Optimal. Leaf size=389 \[ -\frac {b c d^2 \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}+\frac {9 b c^3 d^2 \sqrt {d-c^2 d x^2}}{8 x \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {15}{8} c^4 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{8 x^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{4 x^4}-\frac {15 c^4 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {15 i b c^4 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}-\frac {15 i b c^4 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )}{8 \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.31, antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4785, 4783,
4803, 4268, 2317, 2438, 8, 14, 276} \begin {gather*} \frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{8 x^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{4 x^4}+\frac {15}{8} c^4 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))-\frac {15 c^4 d^2 \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{4 \sqrt {1-c^2 x^2}}+\frac {15 i b c^4 d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}-\frac {15 i b c^4 d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {9 b c^3 d^2 \sqrt {d-c^2 d x^2}}{8 x \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 276
Rule 2317
Rule 2438
Rule 4268
Rule 4783
Rule 4785
Rule 4803
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x^5} \, dx &=-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}-\frac {1}{4} \left (5 c^2 d\right ) \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2}{x^4} \, dx}{4 \sqrt {1-c^2 x^2}}\\ &=\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}+\frac {1}{8} \left (15 c^4 d^2\right ) \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (c^4+\frac {1}{x^4}-\frac {2 c^2}{x^2}\right ) \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1-c^2 x^2}{x^2} \, dx}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}+\frac {b c^5 d^2 x \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {15}{8} c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}-\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-c^2+\frac {1}{x^2}\right ) \, dx}{8 \sqrt {1-c^2 x^2}}+\frac {\left (15 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (15 b c^5 d^2 \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}+\frac {9 b c^3 d^2 \sqrt {d-c^2 d x^2}}{8 x \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {15}{8} c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}+\frac {\left (15 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}+\frac {9 b c^3 d^2 \sqrt {d-c^2 d x^2}}{8 x \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {15}{8} c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}-\frac {15 c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 \sqrt {1-c^2 x^2}}-\frac {\left (15 b c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {\left (15 b c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}+\frac {9 b c^3 d^2 \sqrt {d-c^2 d x^2}}{8 x \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {15}{8} c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}-\frac {15 c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {\left (15 i b c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}-\frac {\left (15 i b c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}+\frac {9 b c^3 d^2 \sqrt {d-c^2 d x^2}}{8 x \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {15}{8} c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 x^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{4 x^4}-\frac {15 c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {15 i b c^4 d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}-\frac {15 i b c^4 d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{8 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 4.01, size = 640, normalized size = 1.65 \begin {gather*} \frac {a d^2 \sqrt {d-c^2 d x^2} \left (-2+9 c^2 x^2+8 c^4 x^4\right )}{8 x^4}+\frac {15}{8} a c^4 d^{5/2} \log (x)-\frac {15}{8} a c^4 d^{5/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b c^4 d^2 \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {1-c^2 x^2} \text {ArcSin}(c x)+\text {ArcSin}(c x) \log \left (1-e^{i \text {ArcSin}(c x)}\right )-\text {ArcSin}(c x) \log \left (1+e^{i \text {ArcSin}(c x)}\right )+i \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-i \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {b c^4 d^3 \sqrt {1-c^2 x^2} \left (-2 \cot \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\text {ArcSin}(c x) \csc ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )-4 \text {ArcSin}(c x) \log \left (1-e^{i \text {ArcSin}(c x)}\right )+4 \text {ArcSin}(c x) \log \left (1+e^{i \text {ArcSin}(c x)}\right )-4 i \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )+4 i \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )+\text {ArcSin}(c x) \sec ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )-2 \tan \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{4 \sqrt {d-c^2 d x^2}}+\frac {b c^4 d^2 \sqrt {d-c^2 d x^2} \left (8 \cot \left (\frac {1}{2} \text {ArcSin}(c x)\right )+6 \text {ArcSin}(c x) \csc ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )-c x \csc ^4\left (\frac {1}{2} \text {ArcSin}(c x)\right )-3 \text {ArcSin}(c x) \csc ^4\left (\frac {1}{2} \text {ArcSin}(c x)\right )-24 \text {ArcSin}(c x) \log \left (1-e^{i \text {ArcSin}(c x)}\right )+24 \text {ArcSin}(c x) \log \left (1+e^{i \text {ArcSin}(c x)}\right )-24 i \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )+24 i \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )-6 \text {ArcSin}(c x) \sec ^2\left (\frac {1}{2} \text {ArcSin}(c x)\right )+3 \text {ArcSin}(c x) \sec ^4\left (\frac {1}{2} \text {ArcSin}(c x)\right )-\frac {16 \sin ^4\left (\frac {1}{2} \text {ArcSin}(c x)\right )}{c^3 x^3}+8 \tan \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{192 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 727, normalized size = 1.87
method | result | size |
default | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{4 d \,x^{4}}+\frac {3 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{8 d \,x^{2}}+\frac {3 a \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8}+\frac {5 a \,c^{4} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8}-\frac {15 a \,c^{4} d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{8}+\frac {15 a \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}\, d^{2}}{8}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{6} \arcsin \left (c x \right ) x^{2}}{c^{2} x^{2}-1}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{5} \sqrt {-c^{2} x^{2}+1}\, x}{c^{2} x^{2}-1}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{4} \arcsin \left (c x \right )}{8 c^{2} x^{2}-8}-\frac {9 b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{3}}{8 x \left (c^{2} x^{2}-1\right )}-\frac {11 b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) c^{2}}{8 x^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c}{12 x^{3} \left (c^{2} x^{2}-1\right )}+\frac {b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{4 x^{4} \left (c^{2} x^{2}-1\right )}+\frac {15 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} c^{4} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) \arcsin \left (c x \right )}{8 c^{2} x^{2}-8}-\frac {15 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} c^{4} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right ) \arcsin \left (c x \right )}{8 c^{2} x^{2}-8}-\frac {15 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} c^{4} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{8 c^{2} x^{2}-8}+\frac {15 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} c^{4} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{8 c^{2} x^{2}-8}\) | \(727\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{x^{5}}\, 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 = -0.00 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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