Optimal. Leaf size=126 \[ \frac{2 b \sqrt{1-c} \sqrt{d} \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{1-c}}\right ),-\frac{1-c}{c+1}\right )}{\sqrt{-c^2-2 c d x^2-d^2 x^4+1}}-\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{x} \]
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Rubi [A] time = 0.0764169, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 4842
Rule 12
Rule 1104
Rule 419
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*}
Mathematica [C] time = 0.230253, size = 140, normalized size = 1.11 \[ -\frac{2 i b d \sqrt{1-\frac{d x^2}{-c-1}} \sqrt{1-\frac{d x^2}{1-c}} \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{-\frac{d}{-c-1}}\right ),\frac{-c-1}{1-c}\right )}{\sqrt{-\frac{d}{-c-1}} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}-\frac{a}{x}-\frac{b \sin ^{-1}\left (c+d x^2\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 114, normalized size = 0.9 \begin{align*} -{\frac{a}{x}}+b \left ( -{\frac{\arcsin \left ( d{x}^{2}+c \right ) }{x}}+2\,{\frac{d}{\sqrt{-{d}^{2}{x}^{4}-2\,cd{x}^{2}-{c}^{2}+1}}\sqrt{1+{\frac{d{x}^{2}}{-1+c}}}\sqrt{1+{\frac{d{x}^{2}}{1+c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{d}{-1+c}}},\sqrt{-1+2\,{\frac{c}{1+c}}} \right ){\frac{1}{\sqrt{-{\frac{d}{-1+c}}}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \arcsin \left (d x^{2} + c\right ) + a}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asin}{\left (c + d x^{2} \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arcsin \left (d x^{2} + c\right ) + a}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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