Optimal. Leaf size=386 \[ -\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt {1-c^2 x^2}}-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.46, 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 = {4695, 4699, 4697, 4709, 4183, 2279, 2391, 8, 270} \[ -\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,-e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,e^{i \sin ^{-1}(c x)}\right )}{2 \sqrt {1-c^2 x^2}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{2 x^2}-\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}+\frac {7 b c^3 d^2 x \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 270
Rule 2279
Rule 2391
Rule 4183
Rule 4695
Rule 4697
Rule 4699
Rule 4709
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 5.70, size = 484, normalized size = 1.25 \[ \frac {-180 a c^2 d^{5/2} x^2 \log (x) \sqrt {d-c^2 d x^2}+180 a c^2 d^{5/2} x^2 \sqrt {d-c^2 d x^2} \log \left (\sqrt {d} \sqrt {d-c^2 d x^2}+d\right )-12 a d^3 \left (c^2 x^2-1\right ) \left (2 c^4 x^4-14 c^2 x^2-3\right )+144 b c^2 d^3 x^2 \sqrt {1-c^2 x^2} \left (-\sqrt {1-c^2 x^2} \sin ^{-1}(c x)-i \left (\text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-\text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )\right )+c x-\sin ^{-1}(c x) \left (\log \left (1-e^{i \sin ^{-1}(c x)}\right )-\log \left (1+e^{i \sin ^{-1}(c x)}\right )\right )\right )-9 b c^2 d^3 x^2 \sqrt {1-c^2 x^2} \left (4 i \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )-4 i \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )+4 \sin ^{-1}(c x) \log \left (1-e^{i \sin ^{-1}(c x)}\right )-4 \sin ^{-1}(c x) \log \left (1+e^{i \sin ^{-1}(c x)}\right )+2 \tan \left (\frac {1}{2} \sin ^{-1}(c x)\right )+2 \cot \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\sin ^{-1}(c x) \csc ^2\left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin ^{-1}(c x) \sec ^2\left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+2 b c^2 d^3 x^2 \sqrt {1-c^2 x^2} \left (-3 \sin ^{-1}(c x) \left (3 \sqrt {1-c^2 x^2}+\cos \left (3 \sin ^{-1}(c x)\right )\right )+9 c x+\sin \left (3 \sin ^{-1}(c x)\right )\right )}{72 x^2 \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} + {\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.42, size = 704, normalized size = 1.82 \[ -\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{2 d \,x^{2}}-\frac {a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{2}-\frac {5 a \,c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{6}+\frac {5 a \,c^{2} d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2}-\frac {5 a \,c^{2} \sqrt {-c^{2} d \,x^{2}+d}\, d^{2}}{2}+\frac {b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c}{2 x \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{6} d^{2} \arcsin \left (c x \right ) x^{4}}{3 c^{2} x^{2}-3}-\frac {8 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{4} d^{2} \arcsin \left (c x \right ) x^{2}}{3 \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{5} d^{2} \sqrt {-c^{2} x^{2}+1}\, x^{3}}{9 c^{2} x^{2}-9}-\frac {7 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} d^{2} \sqrt {-c^{2} x^{2}+1}\, x}{3 \left (c^{2} x^{2}-1\right )}-\frac {5 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{2} d^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{2 c^{2} x^{2}-2}+\frac {5 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{2} d^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{2 c^{2} x^{2}-2}+\frac {11 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{2} d^{2} \arcsin \left (c x \right )}{6 \left (c^{2} x^{2}-1\right )}+\frac {b \,d^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{2 x^{2} \left (c^{2} x^{2}-1\right )}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{2} d^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) \arcsin \left (c x \right )}{2 c^{2} x^{2}-2}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c^{2} d^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right ) \arcsin \left (c x \right )}{2 c^{2} x^{2}-2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b \sqrt {d} \int \frac {{\left (c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{x^{3}}\,{d x} + \frac {1}{6} \, {\left (15 \, c^{2} d^{\frac {5}{2}} \log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right ) - 3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2} - 5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d - 15 \, \sqrt {-c^{2} d x^{2} + d} c^{2} d^{2} - \frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________