Optimal. Leaf size=124 \[ \frac {b (d+e x)^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right ) \sqrt {1-c^2 x^2}}{18 c^3}-\frac {b d \left (2 d^2+\frac {3 e^2}{c^2}\right ) \text {ArcSin}(c x)}{6 e}+\frac {(d+e x)^3 (a+b \text {ArcSin}(c x))}{3 e} \]
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Rubi [A]
time = 0.07, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4827, 757, 794,
222} \begin {gather*} \frac {(d+e x)^3 (a+b \text {ArcSin}(c x))}{3 e}-\frac {b d \text {ArcSin}(c x) \left (\frac {3 e^2}{c^2}+2 d^2\right )}{6 e}+\frac {b \sqrt {1-c^2 x^2} (d+e x)^2}{9 c}+\frac {b \sqrt {1-c^2 x^2} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 757
Rule 794
Rule 4827
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {(b c) \int \frac {(d+e x)^3}{\sqrt {1-c^2 x^2}} \, dx}{3 e}\\ &=\frac {b (d+e x)^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}+\frac {b \int \frac {(d+e x) \left (-3 c^2 d^2-2 e^2-5 c^2 d e x\right )}{\sqrt {1-c^2 x^2}} \, dx}{9 c e}\\ &=\frac {b (d+e x)^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right ) \sqrt {1-c^2 x^2}}{18 c^3}+\frac {(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}-\frac {1}{6} \left (b d \left (\frac {2 c d^2}{e}+\frac {3 e}{c}\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b (d+e x)^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right ) \sqrt {1-c^2 x^2}}{18 c^3}-\frac {b d \left (2 d^2+\frac {3 e^2}{c^2}\right ) \sin ^{-1}(c x)}{6 e}+\frac {(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )}{3 e}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 121, normalized size = 0.98 \begin {gather*} \frac {6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+b \sqrt {1-c^2 x^2} \left (4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 b c \left (6 c^2 d^2 x+2 c^2 e^2 x^3+3 d e \left (-1+2 c^2 x^2\right )\right ) \text {ArcSin}(c x)}{18 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 193, normalized size = 1.56
method | result | size |
derivativedivides | \(\frac {\frac {\left (c e x +d c \right )^{3} a}{3 c^{2} e}+\frac {b \left (\frac {\arcsin \left (c x \right ) c^{3} d^{3}}{3 e}+\arcsin \left (c x \right ) c^{3} d^{2} x +e \arcsin \left (c x \right ) c^{3} d \,x^{2}+\frac {\arcsin \left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \arcsin \left (c x \right )-3 d^{2} c^{2} e \sqrt {-c^{2} x^{2}+1}+3 d c \,e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+e^{3} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3 e}\right )}{c^{2}}}{c}\) | \(193\) |
default | \(\frac {\frac {\left (c e x +d c \right )^{3} a}{3 c^{2} e}+\frac {b \left (\frac {\arcsin \left (c x \right ) c^{3} d^{3}}{3 e}+\arcsin \left (c x \right ) c^{3} d^{2} x +e \arcsin \left (c x \right ) c^{3} d \,x^{2}+\frac {\arcsin \left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \arcsin \left (c x \right )-3 d^{2} c^{2} e \sqrt {-c^{2} x^{2}+1}+3 d c \,e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+e^{3} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3 e}\right )}{c^{2}}}{c}\) | \(193\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 150, normalized size = 1.21 \begin {gather*} \frac {1}{3} \, a x^{3} e^{2} + a d x^{2} e + a d^{2} x + \frac {1}{2} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d e + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{2}}{c} + \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^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.34, size = 136, normalized size = 1.10 \begin {gather*} \frac {6 \, a c^{3} x^{3} e^{2} + 18 \, a c^{3} d x^{2} e + 18 \, a c^{3} d^{2} x + 3 \, {\left (2 \, b c^{3} x^{3} e^{2} + 6 \, b c^{3} d^{2} x + 3 \, {\left (2 \, b c^{3} d x^{2} - b c d\right )} e\right )} \arcsin \left (c x\right ) + {\left (9 \, b c^{2} d x e + 18 \, b c^{2} d^{2} + 2 \, {\left (b c^{2} x^{2} + 2 \, b\right )} e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{18 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 190, normalized size = 1.53 \begin {gather*} \begin {cases} a d^{2} x + a d e x^{2} + \frac {a e^{2} x^{3}}{3} + b d^{2} x \operatorname {asin}{\left (c x \right )} + b d e x^{2} \operatorname {asin}{\left (c x \right )} + \frac {b e^{2} x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b d^{2} \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b d e x \sqrt {- c^{2} x^{2} + 1}}{2 c} + \frac {b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {b d e \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + \frac {2 b e^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\a \left (d^{2} x + d e x^{2} + \frac {e^{2} 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 = 0.41, size = 194, normalized size = 1.56 \begin {gather*} \frac {1}{3} \, a e^{2} x^{3} + b d^{2} x \arcsin \left (c x\right ) + a d^{2} x + \frac {{\left (c^{2} x^{2} - 1\right )} b e^{2} x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d e x}{2 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b d e \arcsin \left (c x\right )}{c^{2}} + \frac {b e^{2} x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2}}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a d e}{c^{2}} + \frac {b d e \arcsin \left (c x\right )}{2 \, c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2}}{9 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e^{2}}{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^2\,\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 (3\,d^2+3\,d\,e\,x+e^2\,x^2\right )}{3}+\frac {b\,d^2\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c}+\frac {2\,b\,d\,e\,\left (\frac {\mathrm {asin}\left (c\,x\right )\,\left (2\,c^2\,x^2-1\right )}{4}+\frac {c\,x\,\sqrt {1-c^2\,x^2}}{4}\right )}{c^2} & \text {\ if\ \ }0<c\\ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^2 \,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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