Optimal. Leaf size=262 \[ \frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{24 c^2}+\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c^5 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^4}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {1-c^2 x^2}}+\frac {b x^2 \sqrt {d-c^2 d x^2}}{32 c^3 \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.28, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {4697, 4707, 4641, 30} \[ \frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{24 c^2}-\frac {x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^4}+\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c^5 \sqrt {1-c^2 x^2}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {1-c^2 x^2}}+\frac {b x^2 \sqrt {d-c^2 d x^2}}{32 c^3 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 4641
Rule 4697
Rule 4707
Rubi steps
\begin {align*} \int x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\sqrt {d-c^2 d x^2} \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{6 \sqrt {1-c^2 x^2}}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int x^5 \, dx}{6 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\sqrt {d-c^2 d x^2} \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{8 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{24 c \sqrt {1-c^2 x^2}}\\ &=\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {1-c^2 x^2}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^4}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\sqrt {d-c^2 d x^2} \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 c^4 \sqrt {1-c^2 x^2}}+\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 c^3 \sqrt {1-c^2 x^2}}\\ &=\frac {b x^2 \sqrt {d-c^2 d x^2}}{32 c^3 \sqrt {1-c^2 x^2}}+\frac {b x^4 \sqrt {d-c^2 d x^2}}{96 c \sqrt {1-c^2 x^2}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}-\frac {x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^4}-\frac {x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{24 c^2}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c^5 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 169, normalized size = 0.65 \[ \frac {\sqrt {d-c^2 d x^2} \left (9 a^2+6 a b c x \sqrt {1-c^2 x^2} \left (8 c^4 x^4-2 c^2 x^2-3\right )+6 b \sin ^{-1}(c x) \left (3 a+b c x \sqrt {1-c^2 x^2} \left (8 c^4 x^4-2 c^2 x^2-3\right )\right )+b^2 c^2 x^2 \left (-8 c^4 x^4+3 c^2 x^2+9\right )+9 b^2 \sin ^{-1}(c x)^2\right )}{288 b c^5 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{4} \arcsin \left (c x\right ) + a x^{4}\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.71, size = 1690, normalized size = 6.45 \[ \text {result too large to display} \]
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 \[ b \sqrt {d} \int \sqrt {c x + 1} \sqrt {-c x + 1} x^{4} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\,{d x} - \frac {1}{48} \, {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{3}}{c^{2} d} - \frac {3 \, \sqrt {-c^{2} d x^{2} + d} x}{c^{4}} + \frac {6 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{4} d} - \frac {3 \, \sqrt {d} \arcsin \left (c x\right )}{c^{5}}\right )} a \]
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 \[ \int x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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