Optimal. Leaf size=237 \[ a x+\frac {2 b \sqrt {1-c} (c+1) \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{c+1}\right )}{\sqrt {d} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}-\frac {2 b \sqrt {1-c} (c+1) \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{c+1}\right )}{\sqrt {d} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}+b x \sin ^{-1}\left (c+d x^2\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4840, 12, 1140, 493, 424, 419} \[ a x+\frac {2 b \sqrt {1-c} (c+1) \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{c+1}\right )}{\sqrt {d} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}-\frac {2 b \sqrt {1-c} (c+1) \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{c+1}\right )}{\sqrt {d} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}+b x \sin ^{-1}\left (c+d x^2\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 419
Rule 424
Rule 493
Rule 1140
Rule 4840
Rubi steps
\begin {align*} \int \left (a+b \sin ^{-1}\left (c+d x^2\right )\right ) \, dx &=a x+b \int \sin ^{-1}\left (c+d x^2\right ) \, dx\\ &=a x+b x \sin ^{-1}\left (c+d x^2\right )-b \int \frac {2 d x^2}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=a x+b x \sin ^{-1}\left (c+d x^2\right )-(2 b d) \int \frac {x^2}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=a x+b x \sin ^{-1}\left (c+d x^2\right )-\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 {x^2}{\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}}\\ &=a x+b x \sin ^{-1}\left (c+d x^2\right )+\frac {\left (2 b (1+c) \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 {\left (2 b (1+c) \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 {\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}}\\ &=a x+b x \sin ^{-1}\left (c+d x^2\right )-\frac {2 b \sqrt {1-c} (1+c) \sqrt {1-\frac {d x^2}{1-c}} \sqrt {1+\frac {d x^2}{1+c}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{1+c}\right )}{\sqrt {d} \sqrt {1-c^2-2 c d x^2-d^2 x^4}}+\frac {2 b \sqrt {1-c} (1+c) \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 {d} \sqrt {1-c^2-2 c d x^2-d^2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 155, normalized size = 0.65 \[ a x+\frac {2 i b (c-1) \sqrt {\frac {c+d x^2-1}{c-1}} \sqrt {\frac {c+d x^2+1}{c+1}} \left (E\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c+1}} x\right )|\frac {c+1}{c-1}\right )-F\left (i \sinh ^{-1}\left (\sqrt {\frac {d}{c+1}} x\right )|\frac {c+1}{c-1}\right )\right )}{\sqrt {\frac {d}{c+1}} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}+b x \sin ^{-1}\left (c+d x^2\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b \arcsin \left (d x^{2} + c\right ) + a, x\right ) \]
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 \[ \int b \arcsin \left (d x^{2} + c\right ) + a\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 153, normalized size = 0.65 \[ a x +b \left (x \arcsin \left (d \,x^{2}+c \right )+\frac {4 d \left (-c^{2}+1\right ) \sqrt {1+\frac {d \,x^{2}}{-1+c}}\, \sqrt {1+\frac {d \,x^{2}}{1+c}}\, \left (\EllipticF \left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )-\EllipticE \left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )\right )}{\sqrt {-\frac {d}{-1+c}}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}\, \left (-2 d c +2 d \right )}\right ) \]
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 = -0.00 \[ \int a+b\,\mathrm {asin}\left (d\,x^2+c\right ) \,d x \]
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 \[ \int \left (a + b \operatorname {asin}{\left (c + d x^{2} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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