Optimal. Leaf size=59 \[ -2 b^2 x+\frac {2 b \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))}{d}+\frac {(c+d x) (a+b \text {ArcSin}(c+d x))^2}{d} \]
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Rubi [A]
time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4887, 4715,
4767, 8} \begin {gather*} \frac {2 b \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))}{d}+\frac {(c+d x) (a+b \text {ArcSin}(c+d x))^2}{d}-2 b^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 4715
Rule 4767
Rule 4887
Rubi steps
\begin {align*} \int \left (a+b \sin ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \sin ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac {(2 b) \text {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {2 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}-\frac {\left (2 b^2\right ) \text {Subst}(\int 1 \, dx,x,c+d x)}{d}\\ &=-2 b^2 x+\frac {2 b \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2}{d}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 61, normalized size = 1.03 \begin {gather*} \frac {-2 b^2 (c+d x)+2 b \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))+(c+d x) (a+b \text {ArcSin}(c+d x))^2}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 92, normalized size = 1.56
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\left (d x +c \right ) \arcsin \left (d x +c \right )^{2}-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+2 a b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(92\) |
default | \(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\left (d x +c \right ) \arcsin \left (d x +c \right )^{2}-2 d x -2 c +2 \arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\right )+2 a b \left (\left (d x +c \right ) \arcsin \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )}{d}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.81, size = 94, normalized size = 1.59 \begin {gather*} \frac {{\left (a^{2} - 2 \, b^{2}\right )} d x + {\left (b^{2} d x + b^{2} c\right )} \arcsin \left (d x + c\right )^{2} + 2 \, {\left (a b d x + a b c\right )} \arcsin \left (d x + c\right ) + 2 \, \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} {\left (b^{2} \arcsin \left (d x + c\right ) + a b\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs.
\(2 (51) = 102\).
time = 0.18, size = 143, normalized size = 2.42 \begin {gather*} \begin {cases} a^{2} x + \frac {2 a b c \operatorname {asin}{\left (c + d x \right )}}{d} + 2 a b x \operatorname {asin}{\left (c + d x \right )} + \frac {2 a b \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{d} + \frac {b^{2} c \operatorname {asin}^{2}{\left (c + d x \right )}}{d} + b^{2} x \operatorname {asin}^{2}{\left (c + d x \right )} - 2 b^{2} x + \frac {2 b^{2} \sqrt {- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname {asin}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asin}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 111, normalized size = 1.88 \begin {gather*} \frac {{\left (d x + c\right )} b^{2} \arcsin \left (d x + c\right )^{2}}{d} + \frac {2 \, {\left (d x + c\right )} a b \arcsin \left (d x + c\right )}{d} + \frac {2 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} b^{2} \arcsin \left (d x + c\right )}{d} + \frac {{\left (d x + c\right )} a^{2}}{d} - \frac {2 \, {\left (d x + c\right )} b^{2}}{d} + \frac {2 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} a b}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 88, normalized size = 1.49 \begin {gather*} a^2\,x+\frac {b^2\,\left ({\mathrm {asin}\left (c+d\,x\right )}^2-2\right )\,\left (c+d\,x\right )}{d}+\frac {2\,a\,b\,\left (\sqrt {1-{\left (c+d\,x\right )}^2}+\mathrm {asin}\left (c+d\,x\right )\,\left (c+d\,x\right )\right )}{d}+\frac {2\,b^2\,\mathrm {asin}\left (c+d\,x\right )\,\sqrt {1-{\left (c+d\,x\right )}^2}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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