Optimal. Leaf size=43 \[ a x+\frac{2 b \sqrt{-d^2 x^4-2 d x^2}}{d x}+b x \sin ^{-1}\left (d x^2+1\right ) \]
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Rubi [A] time = 0.0381969, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4840, 12, 1588} \[ a x+\frac{2 b \sqrt{-d^2 x^4-2 d x^2}}{d x}+b x \sin ^{-1}\left (d x^2+1\right ) \]
Antiderivative was successfully verified.
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Rule 4840
Rule 12
Rule 1588
Rubi steps
\begin{align*} \int \left (a+b \sin ^{-1}\left (1+d x^2\right )\right ) \, dx &=a x+b \int \sin ^{-1}\left (1+d x^2\right ) \, dx\\ &=a x+b x \sin ^{-1}\left (1+d x^2\right )-b \int \frac{2 d x^2}{\sqrt{-2 d x^2-d^2 x^4}} \, dx\\ &=a x+b x \sin ^{-1}\left (1+d x^2\right )-(2 b d) \int \frac{x^2}{\sqrt{-2 d x^2-d^2 x^4}} \, dx\\ &=a x+\frac{2 b \sqrt{-2 d x^2-d^2 x^4}}{d x}+b x \sin ^{-1}\left (1+d x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0256849, size = 41, normalized size = 0.95 \[ a x+\frac{2 b \sqrt{-d x^2 \left (d x^2+2\right )}}{d x}+b x \sin ^{-1}\left (d x^2+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 45, normalized size = 1.1 \begin{align*} ax+b \left ( x\arcsin \left ( d{x}^{2}+1 \right ) -2\,{\frac{x \left ( d{x}^{2}+2 \right ) }{\sqrt{-{d}^{2}{x}^{4}-2\,d{x}^{2}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51233, size = 61, normalized size = 1.42 \begin{align*}{\left (x \arcsin \left (d x^{2} + 1\right ) - \frac{2 \,{\left (d^{\frac{3}{2}} x^{2} + 2 \, \sqrt{d}\right )}}{\sqrt{-d x^{2} - 2} d}\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28855, size = 103, normalized size = 2.4 \begin{align*} \frac{b d x^{2} \arcsin \left (d x^{2} + 1\right ) + a d x^{2} + 2 \, \sqrt{-d^{2} x^{4} - 2 \, d x^{2}} b}{d x} \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 \left (a + b \operatorname{asin}{\left (d x^{2} + 1 \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1397, size = 81, normalized size = 1.88 \begin{align*} -{\left (2 \, d{\left (\frac{\sqrt{2} \sqrt{-d} \mathrm{sgn}\left (x\right )}{d^{2}} - \frac{\sqrt{-d^{2} x^{2} - 2 \, d}}{d^{2} \mathrm{sgn}\left (x\right )}\right )} - x \arcsin \left (d x^{2} + 1\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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