Optimal. Leaf size=301 \[ \frac {8 b d x \sqrt {d-c^2 d x^2}}{315 c^5 \sqrt {1-c^2 x^2}}+\frac {4 b d x^3 \sqrt {d-c^2 d x^2}}{945 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^5 \sqrt {d-c^2 d x^2}}{525 c \sqrt {1-c^2 x^2}}-\frac {10 b c d x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{5 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {ArcSin}(c x))}{7 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {ArcSin}(c x))}{9 c^6 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 301, 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 = {272, 45, 4779,
12, 1167} \begin {gather*} -\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {ArcSin}(c x))}{9 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{7/2} (a+b \text {ArcSin}(c x))}{7 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{5 c^6 d}-\frac {10 b c d x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}+\frac {b d x^5 \sqrt {d-c^2 d x^2}}{525 c \sqrt {1-c^2 x^2}}+\frac {8 b d x \sqrt {d-c^2 d x^2}}{315 c^5 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}+\frac {4 b d x^3 \sqrt {d-c^2 d x^2}}{945 c^3 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 272
Rule 1167
Rule 4779
Rubi steps
\begin {align*} \int x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (-8-20 c^2 x^2-35 c^4 x^4\right )}{315 c^6} \, dx}{\sqrt {1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int x^5 \left (d-c^2 d x^2\right )^{3/2} \, dx\\ &=-\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (-8-20 c^2 x^2-35 c^4 x^4\right ) \, dx}{315 c^5 \sqrt {1-c^2 x^2}}+\frac {1}{2} \left (a+b \sin ^{-1}(c x)\right ) \text {Subst}\left (\int x^2 \left (d-c^2 d x\right )^{3/2} \, dx,x,x^2\right )\\ &=-\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int \left (-8-4 c^2 x^2-3 c^4 x^4+50 c^6 x^6-35 c^8 x^8\right ) \, dx}{315 c^5 \sqrt {1-c^2 x^2}}+\frac {1}{2} \left (a+b \sin ^{-1}(c x)\right ) \text {Subst}\left (\int \left (\frac {\left (d-c^2 d x\right )^{3/2}}{c^4}-\frac {2 \left (d-c^2 d x\right )^{5/2}}{c^4 d}+\frac {\left (d-c^2 d x\right )^{7/2}}{c^4 d^2}\right ) \, dx,x,x^2\right )\\ &=\frac {8 b d x \sqrt {d-c^2 d x^2}}{315 c^5 \sqrt {1-c^2 x^2}}+\frac {4 b d x^3 \sqrt {d-c^2 d x^2}}{945 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^5 \sqrt {d-c^2 d x^2}}{525 c \sqrt {1-c^2 x^2}}-\frac {10 b c d x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{5 c^6 d}+\frac {2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^6 d^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 150, normalized size = 0.50 \begin {gather*} \frac {d \sqrt {d-c^2 d x^2} \left (-315 a \left (1-c^2 x^2\right )^{5/2} \left (8+20 c^2 x^2+35 c^4 x^4\right )+b c x \left (2520+420 c^2 x^2+189 c^4 x^4-2250 c^6 x^6+1225 c^8 x^8\right )-315 b \left (1-c^2 x^2\right )^{5/2} \left (8+20 c^2 x^2+35 c^4 x^4\right ) \text {ArcSin}(c x)\right )}{99225 c^6 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 0.37, size = 1254, normalized size = 4.17
method | result | size |
default | \(a \left (-\frac {x^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{9 c^{2} d}+\frac {-\frac {4 x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{63 c^{2} d}-\frac {8 \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{315 d \,c^{4}}}{c^{2}}\right )+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (256 c^{10} x^{10}-704 c^{8} x^{8}-256 i \sqrt {-c^{2} x^{2}+1}\, x^{9} c^{9}+688 c^{6} x^{6}+576 i \sqrt {-c^{2} x^{2}+1}\, x^{7} c^{7}-280 c^{4} x^{4}-432 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}+41 c^{2} x^{2}+120 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}-9 i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (i+9 \arcsin \left (c x \right )\right ) d}{41472 c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 c^{6} x^{6}-64 i \sqrt {-c^{2} x^{2}+1}\, x^{7} c^{7}+104 c^{4} x^{4}+112 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}-25 c^{2} x^{2}-56 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+7 i \sqrt {-c^{2} x^{2}+1}\, x c +1\right ) \left (i+7 \arcsin \left (c x \right )\right ) d}{25088 c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}-16 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}+13 c^{2} x^{2}+20 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}-5 i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (i+5 \arcsin \left (c x \right )\right ) d}{3200 c^{6} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) \left (\arcsin \left (c x \right )+i\right ) d}{256 c^{6} \left (c^{2} x^{2}-1\right )}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) d}{256 c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i \sqrt {-c^{2} x^{2}+1}\, x c -5 c^{2} x^{2}+1\right ) \left (-i+3 \arcsin \left (c x \right )\right ) d}{1152 c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 i \sqrt {-c^{2} x^{2}+1}\, x^{7} c^{7}+64 c^{8} x^{8}-112 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}-144 c^{6} x^{6}+56 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+104 c^{4} x^{4}-7 i \sqrt {-c^{2} x^{2}+1}\, x c -25 c^{2} x^{2}+1\right ) \left (-i+7 \arcsin \left (c x \right )\right ) d}{25088 c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (256 i \sqrt {-c^{2} x^{2}+1}\, x^{9} c^{9}+256 c^{10} x^{10}-576 i \sqrt {-c^{2} x^{2}+1}\, x^{7} c^{7}-704 c^{8} x^{8}+432 i \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}+688 c^{6} x^{6}-120 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}-280 c^{4} x^{4}+9 i \sqrt {-c^{2} x^{2}+1}\, x c +41 c^{2} x^{2}-1\right ) \left (-i+9 \arcsin \left (c x \right )\right ) d}{41472 c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) \left (17 i+15 \arcsin \left (c x \right )\right ) \cos \left (4 \arcsin \left (c x \right )\right ) d}{14400 c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i x^{2} c^{2}-c x \sqrt {-c^{2} x^{2}+1}-i\right ) \left (2 i+15 \arcsin \left (c x \right )\right ) \sin \left (4 \arcsin \left (c x \right )\right ) d}{3600 c^{6} \left (c^{2} x^{2}-1\right )}\right )\) | \(1254\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 208, normalized size = 0.69 \begin {gather*} -\frac {1}{315} \, {\left (\frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{c^{2} d} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{6} d}\right )} b \arcsin \left (c x\right ) - \frac {1}{315} \, {\left (\frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{c^{2} d} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{6} d}\right )} a + \frac {{\left (1225 \, c^{8} d^{\frac {3}{2}} x^{9} - 2250 \, c^{6} d^{\frac {3}{2}} x^{7} + 189 \, c^{4} d^{\frac {3}{2}} x^{5} + 420 \, c^{2} d^{\frac {3}{2}} x^{3} + 2520 \, d^{\frac {3}{2}} x\right )} b}{99225 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 4.25, size = 219, normalized size = 0.73 \begin {gather*} -\frac {{\left (1225 \, b c^{9} d x^{9} - 2250 \, b c^{7} d x^{7} + 189 \, b c^{5} d x^{5} + 420 \, b c^{3} d x^{3} + 2520 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 315 \, {\left (35 \, a c^{10} d x^{10} - 85 \, a c^{8} d x^{8} + 53 \, a c^{6} d x^{6} + a c^{4} d x^{4} + 4 \, a c^{2} d x^{2} - 8 \, a d + {\left (35 \, b c^{10} d x^{10} - 85 \, b c^{8} d x^{8} + 53 \, b c^{6} d x^{6} + b c^{4} d x^{4} + 4 \, b c^{2} d x^{2} - 8 \, b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{99225 \, {\left (c^{8} x^{2} - c^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[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 x^5\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________