Optimal. Leaf size=374 \[ -2 b^2 d^3 x-\frac {4 b^2 d e^2 x}{3 c^2}-\frac {3}{4} b^2 d^2 e x^2-\frac {3 b^2 e^3 x^2}{32 c^2}-\frac {2}{9} b^2 d e^2 x^3-\frac {1}{32} b^2 e^3 x^4+\frac {2 b d^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {4 b d e^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 c^3}+\frac {3 b d^2 e x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 c}+\frac {3 b e^3 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{16 c^3}+\frac {2 b d e^2 x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 c}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{8 c}-\frac {d^4 (a+b \text {ArcSin}(c x))^2}{4 e}-\frac {3 d^2 e (a+b \text {ArcSin}(c x))^2}{4 c^2}-\frac {3 e^3 (a+b \text {ArcSin}(c x))^2}{32 c^4}+\frac {(d+e x)^4 (a+b \text {ArcSin}(c x))^2}{4 e} \]
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Rubi [A]
time = 0.48, antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {4827, 4847,
4737, 4767, 8, 4795, 30} \begin {gather*} -\frac {3 e^3 (a+b \text {ArcSin}(c x))^2}{32 c^4}+\frac {2 b d^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {3 b d^2 e x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 c}-\frac {3 d^2 e (a+b \text {ArcSin}(c x))^2}{4 c^2}+\frac {2 b d e^2 x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 c}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{8 c}+\frac {4 b d e^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 c^3}+\frac {3 b e^3 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{16 c^3}-\frac {d^4 (a+b \text {ArcSin}(c x))^2}{4 e}+\frac {(d+e x)^4 (a+b \text {ArcSin}(c x))^2}{4 e}-\frac {4 b^2 d e^2 x}{3 c^2}-\frac {3 b^2 e^3 x^2}{32 c^2}-2 b^2 d^3 x-\frac {3}{4} b^2 d^2 e x^2-\frac {2}{9} b^2 d e^2 x^3-\frac {1}{32} b^2 e^3 x^4 \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)^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac {(b c) \int \frac {(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{2 e}\\ &=\frac {(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac {(b c) \int \left (\frac {d^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {4 d^3 e x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {6 d^2 e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {4 d e^3 x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {e^4 x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\right ) \, dx}{2 e}\\ &=\frac {(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\left (2 b c d^3\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {\left (b c d^4\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 e}-\left (3 b c d^2 e\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\left (2 b c d e^2\right ) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{2} \left (b c e^3\right ) \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {2 b d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {3 b d^2 e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac {2 b d e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}+\frac {(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\left (2 b^2 d^3\right ) \int 1 \, dx-\frac {1}{2} \left (3 b^2 d^2 e\right ) \int x \, dx-\frac {\left (3 b d^2 e\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c}-\frac {1}{3} \left (2 b^2 d e^2\right ) \int x^2 \, dx-\frac {\left (4 b d e^2\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{3 c}-\frac {1}{8} \left (b^2 e^3\right ) \int x^3 \, dx-\frac {\left (3 b e^3\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{8 c}\\ &=-2 b^2 d^3 x-\frac {3}{4} b^2 d^2 e x^2-\frac {2}{9} b^2 d e^2 x^3-\frac {1}{32} b^2 e^3 x^4+\frac {2 b d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {4 b d e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac {3 b d^2 e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac {3 b e^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac {2 b d e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac {(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac {\left (4 b^2 d e^2\right ) \int 1 \, dx}{3 c^2}-\frac {\left (3 b e^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 c^3}-\frac {\left (3 b^2 e^3\right ) \int x \, dx}{16 c^2}\\ &=-2 b^2 d^3 x-\frac {4 b^2 d e^2 x}{3 c^2}-\frac {3}{4} b^2 d^2 e x^2-\frac {3 b^2 e^3 x^2}{32 c^2}-\frac {2}{9} b^2 d e^2 x^3-\frac {1}{32} b^2 e^3 x^4+\frac {2 b d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {4 b d e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac {3 b d^2 e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac {3 b e^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac {2 b d e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac {d^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}-\frac {3 e^3 \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+\frac {(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 355, normalized size = 0.95 \begin {gather*} \frac {c \left (72 a^2 c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+6 a b \sqrt {1-c^2 x^2} \left (e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )-b^2 c x \left (3 e^2 (128 d+9 e x)+c^2 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )\right )\right )+6 b \left (3 a \left (-24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )+b c \sqrt {1-c^2 x^2} \left (e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )\right ) \text {ArcSin}(c x)+9 b^2 \left (-24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \text {ArcSin}(c x)^2}{288 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 660, normalized size = 1.76
method | result | size |
derivativedivides | \(\frac {\frac {\left (c e x +d c \right )^{4} a^{2}}{4 c^{3} e}+\frac {b^{2} \left (\frac {e^{3} \left (32 \arcsin \left (c x \right )^{2} c^{4} x^{4}+16 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{3} x^{3}-64 \arcsin \left (c x \right )^{2} c^{2} x^{2}-4 c^{4} x^{4}-40 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +20 \arcsin \left (c x \right )^{2}+20 c^{2} x^{2}-25\right )}{128}+\frac {3 c^{2} d^{2} 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}+\frac {d c \,e^{2} \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-27 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}-42 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+42 c x \right )}{9}+d^{3} c^{3} \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^{3} \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}+3 d c \,e^{2} \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 )\right )}{c^{3}}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) c^{4} d^{4}}{4 e}+\arcsin \left (c x \right ) c^{4} d^{3} x +\frac {3 e \arcsin \left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \arcsin \left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \arcsin \left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{4} d^{4} \arcsin \left (c x \right )-4 d^{3} c^{3} e \sqrt {-c^{2} x^{2}+1}+6 d^{2} c^{2} e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+4 d c \,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 )+e^{4} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4 e}\right )}{c^{3}}}{c}\) | \(660\) |
default | \(\frac {\frac {\left (c e x +d c \right )^{4} a^{2}}{4 c^{3} e}+\frac {b^{2} \left (\frac {e^{3} \left (32 \arcsin \left (c x \right )^{2} c^{4} x^{4}+16 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{3} x^{3}-64 \arcsin \left (c x \right )^{2} c^{2} x^{2}-4 c^{4} x^{4}-40 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +20 \arcsin \left (c x \right )^{2}+20 c^{2} x^{2}-25\right )}{128}+\frac {3 c^{2} d^{2} 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}+\frac {d c \,e^{2} \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-27 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}-42 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+42 c x \right )}{9}+d^{3} c^{3} \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^{3} \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}+3 d c \,e^{2} \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 )\right )}{c^{3}}+\frac {2 a b \left (\frac {\arcsin \left (c x \right ) c^{4} d^{4}}{4 e}+\arcsin \left (c x \right ) c^{4} d^{3} x +\frac {3 e \arcsin \left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \arcsin \left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \arcsin \left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{4} d^{4} \arcsin \left (c x \right )-4 d^{3} c^{3} e \sqrt {-c^{2} x^{2}+1}+6 d^{2} c^{2} e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+4 d c \,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 )+e^{4} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4 e}\right )}{c^{3}}}{c}\) | \(660\) |
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 = 435, normalized size = 1.16 \begin {gather*} \frac {216 \, {\left (2 \, a^{2} - b^{2}\right )} c^{4} d^{2} x^{2} e + 288 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{4} d^{3} x + 9 \, {\left (32 \, b^{2} c^{4} d x^{3} e^{2} + 32 \, b^{2} c^{4} d^{3} x + {\left (8 \, b^{2} c^{4} x^{4} - 3 \, b^{2}\right )} e^{3} + 24 \, {\left (2 \, b^{2} c^{4} d^{2} x^{2} - b^{2} c^{2} d^{2}\right )} e\right )} \arcsin \left (c x\right )^{2} + 18 \, {\left (32 \, a b c^{4} d x^{3} e^{2} + 32 \, a b c^{4} d^{3} x + {\left (8 \, a b c^{4} x^{4} - 3 \, a b\right )} e^{3} + 24 \, {\left (2 \, a b c^{4} d^{2} x^{2} - a b c^{2} d^{2}\right )} e\right )} \arcsin \left (c x\right ) + 9 \, {\left ({\left (8 \, a^{2} - b^{2}\right )} c^{4} x^{4} - 3 \, b^{2} c^{2} x^{2}\right )} e^{3} + 32 \, {\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} d x^{3} - 12 \, b^{2} c^{2} d x\right )} e^{2} + 6 \, {\left (72 \, a b c^{3} d^{2} x e + 96 \, a b c^{3} d^{3} + {\left (72 \, b^{2} c^{3} d^{2} x e + 96 \, b^{2} c^{3} d^{3} + 3 \, {\left (2 \, b^{2} c^{3} x^{3} + 3 \, b^{2} c x\right )} e^{3} + 32 \, {\left (b^{2} c^{3} d x^{2} + 2 \, b^{2} c d\right )} e^{2}\right )} \arcsin \left (c x\right ) + 3 \, {\left (2 \, a b c^{3} x^{3} + 3 \, a b c x\right )} e^{3} + 32 \, {\left (a b c^{3} d x^{2} + 2 \, a b c d\right )} e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{288 \, c^{4}} \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. 743 vs.
\(2 (364) = 728\).
time = 0.60, size = 743, normalized size = 1.99 \begin {gather*} \begin {cases} a^{2} d^{3} x + \frac {3 a^{2} d^{2} e x^{2}}{2} + a^{2} d e^{2} x^{3} + \frac {a^{2} e^{3} x^{4}}{4} + 2 a b d^{3} x \operatorname {asin}{\left (c x \right )} + 3 a b d^{2} e x^{2} \operatorname {asin}{\left (c x \right )} + 2 a b d e^{2} x^{3} \operatorname {asin}{\left (c x \right )} + \frac {a b e^{3} x^{4} \operatorname {asin}{\left (c x \right )}}{2} + \frac {2 a b d^{3} \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {3 a b d^{2} e x \sqrt {- c^{2} x^{2} + 1}}{2 c} + \frac {2 a b d e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {a b e^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{8 c} - \frac {3 a b d^{2} e \operatorname {asin}{\left (c x \right )}}{2 c^{2}} + \frac {4 a b d e^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} + \frac {3 a b e^{3} x \sqrt {- c^{2} x^{2} + 1}}{16 c^{3}} - \frac {3 a b e^{3} \operatorname {asin}{\left (c x \right )}}{16 c^{4}} + b^{2} d^{3} x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d^{3} x + \frac {3 b^{2} d^{2} e x^{2} \operatorname {asin}^{2}{\left (c x \right )}}{2} - \frac {3 b^{2} d^{2} e x^{2}}{4} + b^{2} d e^{2} x^{3} \operatorname {asin}^{2}{\left (c x \right )} - \frac {2 b^{2} d e^{2} x^{3}}{9} + \frac {b^{2} e^{3} x^{4} \operatorname {asin}^{2}{\left (c x \right )}}{4} - \frac {b^{2} e^{3} x^{4}}{32} + \frac {2 b^{2} d^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {3 b^{2} d^{2} e x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{2 c} + \frac {2 b^{2} d e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3 c} + \frac {b^{2} e^{3} x^{3} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{8 c} - \frac {3 b^{2} d^{2} e \operatorname {asin}^{2}{\left (c x \right )}}{4 c^{2}} - \frac {4 b^{2} d e^{2} x}{3 c^{2}} - \frac {3 b^{2} e^{3} x^{2}}{32 c^{2}} + \frac {4 b^{2} d e^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3 c^{3}} + \frac {3 b^{2} e^{3} x \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{16 c^{3}} - \frac {3 b^{2} e^{3} \operatorname {asin}^{2}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 816 vs.
\(2 (334) = 668\).
time = 0.43, size = 816, normalized size = 2.18 \begin {gather*} \frac {1}{4} \, a^{2} e^{3} x^{4} + a^{2} d e^{2} x^{3} + b^{2} d^{3} x \arcsin \left (c x\right )^{2} + 2 \, a b d^{3} x \arcsin \left (c x\right ) + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d e^{2} x \arcsin \left (c x\right )^{2}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} e x \arcsin \left (c x\right )}{2 \, c} + a^{2} d^{3} x - 2 \, b^{2} d^{3} x + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b d e^{2} x \arcsin \left (c x\right )}{c^{2}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{2} e \arcsin \left (c x\right )^{2}}{2 \, c^{2}} + \frac {b^{2} d e^{2} x \arcsin \left (c x\right )^{2}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2} e x}{2 \, c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{c} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} e^{3} x \arcsin \left (c x\right )}{8 \, c^{3}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d e^{2} x}{9 \, c^{2}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} a b d^{2} e \arcsin \left (c x\right )}{c^{2}} + \frac {2 \, a b d e^{2} x \arcsin \left (c x\right )}{c^{2}} + \frac {3 \, b^{2} d^{2} e \arcsin \left (c x\right )^{2}}{4 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} e^{3} \arcsin \left (c x\right )^{2}}{4 \, c^{4}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{c} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b e^{3} x}{8 \, c^{3}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d e^{2} \arcsin \left (c x\right )}{3 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b^{2} e^{3} x \arcsin \left (c x\right )}{16 \, c^{3}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} a^{2} d^{2} e}{2 \, c^{2}} - \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{2} e}{4 \, c^{2}} - \frac {14 \, b^{2} d e^{2} x}{9 \, c^{2}} + \frac {3 \, a b d^{2} e \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} a b e^{3} \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} e^{3} \arcsin \left (c x\right )^{2}}{2 \, c^{4}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d e^{2}}{3 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} a b e^{3} x}{16 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d e^{2} \arcsin \left (c x\right )}{c^{3}} - \frac {3 \, b^{2} d^{2} e}{8 \, c^{2}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} e^{3}}{32 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )} a b e^{3} \arcsin \left (c x\right )}{c^{4}} + \frac {5 \, b^{2} e^{3} \arcsin \left (c x\right )^{2}}{32 \, c^{4}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d e^{2}}{c^{3}} - \frac {5 \, {\left (c^{2} x^{2} - 1\right )} b^{2} e^{3}}{32 \, c^{4}} + \frac {5 \, a b e^{3} \arcsin \left (c x\right )}{16 \, c^{4}} - \frac {17 \, b^{2} e^{3}}{256 \, c^{4}} \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 = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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