Optimal. Leaf size=171 \[ \frac {2 b^2 g \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {2 b g x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c \sqrt {d-c^2 d x^2}}-\frac {g \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c^2 \sqrt {d-c^2 d x^2}}+\frac {f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{3 b c \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.25, antiderivative size = 207, normalized size of antiderivative = 1.21, number of steps
used = 8, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4861, 4847,
4737, 4767, 4715, 267} \begin {gather*} \frac {f \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{3 b c \sqrt {d-c^2 d x^2}}-\frac {g \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c^2 \sqrt {d-c^2 d x^2}}+\frac {2 a b g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {2 b^2 g x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c \sqrt {d-c^2 d x^2}}+\frac {2 b^2 g \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 4715
Rule 4737
Rule 4767
Rule 4847
Rule 4861
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d-c^2 d x^2}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (\frac {f \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}\right ) \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\left (f \sqrt {1-c^2 x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}+\frac {\left (g \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=-\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}+\frac {f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d-c^2 d x^2}}+\frac {\left (2 b g \sqrt {1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{c \sqrt {d-c^2 d x^2}}\\ &=\frac {2 a b g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}-\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}+\frac {f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 g \sqrt {1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{c \sqrt {d-c^2 d x^2}}\\ &=\frac {2 a b g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {2 b^2 g x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt {d-c^2 d x^2}}-\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}+\frac {f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 g \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {2 a b g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}+\frac {2 b^2 g \left (1-c^2 x^2\right )}{c^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 g x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt {d-c^2 d x^2}}-\frac {g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt {d-c^2 d x^2}}+\frac {f \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 291, normalized size = 1.70 \begin {gather*} \frac {3 \sqrt {d} g \left (-1+c^2 x^2\right ) \left (-2 a b c x+a^2 \sqrt {1-c^2 x^2}-2 b^2 \sqrt {1-c^2 x^2}\right )-6 b \sqrt {d} g \left (-1+c^2 x^2\right ) \left (b c x-a \sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)+3 b \sqrt {d} \left (-1+c^2 x^2\right ) \left (-a c f+b g \sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)^2-b^2 c \sqrt {d} f \left (-1+c^2 x^2\right ) \text {ArcSin}(c x)^3-3 a^2 c f \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{3 c^2 \sqrt {d} \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.54, size = 460, normalized size = 2.69
method | result | size |
default | \(-\frac {a^{2} g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+\frac {a^{2} f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} f}{3 c d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) g \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} f}{2 c d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) g \left (\arcsin \left (c x \right )+i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (\arcsin \left (c x \right )-i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(460\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 184, normalized size = 1.08 \begin {gather*} \frac {b^{2} f \arcsin \left (c x\right )^{3}}{3 \, c \sqrt {d}} + 2 \, b^{2} g {\left (\frac {x \arcsin \left (c x\right )}{c \sqrt {d}} + \frac {\sqrt {-c^{2} x^{2} + 1}}{c^{2} \sqrt {d}}\right )} + \frac {a b f \arcsin \left (c x\right )^{2}}{c \sqrt {d}} + \frac {2 \, a b g x}{c \sqrt {d}} + \frac {a^{2} f \arcsin \left (c x\right )}{c \sqrt {d}} - \frac {\sqrt {-c^{2} d x^{2} + d} b^{2} g \arcsin \left (c x\right )^{2}}{c^{2} d} - \frac {2 \, \sqrt {-c^{2} d x^{2} + d} a b g \arcsin \left (c x\right )}{c^{2} d} - \frac {\sqrt {-c^{2} d x^{2} + d} a^{2} g}{c^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (f+g\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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