Optimal. Leaf size=126 \[ \frac {2 b \sqrt {1-c} \sqrt {d} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{c+1}\right )}{\sqrt {-c^2-2 c d x^2-d^2 x^4+1}}-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{x} \]
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Rubi [A] time = 0.08, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4842, 12, 1104, 419} \[ \frac {2 b \sqrt {1-c} \sqrt {d} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{c+1}\right )}{\sqrt {-c^2-2 c d x^2-d^2 x^4+1}}-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 419
Rule 1104
Rule 4842
Rubi steps
\begin {align*} \int \frac {a+b \sin ^{-1}\left (c+d x^2\right )}{x^2} \, dx &=-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{x}+b \int \frac {2 d}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{x}+(2 b d) \int \frac {1}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{x}+\frac {\left (2 b d \sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}\right ) \int \frac {1}{\sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}} \, dx}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}}\\ &=-\frac {a+b \sin ^{-1}\left (c+d x^2\right )}{x}+\frac {2 b \sqrt {1-c} \sqrt {d} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {1+\frac {d x^2}{1+c}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{1+c}\right )}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.24, size = 140, normalized size = 1.11 \[ -\frac {a}{x}-\frac {2 i b d \sqrt {1-\frac {d x^2}{-c-1}} \sqrt {1-\frac {d x^2}{1-c}} F\left (i \sinh ^{-1}\left (\sqrt {-\frac {d}{-c-1}} x\right )|\frac {-c-1}{1-c}\right )}{\sqrt {-\frac {d}{-c-1}} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}-\frac {b \sin ^{-1}\left (c+d x^2\right )}{x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arcsin \left (d x^{2} + c\right ) + a}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arcsin \left (d x^{2} + c\right ) + a}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 114, normalized size = 0.90 \[ -\frac {a}{x}+b \left (-\frac {\arcsin \left (d \,x^{2}+c \right )}{x}+\frac {2 d \sqrt {1+\frac {d \,x^{2}}{-1+c}}\, \sqrt {1+\frac {d \,x^{2}}{1+c}}\, \EllipticF \left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )}{\sqrt {-\frac {d}{-1+c}}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (d\,x^2+c\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asin}{\left (c + d x^{2} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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