Optimal. Leaf size=150 \[ \frac {35 b d^3 x \sqrt {1-c^2 x^2}}{1024 c}+\frac {35 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{1536 c}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{384 c}+\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}+\frac {35 b d^3 \text {ArcSin}(c x)}{1024 c^2}-\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \text {ArcSin}(c x))}{8 c^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4767, 201, 222}
\begin {gather*} -\frac {d^3 \left (1-c^2 x^2\right )^4 (a+b \text {ArcSin}(c x))}{8 c^2}+\frac {35 b d^3 \text {ArcSin}(c x)}{1024 c^2}+\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{384 c}+\frac {35 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{1536 c}+\frac {35 b d^3 x \sqrt {1-c^2 x^2}}{1024 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 4767
Rubi steps
\begin {align*} \int x \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac {\left (b d^3\right ) \int \left (1-c^2 x^2\right )^{7/2} \, dx}{8 c}\\ &=\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac {\left (7 b d^3\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx}{64 c}\\ &=\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{384 c}+\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac {\left (35 b d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{384 c}\\ &=\frac {35 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{1536 c}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{384 c}+\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac {\left (35 b d^3\right ) \int \sqrt {1-c^2 x^2} \, dx}{512 c}\\ &=\frac {35 b d^3 x \sqrt {1-c^2 x^2}}{1024 c}+\frac {35 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{1536 c}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{384 c}+\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac {\left (35 b d^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{1024 c}\\ &=\frac {35 b d^3 x \sqrt {1-c^2 x^2}}{1024 c}+\frac {35 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{1536 c}+\frac {7 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{384 c}+\frac {b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}+\frac {35 b d^3 \sin ^{-1}(c x)}{1024 c^2}-\frac {d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 110, normalized size = 0.73 \begin {gather*} -\frac {d^3 \left (384 a \left (-1+c^2 x^2\right )^4+b c x \sqrt {1-c^2 x^2} \left (-279+326 c^2 x^2-200 c^4 x^4+48 c^6 x^6\right )+3 b \left (93-512 c^2 x^2+768 c^4 x^4-512 c^6 x^6+128 c^8 x^8\right ) \text {ArcSin}(c x)\right )}{3072 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 160, normalized size = 1.07
method | result | size |
derivativedivides | \(\frac {-\frac {d^{3} \left (c^{2} x^{2}-1\right )^{4} a}{8}-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \arcsin \left (c x \right )}{4}-\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {93 \arcsin \left (c x \right )}{1024}+\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}-\frac {25 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{384}+\frac {163 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{1536}-\frac {93 c x \sqrt {-c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\) | \(160\) |
default | \(\frac {-\frac {d^{3} \left (c^{2} x^{2}-1\right )^{4} a}{8}-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {\arcsin \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 c^{4} x^{4} \arcsin \left (c x \right )}{4}-\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {93 \arcsin \left (c x \right )}{1024}+\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{64}-\frac {25 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{384}+\frac {163 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{1536}-\frac {93 c x \sqrt {-c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 358 vs.
\(2 (129) = 258\).
time = 0.48, size = 358, normalized size = 2.39 \begin {gather*} -\frac {1}{8} \, a c^{6} d^{3} x^{8} + \frac {1}{2} \, a c^{4} d^{3} x^{6} - \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 c^{6} d^{3} - \frac {3}{4} \, a c^{2} d^{3} x^{4} + \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 c^{4} 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 c^{2} d^{3} + \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.31, size = 173, normalized size = 1.15 \begin {gather*} -\frac {384 \, a c^{8} d^{3} x^{8} - 1536 \, a c^{6} d^{3} x^{6} + 2304 \, a c^{4} d^{3} x^{4} - 1536 \, a c^{2} d^{3} x^{2} + 3 \, {\left (128 \, b c^{8} d^{3} x^{8} - 512 \, b c^{6} d^{3} x^{6} + 768 \, b c^{4} d^{3} x^{4} - 512 \, b c^{2} d^{3} x^{2} + 93 \, b d^{3}\right )} \arcsin \left (c x\right ) + {\left (48 \, b c^{7} d^{3} x^{7} - 200 \, b c^{5} d^{3} x^{5} + 326 \, b c^{3} d^{3} x^{3} - 279 \, b c d^{3} x\right )} \sqrt {-c^{2} x^{2} + 1}}{3072 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.22, size = 253, normalized size = 1.69 \begin {gather*} \begin {cases} - \frac {a c^{6} d^{3} x^{8}}{8} + \frac {a c^{4} d^{3} x^{6}}{2} - \frac {3 a c^{2} d^{3} x^{4}}{4} + \frac {a d^{3} x^{2}}{2} - \frac {b c^{6} d^{3} x^{8} \operatorname {asin}{\left (c x \right )}}{8} - \frac {b c^{5} d^{3} x^{7} \sqrt {- c^{2} x^{2} + 1}}{64} + \frac {b c^{4} d^{3} x^{6} \operatorname {asin}{\left (c x \right )}}{2} + \frac {25 b c^{3} d^{3} x^{5} \sqrt {- c^{2} x^{2} + 1}}{384} - \frac {3 b c^{2} d^{3} x^{4} \operatorname {asin}{\left (c x \right )}}{4} - \frac {163 b c d^{3} x^{3} \sqrt {- c^{2} x^{2} + 1}}{1536} + \frac {b d^{3} x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {93 b d^{3} x \sqrt {- c^{2} x^{2} + 1}}{1024 c} - \frac {93 b d^{3} \operatorname {asin}{\left (c x \right )}}{1024 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{2}}{2} & \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 = 202, normalized size = 1.35 \begin {gather*} -\frac {1}{8} \, a c^{6} d^{3} x^{8} + \frac {1}{2} \, a c^{4} d^{3} x^{6} - \frac {3}{4} \, a c^{2} d^{3} x^{4} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d^{3} x}{64 \, c} - \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b d^{3} \arcsin \left (c x\right )}{8 \, c^{2}} + \frac {7 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{3} x}{384 \, c} + \frac {35 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{3} x}{1536 \, c} + \frac {35 \, \sqrt {-c^{2} x^{2} + 1} b d^{3} x}{1024 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} a d^{3}}{2 \, c^{2}} + \frac {35 \, b d^{3} \arcsin \left (c x\right )}{1024 \, c^{2}} \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\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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