8 d^{2} x \sqrt {1 + \frac {d}{c x^{2}}}} + \frac {b \sqrt {c}}{8 d 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 d^{\frac {5}{2}}} - \frac {b}{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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