Optimal. Leaf size=149 \[ \frac {1}{4} d x^4 \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cos ^{-1}(c x)\right )-\frac {b e x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b \left (9 c^2 d+5 e\right ) \sin ^{-1}(c x)}{96 c^6}-\frac {b x \sqrt {1-c^2 x^2} \left (9 c^2 d+5 e\right )}{96 c^5}-\frac {b x^3 \sqrt {1-c^2 x^2} \left (9 c^2 d+5 e\right )}{144 c^3} \]
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Rubi [A] time = 0.12, 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, 4732, 12, 459, 321, 216} \[ \frac {1}{4} d x^4 \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cos ^{-1}(c x)\right )-\frac {b x^3 \sqrt {1-c^2 x^2} \left (9 c^2 d+5 e\right )}{144 c^3}-\frac {b x \sqrt {1-c^2 x^2} \left (9 c^2 d+5 e\right )}{96 c^5}+\frac {b \left (9 c^2 d+5 e\right ) \sin ^{-1}(c x)}{96 c^6}-\frac {b e x^5 \sqrt {1-c^2 x^2}}{36 c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 216
Rule 321
Rule 459
Rule 4732
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right ) \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d x^4 \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cos ^{-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 \cos ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cos ^{-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 \cos ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cos ^{-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 \cos ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cos ^{-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 \cos ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cos ^{-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 {1}{4} d x^4 \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{6} e x^6 \left (a+b \cos ^{-1}(c x)\right )+\frac {b \left (9 c^2 d+5 e\right ) \sin ^{-1}(c x)}{96 c^6}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 153, normalized size = 1.03 \[ \frac {1}{4} a d x^4+\frac {1}{6} a e x^6+\frac {5 b e \sin ^{-1}(c x)}{96 c^6}+\frac {3 b d \sin ^{-1}(c x)}{32 c^4}+b d \sqrt {1-c^2 x^2} \left (-\frac {3 x}{32 c^3}-\frac {x^3}{16 c}\right )+b e \sqrt {1-c^2 x^2} \left (-\frac {5 x}{96 c^5}-\frac {5 x^3}{144 c^3}-\frac {x^5}{36 c}\right )+\frac {1}{4} b d x^4 \cos ^{-1}(c x)+\frac {1}{6} b e x^6 \cos ^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 125, normalized size = 0.84 \[ \frac {48 \, a c^{6} e x^{6} + 72 \, a c^{6} d x^{4} + 3 \, {\left (16 \, b c^{6} e x^{6} + 24 \, b c^{6} d x^{4} - 9 \, b c^{2} d - 5 \, b e\right )} \arccos \left (c x\right ) - {\left (8 \, b c^{5} e x^{5} + 2 \, {\left (9 \, b c^{5} d + 5 \, b c^{3} e\right )} x^{3} + 3 \, {\left (9 \, b c^{3} d + 5 \, b c e\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{288 \, c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 171, normalized size = 1.15 \[ \frac {1}{6} \, b x^{6} \arccos \left (c x\right ) e + \frac {1}{6} \, a x^{6} e + \frac {1}{4} \, b d x^{4} \arccos \left (c x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} b x^{5} e}{36 \, c} + \frac {1}{4} \, a d x^{4} - \frac {\sqrt {-c^{2} x^{2} + 1} b d x^{3}}{16 \, c} - \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b x^{3} e}{144 \, c^{3}} - \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b d x}{32 \, c^{3}} - \frac {3 \, b d \arccos \left (c x\right )}{32 \, c^{4}} - \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b x e}{96 \, c^{5}} - \frac {5 \, b \arccos \left (c x\right ) e}{96 \, c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 177, normalized size = 1.19 \[ \frac {\frac {a \left (\frac {1}{6} e \,c^{6} x^{6}+\frac {1}{4} x^{4} c^{6} d \right )}{c^{2}}+\frac {b \left (\frac {\arccos \left (c x \right ) e \,c^{6} x^{6}}{6}+\frac {\arccos \left (c x \right ) c^{6} x^{4} d}{4}+\frac {e \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{6}+\frac {c^{2} d \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}\right )}{c^{2}}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 165, normalized size = 1.11 \[ \frac {1}{6} \, a e x^{6} + \frac {1}{4} \, a d x^{4} + \frac {1}{32} \, {\left (8 \, x^{4} \arccos \left (c x\right ) - {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d + \frac {1}{288} \, {\left (48 \, x^{6} \arccos \left (c x\right ) - {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b e \]
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 \[ \int x^3\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.53, size = 211, normalized size = 1.42 \[ \begin {cases} \frac {a d x^{4}}{4} + \frac {a e x^{6}}{6} + \frac {b d x^{4} \operatorname {acos}{\left (c x \right )}}{4} + \frac {b e x^{6} \operatorname {acos}{\left (c x \right )}}{6} - \frac {b d x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} - \frac {b e x^{5} \sqrt {- c^{2} x^{2} + 1}}{36 c} - \frac {3 b d x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} - \frac {5 b e x^{3} \sqrt {- c^{2} x^{2} + 1}}{144 c^{3}} - \frac {3 b d \operatorname {acos}{\left (c x \right )}}{32 c^{4}} - \frac {5 b e x \sqrt {- c^{2} x^{2} + 1}}{96 c^{5}} - \frac {5 b e \operatorname {acos}{\left (c x \right )}}{96 c^{6}} & \text {for}\: c \neq 0 \\\left (a + \frac {\pi b}{2}\right ) \left (\frac {d x^{4}}{4} + \frac {e x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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