Optimal. Leaf size=361 \[ -\frac {23 b c d^2 x \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}+\frac {11 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {i b d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.31, antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {4787, 4783,
4803, 4268, 2317, 2438, 8, 200} \begin {gather*} d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))-\frac {2 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))}{\sqrt {1-c^2 x^2}}+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))+\frac {i b d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {23 b c d^2 x \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+\frac {11 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 200
Rule 2317
Rule 2438
Rule 4268
Rule 4783
Rule 4787
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} \, dx &=\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+d \int \frac {\left (d-c^2 d x^2\right )^{3/2} \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 (1-c^2 x^2\right )^2 \, dx}{5 \sqrt {1-c^2 x^2}}\\ &=\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+d^2 \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 (1-2 c^2 x^2+c^4 x^4\right ) \, dx}{5 \sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {8 b c d^2 x \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}+\frac {11 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (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}{\sqrt {1-c^2 x^2}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {23 b c d^2 x \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}+\frac {11 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (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 )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {23 b c d^2 x \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}+\frac {11 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2 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 )}{\sqrt {1-c^2 x^2}}-\frac {\left (b 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 )}{\sqrt {1-c^2 x^2}}+\frac {\left (b 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 )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {23 b c d^2 x \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}+\frac {11 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2 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 )}{\sqrt {1-c^2 x^2}}+\frac {\left (i b 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 )}{\sqrt {1-c^2 x^2}}-\frac {\left (i b 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 )}{\sqrt {1-c^2 x^2}}\\ &=-\frac {23 b c d^2 x \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}+\frac {11 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {2 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 )}{\sqrt {1-c^2 x^2}}+\frac {i b d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.10, size = 394, normalized size = 1.09 \begin {gather*} \frac {1}{15} a d^2 \sqrt {d-c^2 d x^2} \left (23-11 c^2 x^2+3 c^4 x^4\right )+a d^{5/2} \log (x)-a d^{5/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b 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 d^2 \sqrt {d-c^2 d x^2} \left (9 c x-3 \text {ArcSin}(c x) \left (3 \sqrt {1-c^2 x^2}+\cos (3 \text {ArcSin}(c x))\right )+\sin (3 \text {ArcSin}(c x))\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d^2 \sqrt {d-c^2 d x^2} \left (450 c x-15 \text {ArcSin}(c x) \left (30 \sqrt {1-c^2 x^2}+5 \cos (3 \text {ArcSin}(c x))-3 \cos (5 \text {ArcSin}(c x))\right )+25 \sin (3 \text {ArcSin}(c x))-9 \sin (5 \text {ArcSin}(c x))\right )}{3600 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 652, normalized size = 1.81
method | result | size |
default | \(\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} a}{5}+\frac {a d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}-a \,d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )+a \sqrt {-c^{2} d \,x^{2}+d}\, d^{2}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}}{25 c^{2} x^{2}-25}-\frac {11 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}}{45 \left (c^{2} x^{2}-1\right )}+\frac {23 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \sqrt {-c^{2} x^{2}+1}\, x c}{15 \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {34 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \arcsin \left (c x \right ) x^{2} c^{2}}{15 \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \arcsin \left (c x \right ) x^{6} c^{6}}{5 c^{2} x^{2}-5}-\frac {14 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \arcsin \left (c x \right ) x^{4} c^{4}}{15 \left (c^{2} x^{2}-1\right )}-\frac {23 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \arcsin \left (c x \right )}{15 \left (c^{2} x^{2}-1\right )}\) | \(652\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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