Optimal. Leaf size=362 \[ -\frac {1}{32} b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {15 b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac {9 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \text {ArcSin}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{8 c}+\frac {1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))^2+\frac {3 x (d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))^2}{8 \left (1-c^2 x^2\right )}+\frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))^3}{8 b c \left (1-c^2 x^2\right )^{3/2}} \]
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Rubi [A]
time = 0.30, antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {4763, 4743,
4741, 4737, 4723, 327, 222, 4767, 201} \begin {gather*} \frac {(c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}+\frac {3 x (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))^2}{8 \left (1-c^2 x^2\right )}+\frac {b \sqrt {1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))}{8 c}-\frac {3 b c x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{4} x (c d x+d)^{3/2} (e-c e x)^{3/2} (a+b \text {ArcSin}(c x))^2+\frac {9 b^2 \text {ArcSin}(c x) (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {15 b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}-\frac {1}{32} b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 327
Rule 4723
Rule 4737
Rule 4741
Rule 4743
Rule 4763
Rule 4767
Rubi steps
\begin {align*} \int (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left ((d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (3 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b c (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac {\left (3 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b c (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac {1}{32} b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {3 b c x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac {(d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{32 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (3 b^2 c^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac {1}{32} b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {15 b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}-\frac {3 b c x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac {(d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (3 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac {1}{32} b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {15 b^2 x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac {9 b^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {b (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac {(d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 1.14, size = 373, normalized size = 1.03 \begin {gather*} \frac {32 b^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^3-96 a^2 d^{3/2} e^{3/2} \sqrt {1-c^2 x^2} \text {ArcTan}\left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+8 b d e \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^2 (12 a+8 b \sin (2 \text {ArcSin}(c x))+b \sin (4 \text {ArcSin}(c x)))+d e \sqrt {d+c d x} \sqrt {e-c e x} \left (160 a^2 c x \sqrt {1-c^2 x^2}-64 a^2 c^3 x^3 \sqrt {1-c^2 x^2}+64 a b \cos (2 \text {ArcSin}(c x))+4 a b \cos (4 \text {ArcSin}(c x))-32 b^2 \sin (2 \text {ArcSin}(c x))-b^2 \sin (4 \text {ArcSin}(c x))\right )+4 b d e \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x) (16 b \cos (2 \text {ArcSin}(c x))+b \cos (4 \text {ArcSin}(c x))+4 a (8 \sin (2 \text {ArcSin}(c x))+\sin (4 \text {ArcSin}(c x))))}{256 c \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx\]
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(-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]
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 {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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