Optimal. Leaf size=241 \[ \frac {b \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \sqrt {1-c^2 x^2}}{315 c^9}-\frac {2 b \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^{3/2}}{945 c^9}+\frac {b \left (21 c^4 d^2+90 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^{5/2}}{525 c^9}-\frac {2 b e \left (9 c^2 d+14 e\right ) \left (1-c^2 x^2\right )^{7/2}}{441 c^9}+\frac {b e^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^9}+\frac {1}{5} d^2 x^5 (a+b \text {ArcSin}(c x))+\frac {2}{7} d e x^7 (a+b \text {ArcSin}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {ArcSin}(c x)) \]
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Rubi [A]
time = 0.23, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {276, 4815, 12,
1265, 911, 1167} \begin {gather*} \frac {1}{5} d^2 x^5 (a+b \text {ArcSin}(c x))+\frac {2}{7} d e x^7 (a+b \text {ArcSin}(c x))+\frac {1}{9} e^2 x^9 (a+b \text {ArcSin}(c x))-\frac {2 b e \left (1-c^2 x^2\right )^{7/2} \left (9 c^2 d+14 e\right )}{441 c^9}+\frac {b e^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^9}+\frac {b \left (1-c^2 x^2\right )^{5/2} \left (21 c^4 d^2+90 c^2 d e+70 e^2\right )}{525 c^9}-\frac {2 b \left (1-c^2 x^2\right )^{3/2} \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{945 c^9}+\frac {b \sqrt {1-c^2 x^2} \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{315 c^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 276
Rule 911
Rule 1167
Rule 1265
Rule 4815
Rubi steps
\begin {align*} \int x^4 \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )}{315 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{315} (b c) \int \frac {x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{630} (b c) \text {Subst}\left (\int \frac {x^2 \left (63 d^2+90 d e x+35 e^2 x^2\right )}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {x^2}{c^2}\right )^2 \left (\frac {63 c^4 d^2+90 c^2 d e+35 e^2}{c^4}-\frac {\left (90 c^2 d e+70 e^2\right ) x^2}{c^4}+\frac {35 e^2 x^4}{c^4}\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{315 c}\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \text {Subst}\left (\int \left (\frac {63 c^4 d^2+90 c^2 d e+35 e^2}{c^8}-\frac {2 \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) x^2}{c^8}+\frac {3 \left (21 c^4 d^2+90 c^2 d e+70 e^2\right ) x^4}{c^8}-\frac {10 e \left (9 c^2 d+14 e\right ) x^6}{c^8}+\frac {35 e^2 x^8}{c^8}\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{315 c}\\ &=\frac {b \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \sqrt {1-c^2 x^2}}{315 c^9}-\frac {2 b \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^{3/2}}{945 c^9}+\frac {b \left (21 c^4 d^2+90 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^{5/2}}{525 c^9}-\frac {2 b e \left (9 c^2 d+14 e\right ) \left (1-c^2 x^2\right )^{7/2}}{441 c^9}+\frac {b e^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^9}+\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 187, normalized size = 0.78 \begin {gather*} \frac {315 a x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )+\frac {b \sqrt {1-c^2 x^2} \left (4480 e^2+160 c^2 e \left (81 d+14 e x^2\right )+24 c^4 \left (441 d^2+270 d e x^2+70 e^2 x^4\right )+4 c^6 \left (1323 d^2 x^2+1215 d e x^4+350 e^2 x^6\right )+c^8 \left (3969 d^2 x^4+4050 d e x^6+1225 e^2 x^8\right )\right )}{c^9}+315 b x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right ) \text {ArcSin}(c x)}{99225} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 339, normalized size = 1.41
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{5} d^{2} c^{9} x^{5}+\frac {2}{7} d \,c^{9} e \,x^{7}+\frac {1}{9} e^{2} c^{9} x^{9}\right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d^{2} c^{9} x^{5}}{5}+\frac {2 \arcsin \left (c x \right ) d \,c^{9} e \,x^{7}}{7}+\frac {\arcsin \left (c x \right ) e^{2} c^{9} x^{9}}{9}-\frac {d^{2} c^{4} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-\frac {2 d \,c^{2} e \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}-\frac {e^{2} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )}{9}\right )}{c^{4}}}{c^{5}}\) | \(339\) |
default | \(\frac {\frac {a \left (\frac {1}{5} d^{2} c^{9} x^{5}+\frac {2}{7} d \,c^{9} e \,x^{7}+\frac {1}{9} e^{2} c^{9} x^{9}\right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d^{2} c^{9} x^{5}}{5}+\frac {2 \arcsin \left (c x \right ) d \,c^{9} e \,x^{7}}{7}+\frac {\arcsin \left (c x \right ) e^{2} c^{9} x^{9}}{9}-\frac {d^{2} c^{4} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-\frac {2 d \,c^{2} e \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}-\frac {e^{2} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )}{9}\right )}{c^{4}}}{c^{5}}\) | \(339\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 314, normalized size = 1.30 \begin {gather*} \frac {1}{9} \, a x^{9} e^{2} + \frac {2}{7} \, a d x^{7} e + \frac {1}{5} \, a d^{2} x^{5} + \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^{2} + \frac {2}{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 d e + \frac {1}{2835} \, {\left (315 \, x^{9} \arcsin \left (c x\right ) + {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b e^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.50, size = 216, normalized size = 0.90 \begin {gather*} \frac {11025 \, a c^{9} x^{9} e^{2} + 28350 \, a c^{9} d x^{7} e + 19845 \, a c^{9} d^{2} x^{5} + 315 \, {\left (35 \, b c^{9} x^{9} e^{2} + 90 \, b c^{9} d x^{7} e + 63 \, b c^{9} d^{2} x^{5}\right )} \arcsin \left (c x\right ) + {\left (3969 \, b c^{8} d^{2} x^{4} + 5292 \, b c^{6} d^{2} x^{2} + 10584 \, b c^{4} d^{2} + 35 \, {\left (35 \, b c^{8} x^{8} + 40 \, b c^{6} x^{6} + 48 \, b c^{4} x^{4} + 64 \, b c^{2} x^{2} + 128 \, b\right )} e^{2} + 810 \, {\left (5 \, b c^{8} d x^{6} + 6 \, b c^{6} d x^{4} + 8 \, b c^{4} d x^{2} + 16 \, b c^{2} d\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{99225 \, c^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.74, size = 415, normalized size = 1.72 \begin {gather*} \begin {cases} \frac {a d^{2} x^{5}}{5} + \frac {2 a d e x^{7}}{7} + \frac {a e^{2} x^{9}}{9} + \frac {b d^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {2 b d e x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {b e^{2} x^{9} \operatorname {asin}{\left (c x \right )}}{9} + \frac {b d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {2 b d e x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} + \frac {b e^{2} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81 c} + \frac {4 b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {12 b d e x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} + \frac {8 b e^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{567 c^{3}} + \frac {8 b d^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} + \frac {16 b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} + \frac {16 b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{945 c^{5}} + \frac {32 b d e \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} + \frac {64 b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{2835 c^{7}} + \frac {128 b e^{2} \sqrt {- c^{2} x^{2} + 1}}{2835 c^{9}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{2} x^{5}}{5} + \frac {2 d e x^{7}}{7} + \frac {e^{2} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 598 vs.
\(2 (215) = 430\).
time = 0.42, size = 598, normalized size = 2.48 \begin {gather*} \frac {1}{9} \, a e^{2} x^{9} + \frac {2}{7} \, a d e x^{7} + \frac {1}{5} \, a d^{2} x^{5} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b d^{2} x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} b d e x \arcsin \left (c x\right )}{7 \, c^{6}} + \frac {b d^{2} x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {6 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d e x \arcsin \left (c x\right )}{7 \, c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b e^{2} x \arcsin \left (c x\right )}{9 \, c^{8}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{2}}{25 \, c^{5}} + \frac {6 \, {\left (c^{2} x^{2} - 1\right )} b d e x \arcsin \left (c x\right )}{7 \, c^{6}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{3} b e^{2} x \arcsin \left (c x\right )}{9 \, c^{8}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2}}{15 \, c^{5}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d e}{49 \, c^{7}} + \frac {2 \, b d e x \arcsin \left (c x\right )}{7 \, c^{6}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b e^{2} x \arcsin \left (c x\right )}{3 \, c^{8}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2}}{5 \, c^{5}} + \frac {6 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d e}{35 \, c^{7}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b e^{2}}{81 \, c^{9}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )} b e^{2} x \arcsin \left (c x\right )}{9 \, c^{8}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e}{7 \, c^{7}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b e^{2}}{63 \, c^{9}} + \frac {b e^{2} x \arcsin \left (c x\right )}{9 \, c^{8}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d e}{7 \, c^{7}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e^{2}}{15 \, c^{9}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2}}{27 \, c^{9}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e^{2}}{9 \, c^{9}} \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 = -0.00 \begin {gather*} \int x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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