Optimal. Leaf size=265 \[ -\frac {25 b c d^2 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 d^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))+\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{32 b c \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4743, 4741,
4737, 30, 14, 267} \begin {gather*} \frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))+\frac {5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{32 b c \sqrt {1-c^2 x^2}}+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} (a+b \text {ArcSin}(c x))+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))-\frac {25 b c d^2 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5 b c^3 d^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 267
Rule 4737
Rule 4741
Rule 4743
Rubi steps
\begin {align*} \int \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} (5 d) \int \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right )^2 \, dx}{6 \sqrt {1-c^2 x^2}}\\ &=\frac {b d^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{8} \left (5 d^2\right ) \int \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{24 \sqrt {1-c^2 x^2}}\\ &=\frac {b d^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{24 \sqrt {1-c^2 x^2}}-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {1-c^2 x^2}}\\ &=-\frac {25 b c d^2 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 d^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {1-c^2 x^2}}+\frac {b d^2 \left (1-c^2 x^2\right )^{5/2} \sqrt {d-c^2 d x^2}}{36 c}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.57, size = 266, normalized size = 1.00 \begin {gather*} \frac {d^2 \left (360 b \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)^2-720 a \sqrt {d} \sqrt {1-c^2 x^2} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\sqrt {d-c^2 d x^2} \left (1584 a c x \sqrt {1-c^2 x^2}-1248 a c^3 x^3 \sqrt {1-c^2 x^2}+384 a c^5 x^5 \sqrt {1-c^2 x^2}+270 b \cos (2 \text {ArcSin}(c x))+27 b \cos (4 \text {ArcSin}(c x))+2 b \cos (6 \text {ArcSin}(c x))\right )+12 b \sqrt {d-c^2 d x^2} \text {ArcSin}(c x) (45 \sin (2 \text {ArcSin}(c x))+9 \sin (4 \text {ArcSin}(c x))+\sin (6 \text {ArcSin}(c x)))\right )}{2304 c \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 = 0.14, size = 691, normalized size = 2.61
method | result | size |
default | \(\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{6}+\frac {5 a d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{24}+\frac {5 a \,d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{16}+\frac {5 a \,d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{16 \sqrt {c^{2} d}}+b \left (-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} d^{2}}{32 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-32 i \sqrt {-c^{2} x^{2}+1}\, x^{6} c^{6}+32 c^{7} x^{7}+48 i \sqrt {-c^{2} x^{2}+1}\, x^{4} c^{4}-64 c^{5} x^{5}-18 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+38 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-6 c x \right ) \left (i+6 \arcsin \left (c x \right )\right ) d^{2}}{2304 c \left (c^{2} x^{2}-1\right )}+\frac {15 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) \left (-i+2 \arcsin \left (c x \right )\right ) d^{2}}{256 c \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 (29 i+96 \arcsin \left (c x \right )\right ) \cos \left (5 \arcsin \left (c x \right )\right ) d^{2}}{4608 c \left (c^{2} x^{2}-1\right )}+\frac {5 \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 (5 i+24 \arcsin \left (c x \right )\right ) \sin \left (5 \arcsin \left (c x \right )\right ) d^{2}}{4608 c \left (c^{2} x^{2}-1\right )}-\frac {3 \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 (11 i+16 \arcsin \left (c x \right )\right ) \cos \left (3 \arcsin \left (c x \right )\right ) d^{2}}{512 c \left (c^{2} x^{2}-1\right )}+\frac {9 \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 (3 i+8 \arcsin \left (c x \right )\right ) \sin \left (3 \arcsin \left (c x \right )\right ) d^{2}}{512 c \left (c^{2} x^{2}-1\right )}\right )\) | \(691\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \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 )\, dx \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 \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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