Optimal. Leaf size=187 \[ -\frac {b c \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {1-c^2 x^2}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{30 x^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{5 d x^5}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{15 d x^3}-\frac {2 b c^5 \sqrt {d-c^2 d x^2} \log (x)}{15 \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {277, 270, 4779,
12, 14} \begin {gather*} -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{5 d x^5}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{15 d x^3}-\frac {b c \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {1-c^2 x^2}}-\frac {2 b c^5 \log (x) \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{30 x^2 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 270
Rule 277
Rule 4779
Rubi steps
\begin {align*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^6} \, dx &=-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-3+c^2 x^2+2 c^4 x^4}{15 x^5} \, dx}{\sqrt {1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int \frac {\sqrt {d-c^2 d x^2}}{x^6} \, dx\\ &=-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 d x^5}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-3+c^2 x^2+2 c^4 x^4}{x^5} \, dx}{15 \sqrt {1-c^2 x^2}}+\frac {1}{5} \left (2 c^2 \left (a+b \sin ^{-1}(c x)\right )\right ) \int \frac {\sqrt {d-c^2 d x^2}}{x^4} \, dx\\ &=-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 d x^5}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 d x^3}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {3}{x^5}+\frac {c^2}{x^3}+\frac {2 c^4}{x}\right ) \, dx}{15 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {1-c^2 x^2}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{30 x^2 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 d x^5}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 d x^3}-\frac {2 b c^5 \sqrt {d-c^2 d x^2} \log (x)}{15 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 162, normalized size = 0.87 \begin {gather*} \frac {\sqrt {d-c^2 d x^2} \left (12 a \left (-1+c^2 x^2\right )^2 \left (3+2 c^2 x^2\right )+b c x \sqrt {1-c^2 x^2} \left (9-6 c^2 x^2-50 c^4 x^4\right )+12 b \left (-1+c^2 x^2\right )^2 \left (3+2 c^2 x^2\right ) \text {ArcSin}(c x)\right )}{180 x^5 \left (-1+c^2 x^2\right )}-\frac {2 b c^5 \sqrt {d-c^2 d x^2} \log (x)}{15 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.34, size = 1903, normalized size = 10.18
method | result | size |
default | \(\text {Expression too large to display}\) | \(1903\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 140, normalized size = 0.75 \begin {gather*} -\frac {1}{60} \, {\left (8 \, c^{4} \sqrt {d} \log \left (x\right ) - \frac {2 \, c^{2} \sqrt {d} x^{2} - 3 \, \sqrt {d}}{x^{4}}\right )} b c - \frac {1}{15} \, b {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2}}{d x^{3}} + \frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{d x^{5}}\right )} \arcsin \left (c x\right ) - \frac {1}{15} \, a {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2}}{d x^{3}} + \frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{d x^{5}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.54, size = 501, normalized size = 2.68 \begin {gather*} \left [\frac {4 \, {\left (b c^{7} x^{7} - b c^{5} x^{5}\right )} \sqrt {d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{4} - 1\right )} \sqrt {d} - d}{c^{2} x^{4} - x^{2}}\right ) - {\left (2 \, b c^{3} x^{3} - {\left (2 \, b c^{3} - 3 \, b c\right )} x^{5} - 3 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 4 \, {\left (2 \, a c^{6} x^{6} - a c^{4} x^{4} - 4 \, a c^{2} x^{2} + {\left (2 \, b c^{6} x^{6} - b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 3 \, b\right )} \arcsin \left (c x\right ) + 3 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{60 \, {\left (c^{2} x^{7} - x^{5}\right )}}, -\frac {8 \, {\left (b c^{7} x^{7} - b c^{5} x^{5}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{2} + 1\right )} \sqrt {-d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) + {\left (2 \, b c^{3} x^{3} - {\left (2 \, b c^{3} - 3 \, b c\right )} x^{5} - 3 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 4 \, {\left (2 \, a c^{6} x^{6} - a c^{4} x^{4} - 4 \, a c^{2} x^{2} + {\left (2 \, b c^{6} x^{6} - b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 3 \, b\right )} \arcsin \left (c x\right ) + 3 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{60 \, {\left (c^{2} x^{7} - x^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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