Optimal. Leaf size=278 \[ \frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {1-c^2 x^2}}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {1-c^2 x^2}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {ArcSin}(c x))}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {ArcSin}(c x))}{9 c^4 d^2} \]
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Rubi [A]
time = 0.12, antiderivative size = 278, 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, 380} \begin {gather*} \frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {ArcSin}(c x))}{9 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {ArcSin}(c x))}{7 c^4 d}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {1-c^2 x^2}}+\frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 272
Rule 380
Rule 4779
Rubi steps
\begin {align*} \int x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{63 c^4} \, dx}{\sqrt {1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int x^3 \left (d-c^2 d x^2\right )^{5/2} \, dx\\ &=-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3 \, dx}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {1}{2} \left (a+b \sin ^{-1}(c x)\right ) \text {Subst}\left (\int x \left (d-c^2 d x\right )^{5/2} \, dx,x,x^2\right )\\ &=-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+15 c^4 x^4-19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3 \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 )^{5/2}}{c^2}-\frac {\left (d-c^2 d x\right )^{7/2}}{c^2 d}\right ) \, dx,x,x^2\right )\\ &=\frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {1-c^2 x^2}}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {1-c^2 x^2}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 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 )^{7/2} \left (a+b \sin ^{-1}(c x)\right )}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^4 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 137, normalized size = 0.49 \begin {gather*} \frac {d^2 \sqrt {d-c^2 d x^2} \left (-63 a \left (1-c^2 x^2\right )^{7/2} \left (2+7 c^2 x^2\right )+b \left (126 c x+21 c^3 x^3-189 c^5 x^5+171 c^7 x^7-49 c^9 x^9\right )-63 b \left (1-c^2 x^2\right )^{7/2} \left (2+7 c^2 x^2\right ) \text {ArcSin}(c x)\right )}{3969 c^4 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.28, size = 1063, normalized size = 3.82
method | result | size |
default | \(a \left (-\frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{9 c^{2} d}-\frac {2 \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{63 d \,c^{4}}\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^{2}}{41472 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {3 \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^{2}}{25088 c^{4} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+3 i \sqrt {-c^{2} x^{2}+1}\, x c +1\right ) \left (i+3 \arcsin \left (c x \right )\right ) d^{2}}{576 c^{4} \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^{2}}{256 c^{4} \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^{2}}{256 c^{4} \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^{2}}{576 c^{4} \left (c^{2} x^{2}-1\right )}-\frac {3 \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^{2}}{25088 c^{4} \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^{2}}{41472 c^{4} \left (c^{2} x^{2}-1\right )}\right )\) | \(1063\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 160, normalized size = 0.58 \begin {gather*} -\frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b \arcsin \left (c x\right ) - \frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a - \frac {{\left (49 \, c^{8} d^{\frac {5}{2}} x^{9} - 171 \, c^{6} d^{\frac {5}{2}} x^{7} + 189 \, c^{4} d^{\frac {5}{2}} x^{5} - 21 \, c^{2} d^{\frac {5}{2}} x^{3} - 126 \, d^{\frac {5}{2}} x\right )} b}{3969 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.68, size = 255, normalized size = 0.92 \begin {gather*} \frac {{\left (49 \, b c^{9} d^{2} x^{9} - 171 \, b c^{7} d^{2} x^{7} + 189 \, b c^{5} d^{2} x^{5} - 21 \, b c^{3} d^{2} x^{3} - 126 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} + 63 \, {\left (7 \, a c^{10} d^{2} x^{10} - 26 \, a c^{8} d^{2} x^{8} + 34 \, a c^{6} d^{2} x^{6} - 16 \, a c^{4} d^{2} x^{4} - a c^{2} d^{2} x^{2} + 2 \, a d^{2} + {\left (7 \, b c^{10} d^{2} x^{10} - 26 \, b c^{8} d^{2} x^{8} + 34 \, b c^{6} d^{2} x^{6} - 16 \, b c^{4} d^{2} x^{4} - b c^{2} d^{2} x^{2} + 2 \, b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{3969 \, {\left (c^{6} x^{2} - c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 x^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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