Optimal. Leaf size=258 \[ \frac {5 b \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x \sqrt {1-c^2 x^2}}{3072 c^7}+\frac {b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{1536 c^5}+\frac {7 b \left (2 c^2 d+e\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{384 c^3}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}-\frac {b \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right ) \text {ArcSin}(c x)}{1024 c^8 e}+\frac {\left (d+e x^2\right )^4 (a+b \text {ArcSin}(c x))}{8 e} \]
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Rubi [A]
time = 0.20, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4813, 427, 542,
396, 222} \begin {gather*} \frac {\left (d+e x^2\right )^4 (a+b \text {ArcSin}(c x))}{8 e}-\frac {b \text {ArcSin}(c x) \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right )}{1024 c^8 e}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}+\frac {7 b x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{384 c^3}+\frac {5 b x \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )}{3072 c^7}+\frac {b x \sqrt {1-c^2 x^2} \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{1536 c^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 396
Rule 427
Rule 542
Rule 4813
Rubi steps
\begin {align*} \int x \left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}-\frac {(b c) \int \frac {\left (d+e x^2\right )^4}{\sqrt {1-c^2 x^2}} \, dx}{8 e}\\ &=\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}+\frac {\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}+\frac {b \int \frac {\left (d+e x^2\right )^2 \left (-d \left (8 c^2 d+e\right )-7 e \left (2 c^2 d+e\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{64 c e}\\ &=\frac {7 b \left (2 c^2 d+e\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{384 c^3}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}+\frac {\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}-\frac {b \int \frac {\left (d+e x^2\right ) \left (d \left (48 c^4 d^2+20 c^2 d e+7 e^2\right )+e \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{384 c^3 e}\\ &=\frac {b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{1536 c^5}+\frac {7 b \left (2 c^2 d+e\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{384 c^3}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}+\frac {\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}+\frac {b \int \frac {-d \left (192 c^6 d^3+184 c^4 d^2 e+132 c^2 d e^2+35 e^3\right )-5 e \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x^2}{\sqrt {1-c^2 x^2}} \, dx}{1536 c^5 e}\\ &=\frac {5 b \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x \sqrt {1-c^2 x^2}}{3072 c^7}+\frac {b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{1536 c^5}+\frac {7 b \left (2 c^2 d+e\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{384 c^3}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}+\frac {\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}-\frac {\left (b \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{1024 c^7 e}\\ &=\frac {5 b \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x \sqrt {1-c^2 x^2}}{3072 c^7}+\frac {b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{1536 c^5}+\frac {7 b \left (2 c^2 d+e\right ) x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2}{384 c^3}+\frac {b x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}-\frac {b \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right ) \sin ^{-1}(c x)}{1024 c^8 e}+\frac {\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 232, normalized size = 0.90 \begin {gather*} \frac {c x \left (384 a c^7 x \left (4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right )+b \sqrt {1-c^2 x^2} \left (105 e^3+10 c^2 e^2 \left (48 d+7 e x^2\right )+8 c^4 e \left (108 d^2+40 d e x^2+7 e^2 x^4\right )+16 c^6 \left (48 d^3+36 d^2 e x^2+16 d e^2 x^4+3 e^3 x^6\right )\right )\right )+3 b \left (-256 c^6 d^3-288 c^4 d^2 e-160 c^2 d e^2-35 e^3+128 c^8 \left (4 d^3 x^2+6 d^2 e x^4+4 d e^2 x^6+e^3 x^8\right )\right ) \text {ArcSin}(c x)}{3072 c^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 376, normalized size = 1.46
method | result | size |
derivativedivides | \(\frac {\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{4} a}{8 c^{6} e}+\frac {b \left (\frac {\arcsin \left (c x \right ) c^{8} d^{4}}{8 e}+\frac {\arcsin \left (c x \right ) c^{8} d^{3} x^{2}}{2}+\frac {3 e \arcsin \left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e^{2} \arcsin \left (c x \right ) c^{8} d \,x^{6}}{2}+\frac {e^{3} \arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {c^{8} d^{4} \arcsin \left (c x \right )+4 c^{6} d^{3} e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+6 c^{4} d^{2} e^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+4 d \,c^{2} e^{3} \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )+e^{4} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )}{8 e}\right )}{c^{6}}}{c^{2}}\) | \(376\) |
default | \(\frac {\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{4} a}{8 c^{6} e}+\frac {b \left (\frac {\arcsin \left (c x \right ) c^{8} d^{4}}{8 e}+\frac {\arcsin \left (c x \right ) c^{8} d^{3} x^{2}}{2}+\frac {3 e \arcsin \left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e^{2} \arcsin \left (c x \right ) c^{8} d \,x^{6}}{2}+\frac {e^{3} \arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {c^{8} d^{4} \arcsin \left (c x \right )+4 c^{6} d^{3} e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+6 c^{4} d^{2} e^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+4 d \,c^{2} e^{3} \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )+e^{4} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )}{8 e}\right )}{c^{6}}}{c^{2}}\) | \(376\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 342, normalized size = 1.33 \begin {gather*} \frac {1}{8} \, a x^{8} e^{3} + \frac {1}{2} \, a d x^{6} e^{2} + \frac {3}{4} \, a d^{2} x^{4} e + \frac {1}{2} \, a d^{3} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{3} + \frac {3}{32} \, {\left (8 \, x^{4} \arcsin \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^{2} e + \frac {1}{96} \, {\left (48 \, x^{6} \arcsin \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 d e^{2} + \frac {1}{3072} \, {\left (384 \, x^{8} \arcsin \left (c x\right ) + {\left (\frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac {56 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac {105 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{8}} - \frac {105 \, \arcsin \left (c x\right )}{c^{9}}\right )} c\right )} b e^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.80, size = 267, normalized size = 1.03 \begin {gather*} \frac {384 \, a c^{8} x^{8} e^{3} + 1536 \, a c^{8} d x^{6} e^{2} + 2304 \, a c^{8} d^{2} x^{4} e + 1536 \, a c^{8} d^{3} x^{2} + 3 \, {\left (512 \, b c^{8} d^{3} x^{2} - 256 \, b c^{6} d^{3} + {\left (128 \, b c^{8} x^{8} - 35 \, b\right )} e^{3} + 32 \, {\left (16 \, b c^{8} d x^{6} - 5 \, b c^{2} d\right )} e^{2} + 96 \, {\left (8 \, b c^{8} d^{2} x^{4} - 3 \, b c^{4} d^{2}\right )} e\right )} \arcsin \left (c x\right ) + {\left (768 \, b c^{7} d^{3} x + {\left (48 \, b c^{7} x^{7} + 56 \, b c^{5} x^{5} + 70 \, b c^{3} x^{3} + 105 \, b c x\right )} e^{3} + 32 \, {\left (8 \, b c^{7} d x^{5} + 10 \, b c^{5} d x^{3} + 15 \, b c^{3} d x\right )} e^{2} + 288 \, {\left (2 \, b c^{7} d^{2} x^{3} + 3 \, b c^{5} d^{2} x\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{3072 \, c^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.20, size = 483, normalized size = 1.87 \begin {gather*} \begin {cases} \frac {a d^{3} x^{2}}{2} + \frac {3 a d^{2} e x^{4}}{4} + \frac {a d e^{2} x^{6}}{2} + \frac {a e^{3} x^{8}}{8} + \frac {b d^{3} x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {3 b d^{2} e x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \frac {b d e^{2} x^{6} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e^{3} x^{8} \operatorname {asin}{\left (c x \right )}}{8} + \frac {b d^{3} x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {3 b d^{2} e x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 c} + \frac {b d e^{2} x^{5} \sqrt {- c^{2} x^{2} + 1}}{12 c} + \frac {b e^{3} x^{7} \sqrt {- c^{2} x^{2} + 1}}{64 c} - \frac {b d^{3} \operatorname {asin}{\left (c x \right )}}{4 c^{2}} + \frac {9 b d^{2} e x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} + \frac {5 b d e^{2} x^{3} \sqrt {- c^{2} x^{2} + 1}}{48 c^{3}} + \frac {7 b e^{3} x^{5} \sqrt {- c^{2} x^{2} + 1}}{384 c^{3}} - \frac {9 b d^{2} e \operatorname {asin}{\left (c x \right )}}{32 c^{4}} + \frac {5 b d e^{2} x \sqrt {- c^{2} x^{2} + 1}}{32 c^{5}} + \frac {35 b e^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{1536 c^{5}} - \frac {5 b d e^{2} \operatorname {asin}{\left (c x \right )}}{32 c^{6}} + \frac {35 b e^{3} x \sqrt {- c^{2} x^{2} + 1}}{1024 c^{7}} - \frac {35 b e^{3} \operatorname {asin}{\left (c x \right )}}{1024 c^{8}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{3} x^{2}}{2} + \frac {3 d^{2} e x^{4}}{4} + \frac {d e^{2} x^{6}}{2} + \frac {e^{3} x^{8}}{8}\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. 597 vs.
\(2 (238) = 476\).
time = 0.42, size = 597, normalized size = 2.31 \begin {gather*} \frac {1}{8} \, a e^{3} x^{8} + \frac {1}{2} \, a d e^{2} x^{6} + \frac {3}{4} \, a d^{2} e x^{4} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{3} x}{4 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{3} \arcsin \left (c x\right )}{2 \, c^{2}} - \frac {3 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} e x}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d^{3}}{2 \, c^{2}} + \frac {b d^{3} \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} e \arcsin \left (c x\right )}{4 \, c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} e x}{32 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d e^{2} x}{12 \, c^{5}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b d^{2} e \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b d e^{2} \arcsin \left (c x\right )}{2 \, c^{6}} - \frac {13 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e^{2} x}{48 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b e^{3} x}{64 \, c^{7}} + \frac {15 \, b d^{2} e \arcsin \left (c x\right )}{32 \, c^{4}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d e^{2} \arcsin \left (c x\right )}{2 \, c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b e^{3} \arcsin \left (c x\right )}{8 \, c^{8}} + \frac {11 \, \sqrt {-c^{2} x^{2} + 1} b d e^{2} x}{32 \, c^{5}} + \frac {25 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e^{3} x}{384 \, c^{7}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b d e^{2} \arcsin \left (c x\right )}{2 \, c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b e^{3} \arcsin \left (c x\right )}{2 \, c^{8}} - \frac {163 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{3} x}{1536 \, c^{7}} + \frac {11 \, b d e^{2} \arcsin \left (c x\right )}{32 \, c^{6}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b e^{3} \arcsin \left (c x\right )}{4 \, c^{8}} + \frac {93 \, \sqrt {-c^{2} x^{2} + 1} b e^{3} x}{1024 \, c^{7}} + \frac {{\left (c^{2} x^{2} - 1\right )} b e^{3} \arcsin \left (c x\right )}{2 \, c^{8}} + \frac {93 \, b e^{3} \arcsin \left (c x\right )}{1024 \, c^{8}} \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\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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