Optimal. Leaf size=128 \[ \frac {2 b d \sqrt {1-c^2 x^2}}{35 c^5}+\frac {b d \left (1-c^2 x^2\right )^{3/2}}{105 c^5}-\frac {8 b d \left (1-c^2 x^2\right )^{5/2}}{175 c^5}+\frac {b d \left (1-c^2 x^2\right )^{7/2}}{49 c^5}+\frac {1}{5} d x^5 (a+b \text {ArcSin}(c x))-\frac {1}{7} c^2 d x^7 (a+b \text {ArcSin}(c x)) \]
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Rubi [A]
time = 0.08, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {14, 4777, 12,
457, 78} \begin {gather*} -\frac {1}{7} c^2 d x^7 (a+b \text {ArcSin}(c x))+\frac {1}{5} d x^5 (a+b \text {ArcSin}(c x))+\frac {b d \left (1-c^2 x^2\right )^{7/2}}{49 c^5}-\frac {8 b d \left (1-c^2 x^2\right )^{5/2}}{175 c^5}+\frac {b d \left (1-c^2 x^2\right )^{3/2}}{105 c^5}+\frac {2 b d \sqrt {1-c^2 x^2}}{35 c^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 78
Rule 457
Rule 4777
Rubi steps
\begin {align*} \int x^4 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{7} c^2 d x^7 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {d x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{5} d x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{7} c^2 d x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{35} (b c d) \int \frac {x^5 \left (7-5 c^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{5} d x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{7} c^2 d x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{70} (b c d) \text {Subst}\left (\int \frac {x^2 \left (7-5 c^2 x\right )}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{5} d x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{7} c^2 d x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{70} (b c d) \text {Subst}\left (\int \left (\frac {2}{c^4 \sqrt {1-c^2 x}}+\frac {\sqrt {1-c^2 x}}{c^4}-\frac {8 \left (1-c^2 x\right )^{3/2}}{c^4}+\frac {5 \left (1-c^2 x\right )^{5/2}}{c^4}\right ) \, dx,x,x^2\right )\\ &=\frac {2 b d \sqrt {1-c^2 x^2}}{35 c^5}+\frac {b d \left (1-c^2 x^2\right )^{3/2}}{105 c^5}-\frac {8 b d \left (1-c^2 x^2\right )^{5/2}}{175 c^5}+\frac {b d \left (1-c^2 x^2\right )^{7/2}}{49 c^5}+\frac {1}{5} d x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{7} c^2 d x^7 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 87, normalized size = 0.68 \begin {gather*} \frac {d \left (-105 a x^5 \left (-7+5 c^2 x^2\right )+\frac {b \sqrt {1-c^2 x^2} \left (152+76 c^2 x^2+57 c^4 x^4-75 c^6 x^6\right )}{c^5}-105 b x^5 \left (-7+5 c^2 x^2\right ) \text {ArcSin}(c x)\right )}{3675} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 130, normalized size = 1.02
method | result | size |
derivativedivides | \(\frac {-d a \left (\frac {1}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {76 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {-c^{2} x^{2}+1}}{3675}\right )}{c^{5}}\) | \(130\) |
default | \(\frac {-d a \left (\frac {1}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {76 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {-c^{2} x^{2}+1}}{3675}\right )}{c^{5}}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 189, normalized size = 1.48 \begin {gather*} -\frac {1}{7} \, a c^{2} d x^{7} + \frac {1}{5} \, a d x^{5} - \frac {1}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{2} d + \frac {1}{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 )} b d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.88, size = 101, normalized size = 0.79 \begin {gather*} -\frac {525 \, a c^{7} d x^{7} - 735 \, a c^{5} d x^{5} + 105 \, {\left (5 \, b c^{7} d x^{7} - 7 \, b c^{5} d x^{5}\right )} \arcsin \left (c x\right ) + {\left (75 \, b c^{6} d x^{6} - 57 \, b c^{4} d x^{4} - 76 \, b c^{2} d x^{2} - 152 \, b d\right )} \sqrt {-c^{2} x^{2} + 1}}{3675 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.74, size = 151, normalized size = 1.18 \begin {gather*} \begin {cases} - \frac {a c^{2} d x^{7}}{7} + \frac {a d x^{5}}{5} - \frac {b c^{2} d x^{7} \operatorname {asin}{\left (c x \right )}}{7} - \frac {b c d x^{6} \sqrt {- c^{2} x^{2} + 1}}{49} + \frac {b d x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {19 b d x^{4} \sqrt {- c^{2} x^{2} + 1}}{1225 c} + \frac {76 b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{3675 c^{3}} + \frac {152 b d \sqrt {- c^{2} x^{2} + 1}}{3675 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d x^{5}}{5} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 195, normalized size = 1.52 \begin {gather*} -\frac {1}{7} \, a c^{2} d x^{7} + \frac {1}{5} \, a d x^{5} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b d x \arcsin \left (c x\right )}{7 \, c^{4}} - \frac {8 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d x \arcsin \left (c x\right )}{35 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )} b d x \arcsin \left (c x\right )}{35 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d}{49 \, c^{5}} + \frac {2 \, b d x \arcsin \left (c x\right )}{35 \, c^{4}} - \frac {8 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d}{175 \, c^{5}} + \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d}{105 \, c^{5}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d}{35 \, 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 = -0.01 \begin {gather*} \int x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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