Optimal. Leaf size=421 \[ -\frac {7 b^2 d x \sqrt {d-c^2 d x^2}}{1152 c^2}-\frac {43 b^2 d x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}+\frac {7 b^2 d \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{18 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2+\frac {d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.45, antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {4787, 4783,
4795, 4737, 4723, 327, 222, 14, 4777, 12, 470} \begin {gather*} \frac {b d x^2 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{16 c \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{16 c^2}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{48 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2+\frac {d \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{18 \sqrt {1-c^2 x^2}}+\frac {7 b^2 d \text {ArcSin}(c x) \sqrt {d-c^2 d x^2}}{1152 c^3 \sqrt {1-c^2 x^2}}-\frac {7 b^2 d x \sqrt {d-c^2 d x^2}}{1152 c^2}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}-\frac {43 b^2 d x^3 \sqrt {d-c^2 d x^2}}{1728} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 222
Rule 327
Rule 470
Rule 4723
Rule 4737
Rule 4777
Rule 4783
Rule 4787
Rule 4795
Rubi steps
\begin {align*} \int x^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d \int x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{16 c^2 \sqrt {1-c^2 x^2}}+\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (3-2 c^2 x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{36 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{64} b^2 d x^3 \sqrt {d-c^2 d x^2}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}+\frac {b d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{27 \sqrt {1-c^2 x^2}}\\ &=\frac {b^2 d x \sqrt {d-c^2 d x^2}}{128 c^2}-\frac {43 b^2 d x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}+\frac {b d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{36 \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{128 c^2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {7 b^2 d x \sqrt {d-c^2 d x^2}}{1152 c^2}-\frac {43 b^2 d x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}-\frac {b^2 d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{128 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{72 c^2 \sqrt {1-c^2 x^2}}\\ &=-\frac {7 b^2 d x \sqrt {d-c^2 d x^2}}{1152 c^2}-\frac {43 b^2 d x^3 \sqrt {d-c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d-c^2 d x^2}+\frac {7 b^2 d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c^3 \sqrt {1-c^2 x^2}}+\frac {b d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c \sqrt {1-c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{48 \sqrt {1-c^2 x^2}}+\frac {b c^3 d x^6 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 \sqrt {1-c^2 x^2}}-\frac {d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 297, normalized size = 0.71 \begin {gather*} \frac {d \sqrt {d-c^2 d x^2} \left (72 a^3+24 a b^2 c^2 x^2 \left (9-21 c^2 x^2+8 c^4 x^4\right )-72 a^2 b c x \sqrt {1-c^2 x^2} \left (3-14 c^2 x^2+8 c^4 x^4\right )+b^3 c x \sqrt {1-c^2 x^2} \left (-21-86 c^2 x^2+32 c^4 x^4\right )+3 b \left (72 a^2-48 a b c x \sqrt {1-c^2 x^2} \left (3-14 c^2 x^2+8 c^4 x^4\right )+b^2 \left (7+72 c^2 x^2-168 c^4 x^4+64 c^6 x^6\right )\right ) \text {ArcSin}(c x)+72 b^2 \left (3 a+b c x \sqrt {1-c^2 x^2} \left (-3+14 c^2 x^2-8 c^4 x^4\right )\right ) \text {ArcSin}(c x)^2+72 b^3 \text {ArcSin}(c x)^3\right )}{3456 b c^3 \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 = 0.50, size = 1320, normalized size = 3.14
method | result | size |
default | \(\text {Expression too large to display}\) | \(1320\) |
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 x^{2} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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