Integrand size = 27, antiderivative size = 430 \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {1-c^2 x^2}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {1-c^2 x^2}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {1-c^2 x^2}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {3 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{512 b c^5 \sqrt {1-c^2 x^2}} \]
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Time = 0.35 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {4787, 4783, 4795, 4737, 30, 14, 272, 45} \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{128 c^2}+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{512 b c^5 \sqrt {1-c^2 x^2}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{256 c^4}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {1-c^2 x^2}}+\frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {1-c^2 x^2}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {1-c^2 x^2}} \]
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Rule 14
Rule 30
Rule 45
Rule 272
Rule 4737
Rule 4783
Rule 4787
Rule 4795
Rubi steps \begin{align*} \text {integral}& = \frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {1}{2} d \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x)) \, dx-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \left (1-c^2 x^2\right )^2 \, dx}{10 \sqrt {1-c^2 x^2}} \\ & = \frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {1}{16} \left (3 d^2\right ) \int x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \, dx-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int x^2 \left (1-c^2 x\right )^2 \, dx,x,x^2\right )}{20 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \left (1-c^2 x^2\right ) \, dx}{16 \sqrt {1-c^2 x^2}} \\ & = \frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{32 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \, dx}{32 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (x^2-2 c^2 x^3+c^4 x^4\right ) \, dx,x,x^2\right )}{20 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (x^5-c^2 x^7\right ) \, dx}{16 \sqrt {1-c^2 x^2}} \\ & = -\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {1-c^2 x^2}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {1-c^2 x^2}}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {\left (3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{128 c \sqrt {1-c^2 x^2}} \\ & = \frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {1-c^2 x^2}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {1-c^2 x^2}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {1-c^2 x^2}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {\left (3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}} \, dx}{256 c^4 \sqrt {1-c^2 x^2}}+\frac {\left (3 b d^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{256 c^3 \sqrt {1-c^2 x^2}} \\ & = \frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {1-c^2 x^2}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {1-c^2 x^2}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {1-c^2 x^2}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {3 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{512 b c^5 \sqrt {1-c^2 x^2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.51 \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (225 a^2+b^2 c^2 x^2 \left (225+75 c^2 x^2-1240 c^4 x^4+1260 c^6 x^6-384 c^8 x^8\right )+30 a b c x \sqrt {1-c^2 x^2} \left (-15-10 c^2 x^2+248 c^4 x^4-336 c^6 x^6+128 c^8 x^8\right )+30 b \left (15 a+b c x \sqrt {1-c^2 x^2} \left (-15-10 c^2 x^2+248 c^4 x^4-336 c^6 x^6+128 c^8 x^8\right )\right ) \arcsin (c x)+225 b^2 \arcsin (c x)^2\right )}{38400 b c^5 \sqrt {1-c^2 x^2}} \]
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Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 1106, normalized size of antiderivative = 2.57
method | result | size |
default | \(\text {Expression too large to display}\) | \(1106\) |
parts | \(\text {Expression too large to display}\) | \(1106\) |
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\[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{4} \,d x } \]
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Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \]
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\[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{4} \,d x } \]
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\[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{4} \,d x } \]
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Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
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