Optimal. Leaf size=40 \[ a x+\frac {b \sqrt {1-(c+d x)^2}}{d}+\frac {b (c+d x) \text {ArcSin}(c+d x)}{d} \]
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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4887, 4715,
267} \begin {gather*} a x+\frac {b (c+d x) \text {ArcSin}(c+d x)}{d}+\frac {b \sqrt {1-(c+d x)^2}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 4715
Rule 4887
Rubi steps
\begin {align*} \int \left (a+b \sin ^{-1}(c+d x)\right ) \, dx &=a x+b \int \sin ^{-1}(c+d x) \, dx\\ &=a x+\frac {b \text {Subst}\left (\int \sin ^{-1}(x) \, dx,x,c+d x\right )}{d}\\ &=a x+\frac {b (c+d x) \sin ^{-1}(c+d x)}{d}-\frac {b \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=a x+\frac {b \sqrt {1-(c+d x)^2}}{d}+\frac {b (c+d x) \sin ^{-1}(c+d x)}{d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(159\) vs. \(2(40)=80\).
time = 0.25, size = 159, normalized size = 3.98 \begin {gather*} a x+b x \text {ArcSin}(c+d x)+\frac {b \left (2 d \sqrt {1-c^2-2 c d x-d^2 x^2}+2 c d \text {ArcTan}\left (\frac {\sqrt {-d^2} x-\sqrt {1-c^2-2 c d x-d^2 x^2}}{c}\right )+c \sqrt {-d^2} \log \left (-1+2 c d x+2 d^2 x^2+2 \sqrt {-d^2} x \sqrt {1-c^2-2 c d x-d^2 x^2}\right )\right )}{2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 36, normalized size = 0.90
method | result | size |
default | \(a x +\frac {b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(36\) |
derivativedivides | \(\frac {\left (d x +c \right ) a +b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 35, normalized size = 0.88 \begin {gather*} a x + \frac {{\left ({\left (d x + c\right )} \arcsin \left (d x + c\right ) + \sqrt {-{\left (d x + c\right )}^{2} + 1}\right )} b}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.07, size = 48, normalized size = 1.20 \begin {gather*} \frac {a d x + {\left (b d x + b c\right )} \arcsin \left (d x + c\right ) + \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} b}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 51, normalized size = 1.28 \begin {gather*} a x + b \left (\begin {cases} \frac {c \operatorname {asin}{\left (c + d x \right )}}{d} + x \operatorname {asin}{\left (c + d x \right )} + \frac {\sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} & \text {for}\: d \neq 0 \\x \operatorname {asin}{\left (c \right )} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 35, normalized size = 0.88 \begin {gather*} a x + \frac {{\left ({\left (d x + c\right )} \arcsin \left (d x + c\right ) + \sqrt {-{\left (d x + c\right )}^{2} + 1}\right )} b}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.61, size = 92, normalized size = 2.30 \begin {gather*} a\,x+b\,x\,\mathrm {asin}\left (c+d\,x\right )+\frac {b\,\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}}{d}+\frac {b\,c\,\ln \left (\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}-\frac {x\,d^2+c\,d}{\sqrt {-d^2}}\right )}{\sqrt {-d^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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