Optimal. Leaf size=142 \[ -2 b^2 d x-\frac {1}{4} b^2 e x^2+\frac {2 b d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {b e x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 c}-\frac {d^2 (a+b \text {ArcSin}(c x))^2}{2 e}-\frac {e (a+b \text {ArcSin}(c x))^2}{4 c^2}+\frac {(d+e x)^2 (a+b \text {ArcSin}(c x))^2}{2 e} \]
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Rubi [A]
time = 0.20, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4827, 4847,
4737, 4767, 8, 4795, 30} \begin {gather*} \frac {2 b d \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {b e x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 c}-\frac {e (a+b \text {ArcSin}(c x))^2}{4 c^2}-\frac {d^2 (a+b \text {ArcSin}(c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \text {ArcSin}(c x))^2}{2 e}-2 b^2 d x-\frac {1}{4} b^2 e x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 4737
Rule 4767
Rule 4795
Rule 4827
Rule 4847
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}-\frac {(b c) \int \frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{e}\\ &=\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}-\frac {(b c) \int \left (\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {2 d e x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\right ) \, dx}{e}\\ &=\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}-(2 b c d) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {\left (b c d^2\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{e}-(b c e) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {2 b d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {b e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}-\left (2 b^2 d\right ) \int 1 \, dx-\frac {1}{2} \left (b^2 e\right ) \int x \, dx-\frac {(b e) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c}\\ &=-2 b^2 d x-\frac {1}{4} b^2 e x^2+\frac {2 b d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {b e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}-\frac {e \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac {(d+e x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 e}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 147, normalized size = 1.04 \begin {gather*} \frac {c \left (2 a^2 c x (2 d+e x)-b^2 c x (8 d+e x)+2 a b (4 d+e x) \sqrt {1-c^2 x^2}\right )+2 b \left (4 a c^2 d x+b c (4 d+e x) \sqrt {1-c^2 x^2}+a e \left (-1+2 c^2 x^2\right )\right ) \text {ArcSin}(c x)+b^2 \left (4 c^2 d x+e \left (-1+2 c^2 x^2\right )\right ) \text {ArcSin}(c x)^2}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 198, normalized size = 1.39
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {e \left (2 \arcsin \left (c x \right )^{2} c^{2} x^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}\right )}{c}+\frac {2 a b \left (\arcsin \left (c x \right ) d \,c^{2} x +\frac {\arcsin \left (c x \right ) e \,c^{2} x^{2}}{2}+d c \sqrt {-c^{2} x^{2}+1}-\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}\right )}{c}}{c}\) | \(198\) |
default | \(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+\frac {e \left (2 \arcsin \left (c x \right )^{2} c^{2} x^{2}+2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x -\arcsin \left (c x \right )^{2}-c^{2} x^{2}\right )}{4}\right )}{c}+\frac {2 a b \left (\arcsin \left (c x \right ) d \,c^{2} x +\frac {\arcsin \left (c x \right ) e \,c^{2} x^{2}}{2}+d c \sqrt {-c^{2} x^{2}+1}-\frac {e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}\right )}{c}}{c}\) | \(198\) |
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 = 2.13, size = 163, normalized size = 1.15 \begin {gather*} \frac {{\left (2 \, a^{2} - b^{2}\right )} c^{2} x^{2} e + 4 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{2} d x + {\left (4 \, b^{2} c^{2} d x + {\left (2 \, b^{2} c^{2} x^{2} - b^{2}\right )} e\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (4 \, a b c^{2} d x + {\left (2 \, a b c^{2} x^{2} - a b\right )} e\right )} \arcsin \left (c x\right ) + 2 \, {\left (a b c x e + 4 \, a b c d + {\left (b^{2} c x e + 4 \, b^{2} c d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{4 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 233, normalized size = 1.64 \begin {gather*} \begin {cases} a^{2} d x + \frac {a^{2} e x^{2}}{2} + 2 a b d x \operatorname {asin}{\left (c x \right )} + a b e x^{2} \operatorname {asin}{\left (c x \right )} + \frac {2 a b d \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {a b e x \sqrt {- c^{2} x^{2} + 1}}{2 c} - \frac {a b e \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + b^{2} d x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d x + \frac {b^{2} e x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {b^{2} e x^{2}}{4} + \frac {2 b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {b^{2} e x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2 c} - \frac {b^{2} e \operatorname {asin}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\a^{2} \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 244, normalized size = 1.72 \begin {gather*} b^{2} d x \arcsin \left (c x\right )^{2} + 2 \, a b d x \arcsin \left (c x\right ) + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2} e x \arcsin \left (c x\right )}{2 \, c} + a^{2} d x - 2 \, b^{2} d x + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e \arcsin \left (c x\right )^{2}}{2 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} a b e x}{2 \, c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a b e \arcsin \left (c x\right )}{c^{2}} + \frac {b^{2} e \arcsin \left (c x\right )^{2}}{4 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a^{2} e}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e}{4 \, c^{2}} + \frac {a b e \arcsin \left (c x\right )}{2 \, c^{2}} - \frac {b^{2} e}{8 \, c^{2}} \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 {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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