Optimal. Leaf size=386 \[ -\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )}{2 \sqrt {1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.30, antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4785, 4787,
4783, 4803, 4268, 2317, 2438, 8, 276} \begin {gather*} -\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {5 c^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 {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{2 x^2}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right )}{2 \sqrt {1-c^2 x^2}}-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 276
Rule 2317
Rule 2438
Rule 4268
Rule 4783
Rule 4785
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^3} \, dx &=-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac {1}{2} \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} \, 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^2} \, dx}{2 \sqrt {1-c^2 x^2}}\\ &=-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac {1}{2} \left (5 c^2 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 (-2 c^2+\frac {1}{x^2}+c^4 x^2\right ) \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{6 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2}}{6 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac {\left (5 c^2 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}{2 \sqrt {1-c^2 x^2}}+\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{2 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac {\left (5 c^2 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 )}{2 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {5 c^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 (5 b c^2 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 )}{2 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c^2 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 )}{2 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {5 c^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 (5 i b c^2 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 )}{2 \sqrt {1-c^2 x^2}}+\frac {\left (5 i b c^2 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 )}{2 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}+\frac {5 c^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 {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt {1-c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 3.02, size = 484, normalized size = 1.25 \begin {gather*} \frac {-12 a d^3 \left (-1+c^2 x^2\right ) \left (-3-14 c^2 x^2+2 c^4 x^4\right )-180 a c^2 d^{5/2} x^2 \sqrt {d-c^2 d x^2} \log (x)+180 a c^2 d^{5/2} x^2 \sqrt {d-c^2 d x^2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+144 b c^2 d^3 x^2 \sqrt {1-c^2 x^2} \left (c x-\sqrt {1-c^2 x^2} \text {ArcSin}(c x)-\text {ArcSin}(c x) \left (\log \left (1-e^{i \text {ArcSin}(c x)}\right )-\log \left (1+e^{i \text {ArcSin}(c x)}\right )\right )-i \left (\text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )\right )\right )+2 b c^2 d^3 x^2 \sqrt {1-c^2 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 )-9 b c^2 d^3 x^2 \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 )}{72 x^2 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.26, size = 704, normalized size = 1.82
method | result | size |
default | \(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{2 d \,x^{2}}-\frac {a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{2}-\frac {5 a \,c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{6}+\frac {5 a \,c^{2} d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2}-\frac {5 a \,c^{2} \sqrt {-c^{2} d \,x^{2}+d}\, d^{2}}{2}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{5} d^{2} \sqrt {-c^{2} x^{2}+1}\, x^{3}}{9 c^{2} x^{2}-9}-\frac {7 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} d^{2} \sqrt {-c^{2} x^{2}+1}\, x}{3 \left (c^{2} x^{2}-1\right )}+\frac {5 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{2} d^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{2 c^{2} x^{2}-2}-\frac {5 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{2} d^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{2 c^{2} x^{2}-2}+\frac {11 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} d^{2} \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right )}+\frac {b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{2 x^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{6} d^{2} \arcsin \left (c x \right ) x^{4}}{3 c^{2} x^{2}-3}-\frac {8 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} d^{2} \arcsin \left (c x \right ) x^{2}}{3 \left (c^{2} x^{2}-1\right )}+\frac {b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c}{2 x \left (c^{2} x^{2}-1\right )}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{2} d^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{2 c^{2} x^{2}-2}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{2} d^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{2 c^{2} x^{2}-2}\) | \(704\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________