Optimal. Leaf size=374 \[ -\frac {3 e^3 \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+\frac {2 b d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {3 b d^2 e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac {2 b d e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {4 b d e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac {3 b e^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}-\frac {d^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}+\frac {(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac {4 b^2 d e^2 x}{3 c^2}-\frac {3 b^2 e^3 x^2}{32 c^2}-2 b^2 d^3 x-\frac {3}{4} b^2 d^2 e x^2-\frac {2}{9} b^2 d e^2 x^3-\frac {1}{32} b^2 e^3 x^4 \]
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Rubi [A] time = 0.71, 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 = {4743, 4763, 4641, 4677, 8, 4707, 30} \[ \frac {3 b d^2 e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac {3 d^2 e \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac {2 b d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {4 b d e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac {2 b d e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac {b e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {3 b e^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}-\frac {3 e^3 \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}-\frac {d^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}+\frac {(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac {4 b^2 d e^2 x}{3 c^2}-\frac {3 b^2 e^3 x^2}{32 c^2}-\frac {3}{4} b^2 d^2 e x^2-2 b^2 d^3 x-\frac {2}{9} b^2 d e^2 x^3-\frac {1}{32} b^2 e^3 x^4 \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 4641
Rule 4677
Rule 4707
Rule 4743
Rule 4763
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 = 0.64, size = 355, normalized size = 0.95 \[ \frac {c \left (72 a^2 c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+6 a b \sqrt {1-c^2 x^2} \left (c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )+e^2 (64 d+9 e x)\right )-b^2 c x \left (c^2 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )+3 e^2 (128 d+9 e x)\right )\right )+6 b \sin ^{-1}(c x) \left (3 a \left (8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-24 c^2 d^2 e-3 e^3\right )+b c \sqrt {1-c^2 x^2} \left (c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )+e^2 (64 d+9 e x)\right )\right )+9 b^2 \sin ^{-1}(c x)^2 \left (8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-24 c^2 d^2 e-3 e^3\right )}{288 c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 446, normalized size = 1.19 \[ \frac {9 \, {\left (8 \, a^{2} - b^{2}\right )} c^{4} e^{3} x^{4} + 32 \, {\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} d e^{2} x^{3} + 27 \, {\left (8 \, {\left (2 \, a^{2} - b^{2}\right )} c^{4} d^{2} e - b^{2} c^{2} e^{3}\right )} x^{2} + 9 \, {\left (8 \, b^{2} c^{4} e^{3} x^{4} + 32 \, b^{2} c^{4} d e^{2} x^{3} + 48 \, b^{2} c^{4} d^{2} e x^{2} + 32 \, b^{2} c^{4} d^{3} x - 24 \, b^{2} c^{2} d^{2} e - 3 \, b^{2} e^{3}\right )} \arcsin \left (c x\right )^{2} + 96 \, {\left (3 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{4} d^{3} - 4 \, b^{2} c^{2} d e^{2}\right )} x + 18 \, {\left (8 \, a b c^{4} e^{3} x^{4} + 32 \, a b c^{4} d e^{2} x^{3} + 48 \, a b c^{4} d^{2} e x^{2} + 32 \, a b c^{4} d^{3} x - 24 \, a b c^{2} d^{2} e - 3 \, a b e^{3}\right )} \arcsin \left (c x\right ) + 6 \, {\left (6 \, a b c^{3} e^{3} x^{3} + 32 \, a b c^{3} d e^{2} x^{2} + 96 \, a b c^{3} d^{3} + 64 \, a b c d e^{2} + 9 \, {\left (8 \, a b c^{3} d^{2} e + a b c e^{3}\right )} x + {\left (6 \, b^{2} c^{3} e^{3} x^{3} + 32 \, b^{2} c^{3} d e^{2} x^{2} + 96 \, b^{2} c^{3} d^{3} + 64 \, b^{2} c d e^{2} + 9 \, {\left (8 \, b^{2} c^{3} d^{2} e + b^{2} c e^{3}\right )} x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{288 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.48, size = 800, normalized size = 2.14 \[ b^{2} d^{3} x \arcsin \left (c x\right )^{2} + 2 \, a b d^{3} x \arcsin \left (c x\right ) + \frac {1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} x \arcsin \left (c x\right ) e}{2 \, c} + a^{2} d^{3} x - 2 \, b^{2} d^{3} x + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d x \arcsin \left (c x\right )^{2} e^{2}}{c^{2}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{2} \arcsin \left (c x\right )^{2} e}{2 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{3} \arcsin \left (c x\right )}{c} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2} x e}{2 \, c} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b d x \arcsin \left (c x\right ) e^{2}}{c^{2}} + \frac {b^{2} d x \arcsin \left (c x\right )^{2} e^{2}}{c^{2}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} a b d^{2} \arcsin \left (c x\right ) e}{c^{2}} + \frac {3 \, b^{2} d^{2} \arcsin \left (c x\right )^{2} e}{4 \, c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d^{3}}{c} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d x e^{2}}{9 \, c^{2}} + \frac {2 \, a b d x \arcsin \left (c x\right ) e^{2}}{c^{2}} + \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 {3 \, a b d^{2} \arcsin \left (c x\right ) e}{2 \, c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} x \arcsin \left (c x\right ) e^{3}}{8 \, c^{3}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d \arcsin \left (c x\right ) e^{2}}{3 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} \arcsin \left (c x\right )^{2} e^{3}}{4 \, c^{4}} - \frac {14 \, b^{2} d x e^{2}}{9 \, c^{2}} - \frac {3 \, b^{2} d^{2} e}{8 \, c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b x e^{3}}{8 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b^{2} x \arcsin \left (c x\right ) e^{3}}{16 \, c^{3}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d e^{2}}{3 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right ) e^{2}}{c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} a b \arcsin \left (c x\right ) e^{3}}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} \arcsin \left (c x\right )^{2} e^{3}}{2 \, c^{4}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} a b x e^{3}}{16 \, c^{3}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d e^{2}}{c^{3}} - \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 \arcsin \left (c x\right ) e^{3}}{c^{4}} + \frac {5 \, b^{2} \arcsin \left (c x\right )^{2} e^{3}}{32 \, c^{4}} - \frac {5 \, {\left (c^{2} x^{2} - 1\right )} b^{2} e^{3}}{32 \, c^{4}} + \frac {5 \, a b \arcsin \left (c x\right ) e^{3}}{16 \, c^{4}} - \frac {17 \, b^{2} e^{3}}{256 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 660, normalized size = 1.76 \[ \frac {\frac {\left (c e x +d c \right )^{4} a^{2}}{4 c^{3} e}+\frac {b^{2} \left (\frac {e^{3} \left (8 \arcsin \left (c x \right )^{2} c^{4} x^{4}+4 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{3} x^{3}-16 \arcsin \left (c x \right )^{2} c^{2} x^{2}-c^{4} x^{4}-10 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x +5 \arcsin \left (c x \right )^{2}+5 c^{2} x^{2}-4\right )}{32}+\frac {3 d^{2} c^{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 \arcsin \left (c x \right )^{2} c^{3} x^{3}+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}+c^{3} d^{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 {e^{3} \arcsin \left (c x \right ) c^{4} x^{4}}{4}+e^{2} \arcsin \left (c x \right ) c^{4} x^{3} d +\frac {3 e \arcsin \left (c x \right ) c^{4} x^{2} d^{2}}{2}+\arcsin \left (c x \right ) c^{4} x \,d^{3}+\frac {\arcsin \left (c x \right ) c^{4} d^{4}}{4 e}-\frac {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 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 )+6 c^{2} d^{2} e^{2} \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )-4 c^{3} d^{3} e \sqrt {-c^{2} x^{2}+1}+c^{4} d^{4} \arcsin \left (c x \right )}{4 e}\right )}{c^{3}}}{c} \]
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 \[ \frac {1}{4} \, a^{2} e^{3} x^{4} + a^{2} d e^{2} x^{3} + b^{2} d^{3} x \arcsin \left (c x\right )^{2} + \frac {3}{2} \, a^{2} d^{2} e x^{2} + \frac {3}{2} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} a b d^{2} e + \frac {2}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d e^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} a b e^{3} - 2 \, b^{2} d^{3} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b d^{3}}{c} + \frac {1}{4} \, {\left (b^{2} e^{3} x^{4} + 4 \, b^{2} d e^{2} x^{3} + 6 \, b^{2} d^{2} e x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + \int \frac {{\left (b^{2} c e^{3} x^{4} + 4 \, b^{2} c d e^{2} x^{3} + 6 \, b^{2} c d^{2} e x^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{2 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.49, size = 743, normalized size = 1.99 \[ \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} \]
Verification of antiderivative is not currently implemented for this CAS.
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