Optimal. Leaf size=502 \[ \frac{5 (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}+\frac{5 x (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac{5 x (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac{b \sqrt{1-c^2 x^2} (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5 b (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt{1-c^2 x^2}}-\frac{5 b c x^2 (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac{1}{6} x (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{65 b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}-\frac{245 b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2}}{1152 \left (1-c^2 x^2\right )^2}+\frac{115 b^2 (c d x+d)^{5/2} (e-c e x)^{5/2} \sin ^{-1}(c x)}{1152 c \left (1-c^2 x^2\right )^{5/2}}-\frac{1}{108} b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2} \]
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Rubi [A] time = 0.567899, antiderivative size = 502, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.281, Rules used = {4673, 4649, 4647, 4641, 4627, 321, 216, 4677, 195} \[ \frac{5 (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}+\frac{5 x (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac{5 x (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac{b \sqrt{1-c^2 x^2} (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{5 b (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt{1-c^2 x^2}}-\frac{5 b c x^2 (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac{1}{6} x (c d x+d)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{65 b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}-\frac{245 b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2}}{1152 \left (1-c^2 x^2\right )^2}+\frac{115 b^2 (c d x+d)^{5/2} (e-c e x)^{5/2} \sin ^{-1}(c x)}{1152 c \left (1-c^2 x^2\right )^{5/2}}-\frac{1}{108} b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 4649
Rule 4647
Rule 4641
Rule 4627
Rule 321
Rule 216
Rule 4677
Rule 195
Rubi steps
\begin{align*} \int (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{\left ((d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\left (1-c^2 x^2\right )^{5/2}}\\ &=\frac{1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{\left (5 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{6 \left (1-c^2 x^2\right )^{5/2}}-\frac{\left (b c (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \left (1-c^2 x^2\right )^{5/2}}\\ &=\frac{b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac{\left (5 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{8 \left (1-c^2 x^2\right )^{5/2}}-\frac{\left (b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx}{18 \left (1-c^2 x^2\right )^{5/2}}-\frac{\left (5 b c (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{12 \left (1-c^2 x^2\right )^{5/2}}\\ &=-\frac{1}{108} b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}+\frac{5 b (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt{1-c^2 x^2}}+\frac{b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac{5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac{\left (5 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{5/2}}-\frac{\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{108 \left (1-c^2 x^2\right )^{5/2}}-\frac{\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{48 \left (1-c^2 x^2\right )^{5/2}}-\frac{\left (5 b c (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 \left (1-c^2 x^2\right )^{5/2}}\\ &=-\frac{1}{108} b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}-\frac{65 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}-\frac{5 b c x^2 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac{5 b (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt{1-c^2 x^2}}+\frac{b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac{5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac{5 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}-\frac{\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \sqrt{1-c^2 x^2} \, dx}{144 \left (1-c^2 x^2\right )^{5/2}}-\frac{\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \sqrt{1-c^2 x^2} \, dx}{64 \left (1-c^2 x^2\right )^{5/2}}+\frac{\left (5 b^2 c^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{5/2}}\\ &=-\frac{1}{108} b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}-\frac{245 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1152 \left (1-c^2 x^2\right )^2}-\frac{65 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}-\frac{5 b c x^2 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac{5 b (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt{1-c^2 x^2}}+\frac{b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac{5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac{5 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}-\frac{\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{288 \left (1-c^2 x^2\right )^{5/2}}-\frac{\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{128 \left (1-c^2 x^2\right )^{5/2}}+\frac{\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{32 \left (1-c^2 x^2\right )^{5/2}}\\ &=-\frac{1}{108} b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}-\frac{245 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1152 \left (1-c^2 x^2\right )^2}-\frac{65 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}+\frac{115 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2} \sin ^{-1}(c x)}{1152 c \left (1-c^2 x^2\right )^{5/2}}-\frac{5 b c x^2 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac{5 b (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt{1-c^2 x^2}}+\frac{b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac{5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac{5 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 2.83636, size = 450, normalized size = 0.9 \[ \frac{d^2 e^2 \left (\sqrt{c d x+d} \sqrt{e-c e x} \left (2304 a^2 c^5 x^5 \sqrt{1-c^2 x^2}-7488 a^2 c^3 x^3 \sqrt{1-c^2 x^2}+9504 a^2 c x \sqrt{1-c^2 x^2}+3240 a b \cos \left (2 \sin ^{-1}(c x)\right )+324 a b \cos \left (4 \sin ^{-1}(c x)\right )+24 a b \cos \left (6 \sin ^{-1}(c x)\right )-1620 b^2 \sin \left (2 \sin ^{-1}(c x)\right )-81 b^2 \sin \left (4 \sin ^{-1}(c x)\right )-4 b^2 \sin \left (6 \sin ^{-1}(c x)\right )\right )-4320 a^2 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )+72 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left (60 a+45 b \sin \left (2 \sin ^{-1}(c x)\right )+9 b \sin \left (4 \sin ^{-1}(c x)\right )+b \sin \left (6 \sin ^{-1}(c x)\right )\right )+12 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left (540 a \sin \left (2 \sin ^{-1}(c x)\right )+108 a \sin \left (4 \sin ^{-1}(c x)\right )+12 a \sin \left (6 \sin ^{-1}(c x)\right )+270 b \cos \left (2 \sin ^{-1}(c x)\right )+27 b \cos \left (4 \sin ^{-1}(c x)\right )+2 b \cos \left (6 \sin ^{-1}(c x)\right )\right )+1440 b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3\right )}{13824 c \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.256, size = 0, normalized size = 0. \begin{align*} \int \left ( cdx+d \right ) ^{{\frac{5}{2}}} \left ( -cex+e \right ) ^{{\frac{5}{2}}} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c^{4} d^{2} e^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} +{\left (b^{2} c^{4} d^{2} e^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} e^{2} x^{2} + b^{2} d^{2} e^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{4} d^{2} e^{2} x^{4} - 2 \, a b c^{2} d^{2} e^{2} x^{2} + a b d^{2} e^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{c d x + d} \sqrt{-c e x + e}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c d x + d\right )}^{\frac{5}{2}}{\left (-c e x + e\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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