Optimal. Leaf size=81 \[ \frac {b \left (3 c^2 d+e\right ) \sqrt {1-c^2 x^2}}{3 c^3}-\frac {b e \left (1-c^2 x^2\right )^{3/2}}{9 c^3}+d x (a+b \text {ArcSin}(c x))+\frac {1}{3} e x^3 (a+b \text {ArcSin}(c x)) \]
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Rubi [A]
time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4755, 455, 45}
\begin {gather*} d x (a+b \text {ArcSin}(c x))+\frac {1}{3} e x^3 (a+b \text {ArcSin}(c x))+\frac {b \sqrt {1-c^2 x^2} \left (3 c^2 d+e\right )}{3 c^3}-\frac {b e \left (1-c^2 x^2\right )^{3/2}}{9 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 455
Rule 4755
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x \left (d+\frac {e x^2}{3}\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=d x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{2} (b c) \text {Subst}\left (\int \frac {d+\frac {e x}{3}}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=d x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{2} (b c) \text {Subst}\left (\int \left (\frac {3 c^2 d+e}{3 c^2 \sqrt {1-c^2 x}}-\frac {e \sqrt {1-c^2 x}}{3 c^2}\right ) \, dx,x,x^2\right )\\ &=\frac {b \left (3 c^2 d+e\right ) \sqrt {1-c^2 x^2}}{3 c^3}-\frac {b e \left (1-c^2 x^2\right )^{3/2}}{9 c^3}+d x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 71, normalized size = 0.88 \begin {gather*} \frac {1}{9} \left (3 a x \left (3 d+e x^2\right )+\frac {b \sqrt {1-c^2 x^2} \left (2 e+c^2 \left (9 d+e x^2\right )\right )}{c^3}+3 b x \left (3 d+e x^2\right ) \text {ArcSin}(c x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 111, normalized size = 1.37
method | result | size |
derivativedivides | \(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\arcsin \left (c x \right ) d \,c^{3} x +\frac {\arcsin \left (c x \right ) e \,c^{3} x^{3}}{3}+d \,c^{2} \sqrt {-c^{2} x^{2}+1}-\frac {e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{2}}}{c}\) | \(111\) |
default | \(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\arcsin \left (c x \right ) d \,c^{3} x +\frac {\arcsin \left (c x \right ) e \,c^{3} x^{3}}{3}+d \,c^{2} \sqrt {-c^{2} x^{2}+1}-\frac {e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{2}}}{c}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 93, normalized size = 1.15 \begin {gather*} \frac {1}{3} \, a x^{3} e + a d x + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.11, size = 86, normalized size = 1.06 \begin {gather*} \frac {3 \, a c^{3} x^{3} e + 9 \, a c^{3} d x + 3 \, {\left (b c^{3} x^{3} e + 3 \, b c^{3} d x\right )} \arcsin \left (c x\right ) + {\left (9 \, b c^{2} d + {\left (b c^{2} x^{2} + 2 \, b\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{9 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 109, normalized size = 1.35 \begin {gather*} \begin {cases} a d x + \frac {a e x^{3}}{3} + b d x \operatorname {asin}{\left (c x \right )} + \frac {b e x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b d \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b e x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {2 b e \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\a \left (d x + \frac {e x^{3}}{3}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 109, normalized size = 1.35 \begin {gather*} \frac {1}{3} \, a e x^{3} + b d x \arcsin \left (c x\right ) + a d x + \frac {{\left (c^{2} x^{2} - 1\right )} b e x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {b e x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d}{c} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e}{9 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e}{3 \, c^{3}} \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*} \left \{\begin {array}{cl} b\,e\,\left (\frac {\sqrt {\frac {1}{c^2}-x^2}\,\left (\frac {2}{c^2}+x^2\right )}{9}+\frac {x^3\,\mathrm {asin}\left (c\,x\right )}{3}\right )+\frac {a\,x\,\left (e\,x^2+3\,d\right )}{3}+\frac {b\,d\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c} & \text {\ if\ \ }0<c\\ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x & \text {\ if\ \ }\neg 0<c \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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