Optimal. Leaf size=613 \[ -\frac{b c^3 d^2 x^4 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{1}{4} c^2 d^2 x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{4 b c^2 d^2 x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{3 b c d^2 x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{4 b d^2 x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{5 d^2 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c \sqrt{1-c^2 x^2}}-\frac{2 d^2 \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{3}{8} d^2 x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{32} b^2 c^2 d^2 x^3 \sqrt{c d x+d} \sqrt{e-c e x}+\frac{4 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x}}{27 c}+\frac{15 b^2 d^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}-\frac{15}{64} b^2 d^2 x \sqrt{c d x+d} \sqrt{e-c e x}+\frac{8 b^2 d^2 \sqrt{c d x+d} \sqrt{e-c e x}}{9 c} \]
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Rubi [A] time = 1.00997, antiderivative size = 613, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 13, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.406, Rules used = {4673, 4763, 4647, 4641, 4627, 321, 216, 4677, 4645, 444, 43, 4697, 4707} \[ -\frac{b c^3 d^2 x^4 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{1}{4} c^2 d^2 x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{4 b c^2 d^2 x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{3 b c d^2 x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{4 b d^2 x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{5 d^2 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c \sqrt{1-c^2 x^2}}-\frac{2 d^2 \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{3}{8} d^2 x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{32} b^2 c^2 d^2 x^3 \sqrt{c d x+d} \sqrt{e-c e x}+\frac{4 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x}}{27 c}+\frac{15 b^2 d^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}-\frac{15}{64} b^2 d^2 x \sqrt{c d x+d} \sqrt{e-c e x}+\frac{8 b^2 d^2 \sqrt{c d x+d} \sqrt{e-c e x}}{9 c} \]
Antiderivative was successfully verified.
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Rule 4673
Rule 4763
Rule 4647
Rule 4641
Rule 4627
Rule 321
Rule 216
Rule 4677
Rule 4645
Rule 444
Rule 43
Rule 4697
Rule 4707
Rubi steps
\begin{align*} \int (d+c d x)^{5/2} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{\left (\sqrt{d+c d x} \sqrt{e-c e x}\right ) \int (d+c d x)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (\sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \left (d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+2 c d^2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+c^2 d^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (2 c d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (c^2 d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{1}{2} d^2 x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} c^2 d^2 x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{\left (d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}+\frac{\left (4 b d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt{1-c^2 x^2}}-\frac{\left (b c d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{1-c^2 x^2}}+\frac{\left (c^2 d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{4 \sqrt{1-c^2 x^2}}-\frac{\left (b c^3 d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \sqrt{1-c^2 x^2}}\\ &=\frac{4 b d^2 x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}-\frac{b c d^2 x^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}-\frac{4 b c^2 d^2 x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^4 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{3}{8} d^2 x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} c^2 d^2 x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{d^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt{1-c^2 x^2}}+\frac{\left (d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{\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^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt{1-c^2 x^2}}-\frac{\left (4 b^2 c d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{x \left (1-\frac{c^2 x^2}{3}\right )}{\sqrt{1-c^2 x^2}} \, dx}{3 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^2 d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{2 \sqrt{1-c^2 x^2}}+\frac{\left (b^2 c^4 d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{x^4}{\sqrt{1-c^2 x^2}} \, dx}{8 \sqrt{1-c^2 x^2}}\\ &=-\frac{1}{4} b^2 d^2 x \sqrt{d+c d x} \sqrt{e-c e x}-\frac{1}{32} b^2 c^2 d^2 x^3 \sqrt{d+c d x} \sqrt{e-c e x}+\frac{4 b d^2 x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}-\frac{3 b c d^2 x^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}-\frac{4 b c^2 d^2 x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^4 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{3}{8} d^2 x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} c^2 d^2 x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{5 d^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c \sqrt{1-c^2 x^2}}+\frac{\left (b^2 d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{4 \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 c d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{c^2 x}{3}}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{3 \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 c^2 d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{32 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 c^2 d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2}} \, dx}{8 \sqrt{1-c^2 x^2}}\\ &=-\frac{15}{64} b^2 d^2 x \sqrt{d+c d x} \sqrt{e-c e x}-\frac{1}{32} b^2 c^2 d^2 x^3 \sqrt{d+c d x} \sqrt{e-c e x}+\frac{b^2 d^2 \sqrt{d+c d x} \sqrt{e-c e x} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}+\frac{4 b d^2 x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}-\frac{3 b c d^2 x^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}-\frac{4 b c^2 d^2 x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^4 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{3}{8} d^2 x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} c^2 d^2 x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{5 d^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c \sqrt{1-c^2 x^2}}+\frac{\left (3 b^2 d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{64 \sqrt{1-c^2 x^2}}-\frac{\left (b^2 d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{16 \sqrt{1-c^2 x^2}}-\frac{\left (2 b^2 c d^2 \sqrt{d+c d x} \sqrt{e-c e x}\right ) \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt{1-c^2 x}}+\frac{1}{3} \sqrt{1-c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt{1-c^2 x^2}}\\ &=\frac{8 b^2 d^2 \sqrt{d+c d x} \sqrt{e-c e x}}{9 c}-\frac{15}{64} b^2 d^2 x \sqrt{d+c d x} \sqrt{e-c e x}-\frac{1}{32} b^2 c^2 d^2 x^3 \sqrt{d+c d x} \sqrt{e-c e x}+\frac{4 b^2 d^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )}{27 c}+\frac{15 b^2 d^2 \sqrt{d+c d x} \sqrt{e-c e x} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{4 b d^2 x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}-\frac{3 b c d^2 x^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}-\frac{4 b c^2 d^2 x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^4 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt{1-c^2 x^2}}+\frac{3}{8} d^2 x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{4} c^2 d^2 x^3 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{2 d^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{5 d^2 \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 2.29064, size = 555, normalized size = 0.91 \[ \frac{d^2 \sqrt{c d x+d} \sqrt{e-c e x} \left (3 \left (576 a^2 c^3 x^3 \sqrt{1-c^2 x^2}+1536 a^2 c^2 x^2 \sqrt{1-c^2 x^2}+864 a^2 c x \sqrt{1-c^2 x^2}-1536 a^2 \sqrt{1-c^2 x^2}-1024 a b c^3 x^3+3072 a b c x-36 a b \cos \left (4 \sin ^{-1}(c x)\right )+2304 b^2 \sqrt{1-c^2 x^2}-288 b^2 \sin \left (2 \sin ^{-1}(c x)\right )+9 b^2 \sin \left (4 \sin ^{-1}(c x)\right )\right )+1728 a b \cos \left (2 \sin ^{-1}(c x)\right )+256 b^2 \cos \left (3 \sin ^{-1}(c x)\right )\right )-4320 a^2 d^{5/2} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )-72 b d^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left (-60 a+48 b \sqrt{1-c^2 x^2}-24 b \sin \left (2 \sin ^{-1}(c x)\right )+3 b \sin \left (4 \sin ^{-1}(c x)\right )+16 b \cos \left (3 \sin ^{-1}(c x)\right )\right )+12 b d^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left (768 a c^2 x^2 \sqrt{1-c^2 x^2}-768 a \sqrt{1-c^2 x^2}+288 a \sin \left (2 \sin ^{-1}(c x)\right )-36 a \sin \left (4 \sin ^{-1}(c x)\right )+576 b c x+64 b \sin \left (3 \sin ^{-1}(c x)\right )+144 b \cos \left (2 \sin ^{-1}(c x)\right )-9 b \cos \left (4 \sin ^{-1}(c x)\right )\right )+1440 b^2 d^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{6912 c \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.311, size = 0, normalized size = 0. \begin{align*} \int \left ( cdx+d \right ) ^{{\frac{5}{2}}} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}\sqrt{-cex+e}\, 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^{2} d^{2} x^{2} + 2 \, a^{2} c d^{2} x + a^{2} d^{2} +{\left (b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d^{2} x^{2} + 2 \, a b c d^{2} x + a b d^{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}} \sqrt{-c e x + e}{\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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