Optimal. Leaf size=149 \[ \frac {b \left (9 c^2 d+5 e\right ) x \sqrt {1-c^2 x^2}}{96 c^5}+\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e x^5 \sqrt {1-c^2 x^2}}{36 c}-\frac {b \left (9 c^2 d+5 e\right ) \text {ArcSin}(c x)}{96 c^6}+\frac {1}{4} d x^4 (a+b \text {ArcSin}(c x))+\frac {1}{6} e x^6 (a+b \text {ArcSin}(c x)) \]
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Rubi [A]
time = 0.09, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {14, 4815, 12,
470, 327, 222} \begin {gather*} \frac {1}{4} d x^4 (a+b \text {ArcSin}(c x))+\frac {1}{6} e x^6 (a+b \text {ArcSin}(c x))-\frac {b \text {ArcSin}(c x) \left (9 c^2 d+5 e\right )}{96 c^6}+\frac {b e x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b x \sqrt {1-c^2 x^2} \left (9 c^2 d+5 e\right )}{96 c^5}+\frac {b x^3 \sqrt {1-c^2 x^2} \left (9 c^2 d+5 e\right )}{144 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 222
Rule 327
Rule 470
Rule 4815
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x^4 \left (3 d+2 e x^2\right )}{12 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{12} (b c) \int \frac {x^4 \left (3 d+2 e x^2\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b e x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{36} \left (b c \left (9 d+\frac {5 e}{c^2}\right )\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b \left (9 c^2 d+5 e\right )\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{48 c^3}\\ &=\frac {b \left (9 c^2 d+5 e\right ) x \sqrt {1-c^2 x^2}}{96 c^5}+\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b \left (9 c^2 d+5 e\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{96 c^5}\\ &=\frac {b \left (9 c^2 d+5 e\right ) x \sqrt {1-c^2 x^2}}{96 c^5}+\frac {b \left (9 c^2 d+5 e\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e x^5 \sqrt {1-c^2 x^2}}{36 c}-\frac {b \left (9 c^2 d+5 e\right ) \sin ^{-1}(c x)}{96 c^6}+\frac {1}{4} d x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 116, normalized size = 0.78 \begin {gather*} \frac {24 a c^6 x^4 \left (3 d+2 e x^2\right )+b c x \sqrt {1-c^2 x^2} \left (15 e+c^2 \left (27 d+10 e x^2\right )+2 c^4 \left (9 d x^2+4 e x^4\right )\right )+3 b \left (-9 c^2 d-5 e+8 c^6 \left (3 d x^4+2 e x^6\right )\right ) \text {ArcSin}(c x)}{288 c^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 177, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{4} d \,c^{6} x^{4}+\frac {1}{6} e \,c^{6} x^{6}\right )}{c^{2}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d \,c^{6} x^{4}}{4}+\frac {\arcsin \left (c x \right ) e \,c^{6} x^{6}}{6}-\frac {d \,c^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}-\frac {e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{6}\right )}{c^{2}}}{c^{4}}\) | \(177\) |
default | \(\frac {\frac {a \left (\frac {1}{4} d \,c^{6} x^{4}+\frac {1}{6} e \,c^{6} x^{6}\right )}{c^{2}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d \,c^{6} x^{4}}{4}+\frac {\arcsin \left (c x \right ) e \,c^{6} x^{6}}{6}-\frac {d \,c^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}-\frac {e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{6}\right )}{c^{2}}}{c^{4}}\) | \(177\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 165, normalized size = 1.11 \begin {gather*} \frac {1}{6} \, a x^{6} e + \frac {1}{4} \, a d x^{4} + \frac {1}{32} \, {\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 )} b d + \frac {1}{288} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.65, size = 126, normalized size = 0.85 \begin {gather*} \frac {48 \, a c^{6} x^{6} e + 72 \, a c^{6} d x^{4} + 3 \, {\left (24 \, b c^{6} d x^{4} - 9 \, b c^{2} d + {\left (16 \, b c^{6} x^{6} - 5 \, b\right )} e\right )} \arcsin \left (c x\right ) + {\left (18 \, b c^{5} d x^{3} + 27 \, b c^{3} d x + {\left (8 \, b c^{5} x^{5} + 10 \, b c^{3} x^{3} + 15 \, b c x\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{288 \, c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.53, size = 206, normalized size = 1.38 \begin {gather*} \begin {cases} \frac {a d x^{4}}{4} + \frac {a e x^{6}}{6} + \frac {b d x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b e x^{6} \operatorname {asin}{\left (c x \right )}}{6} + \frac {b d x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b e x^{5} \sqrt {- c^{2} x^{2} + 1}}{36 c} + \frac {3 b d x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {5 b e x^{3} \sqrt {- c^{2} x^{2} + 1}}{144 c^{3}} - \frac {3 b d \operatorname {asin}{\left (c x \right )}}{32 c^{4}} + \frac {5 b e x \sqrt {- c^{2} x^{2} + 1}}{96 c^{5}} - \frac {5 b e \operatorname {asin}{\left (c x \right )}}{96 c^{6}} & \text {for}\: c \neq 0 \\a \left (\frac {d x^{4}}{4} + \frac {e x^{6}}{6}\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 = 254, normalized size = 1.70 \begin {gather*} \frac {1}{6} \, a e x^{6} + \frac {1}{4} \, a d x^{4} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d x}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d \arcsin \left (c x\right )}{4 \, c^{4}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b d x}{32 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e x}{36 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b e \arcsin \left (c x\right )}{6 \, c^{6}} - \frac {13 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e x}{144 \, c^{5}} + \frac {5 \, b d \arcsin \left (c x\right )}{32 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b e \arcsin \left (c x\right )}{2 \, c^{6}} + \frac {11 \, \sqrt {-c^{2} x^{2} + 1} b e x}{96 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )} b e \arcsin \left (c x\right )}{2 \, c^{6}} + \frac {11 \, b e \arcsin \left (c x\right )}{96 \, c^{6}} \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 x^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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