Optimal. Leaf size=335 \[ -2 b^2 d^2 x-\frac {8 b^2 d e x}{9 c^2}-\frac {16 b^2 e^2 x}{75 c^4}-\frac {4}{27} b^2 d e x^3-\frac {8 b^2 e^2 x^3}{225 c^2}-\frac {2}{125} b^2 e^2 x^5+\frac {2 b d^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {8 b d e \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c^3}+\frac {16 b e^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{75 c^5}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c}+\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{75 c^3}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{25 c}+d^2 x (a+b \text {ArcSin}(c x))^2+\frac {2}{3} d e x^3 (a+b \text {ArcSin}(c x))^2+\frac {1}{5} e^2 x^5 (a+b \text {ArcSin}(c x))^2 \]
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Rubi [A]
time = 0.39, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {4757, 4715,
4767, 8, 4723, 4795, 30} \begin {gather*} \frac {2 b d^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{c}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{25 c}+\frac {16 b e^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{75 c^5}+\frac {8 b d e \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 c^3}+\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{75 c^3}+d^2 x (a+b \text {ArcSin}(c x))^2+\frac {2}{3} d e x^3 (a+b \text {ArcSin}(c x))^2+\frac {1}{5} e^2 x^5 (a+b \text {ArcSin}(c x))^2-\frac {16 b^2 e^2 x}{75 c^4}-\frac {8 b^2 d e x}{9 c^2}-\frac {8 b^2 e^2 x^3}{225 c^2}-2 b^2 d^2 x-\frac {4}{27} b^2 d e x^3-\frac {2}{125} b^2 e^2 x^5 \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 4715
Rule 4723
Rule 4757
Rule 4767
Rule 4795
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 \left (a+b \sin ^{-1}(c x)\right )^2+2 d e x^2 \left (a+b \sin ^{-1}(c x)\right )^2+e^2 x^4 \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+(2 d e) \int x^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+e^2 \int x^4 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ &=d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b c d^2\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{3} (4 b c d e) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{5} \left (2 b c e^2\right ) \int \frac {x^5 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {2 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b^2 d^2\right ) \int 1 \, dx-\frac {1}{9} \left (4 b^2 d e\right ) \int x^2 \, dx-\frac {(8 b d e) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{9 c}-\frac {1}{25} \left (2 b^2 e^2\right ) \int x^4 \, dx-\frac {\left (8 b e^2\right ) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{25 c}\\ &=-2 b^2 d^2 x-\frac {4}{27} b^2 d e x^3-\frac {2}{125} b^2 e^2 x^5+\frac {2 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {8 b d e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (8 b^2 d e\right ) \int 1 \, dx}{9 c^2}-\frac {\left (16 b e^2\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{75 c^3}-\frac {\left (8 b^2 e^2\right ) \int x^2 \, dx}{75 c^2}\\ &=-2 b^2 d^2 x-\frac {8 b^2 d e x}{9 c^2}-\frac {4}{27} b^2 d e x^3-\frac {8 b^2 e^2 x^3}{225 c^2}-\frac {2}{125} b^2 e^2 x^5+\frac {2 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {8 b d e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {16 b e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (16 b^2 e^2\right ) \int 1 \, dx}{75 c^4}\\ &=-2 b^2 d^2 x-\frac {8 b^2 d e x}{9 c^2}-\frac {16 b^2 e^2 x}{75 c^4}-\frac {4}{27} b^2 d e x^3-\frac {8 b^2 e^2 x^3}{225 c^2}-\frac {2}{125} b^2 e^2 x^5+\frac {2 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac {8 b d e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac {16 b e^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^5}+\frac {4 b d e x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac {8 b e^2 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{75 c^3}+\frac {2 b e^2 x^4 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {2}{3} d e x^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 291, normalized size = 0.87 \begin {gather*} \frac {225 a^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+30 a b \sqrt {1-c^2 x^2} \left (24 e^2+4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )-2 b^2 c x \left (360 e^2+60 c^2 e \left (25 d+e x^2\right )+c^4 \left (3375 d^2+250 d e x^2+27 e^2 x^4\right )\right )+30 b \left (15 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b \sqrt {1-c^2 x^2} \left (24 e^2+4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )\right ) \text {ArcSin}(c x)+225 b^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \text {ArcSin}(c x)^2}{3375 c^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(634\) vs.
\(2(299)=598\).
time = 0.15, size = 635, normalized size = 1.90
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b^{2} \left (\frac {e^{2} \left (675 \arcsin \left (c x \right )^{2} c^{5} x^{5}+270 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2250 c^{3} x^{3} \arcsin \left (c x \right )^{2}-54 c^{5} x^{5}-1140 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+3375 c x \arcsin \left (c x \right )^{2}+380 c^{3} x^{3}+4470 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-4470 c x \right )}{3375}+\frac {2 c^{2} d e \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-27 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}-42 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+42 c x \right )}{27}+\frac {2 e^{2} \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-27 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}-42 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+42 c x \right )}{27}+c^{4} d^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+2 c^{2} d e \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+e^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{4}}+\frac {2 a b \left (\arcsin \left (c x \right ) d^{2} c^{5} x +\frac {2 \arcsin \left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) e^{2} c^{5} x^{5}}{5}+d^{2} c^{4} \sqrt {-c^{2} x^{2}+1}-\frac {2 d \,c^{2} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-\frac {e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{4}}}{c}\) | \(635\) |
default | \(\frac {\frac {a^{2} \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b^{2} \left (\frac {e^{2} \left (675 \arcsin \left (c x \right )^{2} c^{5} x^{5}+270 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2250 c^{3} x^{3} \arcsin \left (c x \right )^{2}-54 c^{5} x^{5}-1140 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+3375 c x \arcsin \left (c x \right )^{2}+380 c^{3} x^{3}+4470 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-4470 c x \right )}{3375}+\frac {2 c^{2} d e \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-27 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}-42 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+42 c x \right )}{27}+\frac {2 e^{2} \left (9 c^{3} x^{3} \arcsin \left (c x \right )^{2}+6 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-27 c x \arcsin \left (c x \right )^{2}-2 c^{3} x^{3}-42 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+42 c x \right )}{27}+c^{4} d^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+2 c^{2} d e \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )+e^{2} \left (c x \arcsin \left (c x \right )^{2}-2 c x +2 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\right )\right )}{c^{4}}+\frac {2 a b \left (\arcsin \left (c x \right ) d^{2} c^{5} x +\frac {2 \arcsin \left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) e^{2} c^{5} x^{5}}{5}+d^{2} c^{4} \sqrt {-c^{2} x^{2}+1}-\frac {2 d \,c^{2} e \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-\frac {e^{2} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{4}}}{c}\) | \(635\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 437, normalized size = 1.30 \begin {gather*} \frac {1}{5} \, b^{2} x^{5} \arcsin \left (c x\right )^{2} e^{2} + \frac {2}{3} \, b^{2} d x^{3} \arcsin \left (c x\right )^{2} e + \frac {1}{5} \, a^{2} x^{5} e^{2} + b^{2} d^{2} x \arcsin \left (c x\right )^{2} + \frac {2}{3} \, a^{2} d x^{3} e - 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {4}{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 )} a b d e + \frac {4}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} d e + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b d^{2}}{c} + \frac {2}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b e^{2} + \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arcsin \left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} e^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.30, size = 342, normalized size = 1.02 \begin {gather*} \frac {3375 \, {\left (a^{2} - 2 \, b^{2}\right )} c^{5} d^{2} x + 225 \, {\left (3 \, b^{2} c^{5} x^{5} e^{2} + 10 \, b^{2} c^{5} d x^{3} e + 15 \, b^{2} c^{5} d^{2} x\right )} \arcsin \left (c x\right )^{2} + 450 \, {\left (3 \, a b c^{5} x^{5} e^{2} + 10 \, a b c^{5} d x^{3} e + 15 \, a b c^{5} d^{2} x\right )} \arcsin \left (c x\right ) + 3 \, {\left (9 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{5} x^{5} - 40 \, b^{2} c^{3} x^{3} - 240 \, b^{2} c x\right )} e^{2} + 250 \, {\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{5} d x^{3} - 12 \, b^{2} c^{3} d x\right )} e + 30 \, {\left (225 \, a b c^{4} d^{2} + {\left (225 \, b^{2} c^{4} d^{2} + 3 \, {\left (3 \, b^{2} c^{4} x^{4} + 4 \, b^{2} c^{2} x^{2} + 8 \, b^{2}\right )} e^{2} + 50 \, {\left (b^{2} c^{4} d x^{2} + 2 \, b^{2} c^{2} d\right )} e\right )} \arcsin \left (c x\right ) + 3 \, {\left (3 \, a b c^{4} x^{4} + 4 \, a b c^{2} x^{2} + 8 \, a b\right )} e^{2} + 50 \, {\left (a b c^{4} d x^{2} + 2 \, a b c^{2} d\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{3375 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.71, size = 595, normalized size = 1.78 \begin {gather*} \begin {cases} a^{2} d^{2} x + \frac {2 a^{2} d e x^{3}}{3} + \frac {a^{2} e^{2} x^{5}}{5} + 2 a b d^{2} x \operatorname {asin}{\left (c x \right )} + \frac {4 a b d e x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {2 a b e^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {2 a b d^{2} \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {4 a b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} + \frac {2 a b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {8 a b d e \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {8 a b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {16 a b e^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} + b^{2} d^{2} x \operatorname {asin}^{2}{\left (c x \right )} - 2 b^{2} d^{2} x + \frac {2 b^{2} d e x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} - \frac {4 b^{2} d e x^{3}}{27} + \frac {b^{2} e^{2} x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} - \frac {2 b^{2} e^{2} x^{5}}{125} + \frac {2 b^{2} d^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{c} + \frac {4 b^{2} d e x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c} + \frac {2 b^{2} e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{25 c} - \frac {8 b^{2} d e x}{9 c^{2}} - \frac {8 b^{2} e^{2} x^{3}}{225 c^{2}} + \frac {8 b^{2} d e \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{9 c^{3}} + \frac {8 b^{2} e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{75 c^{3}} - \frac {16 b^{2} e^{2} x}{75 c^{4}} + \frac {16 b^{2} e^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{75 c^{5}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{2} x + \frac {2 d e x^{3}}{3} + \frac {e^{2} x^{5}}{5}\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. 683 vs.
\(2 (299) = 598\).
time = 0.43, size = 683, normalized size = 2.04 \begin {gather*} \frac {1}{5} \, a^{2} e^{2} x^{5} + \frac {2}{3} \, a^{2} d e x^{3} + b^{2} d^{2} x \arcsin \left (c x\right )^{2} + 2 \, a b d^{2} x \arcsin \left (c x\right ) + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d e x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + a^{2} d^{2} x - 2 \, b^{2} d^{2} x + \frac {4 \, {\left (c^{2} x^{2} - 1\right )} a b d e x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, b^{2} d e x \arcsin \left (c x\right )^{2}}{3 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} e^{2} x \arcsin \left (c x\right )^{2}}{5 \, c^{4}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{c} - \frac {4 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d e x}{27 \, c^{2}} + \frac {4 \, a b d e x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b e^{2} x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b^{2} e^{2} x \arcsin \left (c x\right )^{2}}{5 \, c^{4}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{c} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d e \arcsin \left (c x\right )}{9 \, c^{3}} - \frac {28 \, b^{2} d e x}{27 \, c^{2}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} e^{2} x}{125 \, c^{4}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )} a b e^{2} x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {b^{2} e^{2} x \arcsin \left (c x\right )^{2}}{5 \, c^{4}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d e}{9 \, c^{3}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d e \arcsin \left (c x\right )}{3 \, c^{3}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} e^{2} \arcsin \left (c x\right )}{25 \, c^{5}} - \frac {76 \, {\left (c^{2} x^{2} - 1\right )} b^{2} e^{2} x}{1125 \, c^{4}} + \frac {2 \, a b e^{2} x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b d e}{3 \, c^{3}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b e^{2}}{25 \, c^{5}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} e^{2} \arcsin \left (c x\right )}{15 \, c^{5}} - \frac {298 \, b^{2} e^{2} x}{1125 \, c^{4}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b e^{2}}{15 \, c^{5}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b^{2} e^{2} \arcsin \left (c x\right )}{5 \, c^{5}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} a b e^{2}}{5 \, c^{5}} \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 (e\,x^2+d\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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