Optimal. Leaf size=276 \[ \frac{x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac{6 a c}{d^2}+\frac{b^2}{d^2}+\frac{3 c^2}{d^4}\right )}{d^6 \sqrt{1-d^2 x^2}}-\frac{3 \sin ^{-1}(d x) \left (8 a^2 c d^4+8 a b^2 d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right )}{8 d^7}+\frac{c x \sqrt{1-d^2 x^2} \left (12 a c d^2+12 b^2 d^2+7 c^2\right )}{8 d^6}+\frac{b \sqrt{1-d^2 x^2} \left (6 a c d^2+b^2 d^2+5 c^2\right )}{d^6}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4} \]
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Rubi [A] time = 0.597837, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {899, 1814, 1815, 641, 216} \[ \frac{x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac{6 a c}{d^2}+\frac{b^2}{d^2}+\frac{3 c^2}{d^4}\right )}{d^6 \sqrt{1-d^2 x^2}}-\frac{3 \sin ^{-1}(d x) \left (8 a^2 c d^4+8 a b^2 d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right )}{8 d^7}+\frac{c x \sqrt{1-d^2 x^2} \left (12 a c d^2+12 b^2 d^2+7 c^2\right )}{8 d^6}+\frac{b \sqrt{1-d^2 x^2} \left (6 a c d^2+b^2 d^2+5 c^2\right )}{d^6}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4} \]
Antiderivative was successfully verified.
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Rule 899
Rule 1814
Rule 1815
Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx &=\int \frac{\left (a+b x+c x^2\right )^3}{\left (1-d^2 x^2\right )^{3/2}} \, dx\\ &=\frac{b \left (3 a^2+\frac{3 c^2}{d^4}+\frac{b^2}{d^2}+\frac{6 a c}{d^2}\right ) d^4+\left (c+a d^2\right ) \left (c^2+3 b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^6 \sqrt{1-d^2 x^2}}-\int \frac{\frac{c^3+3 a c^2 d^2+3 a b^2 d^4+3 c d^2 \left (b^2+a^2 d^2\right )}{d^6}+\frac{b \left (b^2+3 c \left (2 a+\frac{c}{d^2}\right )\right ) x}{d^2}+\frac{c \left (3 b^2+c \left (3 a+\frac{c}{d^2}\right )\right ) x^2}{d^2}+\frac{3 b c^2 x^3}{d^2}+\frac{c^3 x^4}{d^2}}{\sqrt{1-d^2 x^2}} \, dx\\ &=\frac{b \left (3 a^2+\frac{3 c^2}{d^4}+\frac{b^2}{d^2}+\frac{6 a c}{d^2}\right ) d^4+\left (c+a d^2\right ) \left (c^2+3 b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^6 \sqrt{1-d^2 x^2}}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4}+\frac{\int \frac{-\frac{4 \left (c^3+3 a c^2 d^2+3 a b^2 d^4+3 c d^2 \left (b^2+a^2 d^2\right )\right )}{d^4}-4 b \left (b^2+3 c \left (2 a+\frac{c}{d^2}\right )\right ) x-c \left (12 b^2+c \left (12 a+\frac{7 c}{d^2}\right )\right ) x^2-12 b c^2 x^3}{\sqrt{1-d^2 x^2}} \, dx}{4 d^2}\\ &=\frac{b \left (3 a^2+\frac{3 c^2}{d^4}+\frac{b^2}{d^2}+\frac{6 a c}{d^2}\right ) d^4+\left (c+a d^2\right ) \left (c^2+3 b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^6 \sqrt{1-d^2 x^2}}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4}-\frac{\int \frac{12 \left (3 b^2 \left (c+a d^2\right )+c \left (3 a c+\frac{c^2}{d^2}+3 a^2 d^2\right )\right )+12 b \left (5 c^2+b^2 d^2+6 a c d^2\right ) x+3 c \left (7 c^2+12 b^2 d^2+12 a c d^2\right ) x^2}{\sqrt{1-d^2 x^2}} \, dx}{12 d^4}\\ &=\frac{b \left (3 a^2+\frac{3 c^2}{d^4}+\frac{b^2}{d^2}+\frac{6 a c}{d^2}\right ) d^4+\left (c+a d^2\right ) \left (c^2+3 b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^6 \sqrt{1-d^2 x^2}}+\frac{c \left (7 c^2+12 b^2 d^2+12 a c d^2\right ) x \sqrt{1-d^2 x^2}}{8 d^6}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4}+\frac{\int \frac{-9 \left (5 c^3+12 b^2 c d^2+12 a c^2 d^2+8 a b^2 d^4+8 a^2 c d^4\right )-24 b d^2 \left (5 c^2+b^2 d^2+6 a c d^2\right ) x}{\sqrt{1-d^2 x^2}} \, dx}{24 d^6}\\ &=\frac{b \left (3 a^2+\frac{3 c^2}{d^4}+\frac{b^2}{d^2}+\frac{6 a c}{d^2}\right ) d^4+\left (c+a d^2\right ) \left (c^2+3 b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^6 \sqrt{1-d^2 x^2}}+\frac{b \left (5 c^2+b^2 d^2+6 a c d^2\right ) \sqrt{1-d^2 x^2}}{d^6}+\frac{c \left (7 c^2+12 b^2 d^2+12 a c d^2\right ) x \sqrt{1-d^2 x^2}}{8 d^6}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4}-\frac{\left (3 \left (5 c^3+12 b^2 c d^2+12 a c^2 d^2+8 a b^2 d^4+8 a^2 c d^4\right )\right ) \int \frac{1}{\sqrt{1-d^2 x^2}} \, dx}{8 d^6}\\ &=\frac{b \left (3 a^2+\frac{3 c^2}{d^4}+\frac{b^2}{d^2}+\frac{6 a c}{d^2}\right ) d^4+\left (c+a d^2\right ) \left (c^2+3 b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^6 \sqrt{1-d^2 x^2}}+\frac{b \left (5 c^2+b^2 d^2+6 a c d^2\right ) \sqrt{1-d^2 x^2}}{d^6}+\frac{c \left (7 c^2+12 b^2 d^2+12 a c d^2\right ) x \sqrt{1-d^2 x^2}}{8 d^6}+\frac{b c^2 x^2 \sqrt{1-d^2 x^2}}{d^4}+\frac{c^3 x^3 \sqrt{1-d^2 x^2}}{4 d^4}-\frac{3 \left (5 c^3+12 b^2 c d^2+12 a c^2 d^2+8 a b^2 d^4+8 a^2 c d^4\right ) \sin ^{-1}(d x)}{8 d^7}\\ \end{align*}
Mathematica [A] time = 0.24352, size = 239, normalized size = 0.87 \[ \frac{-3 \sqrt{1-d^2 x^2} \sin ^{-1}(d x) \left (8 a^2 c d^4+8 a b^2 d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right )-8 b \left (-3 a^2 d^5+6 a c d^3 \left (d^2 x^2-2\right )+c^2 d \left (d^4 x^4+4 d^2 x^2-8\right )\right )+d x \left (24 a^2 c d^4+8 a^3 d^6-12 a c^2 d^2 \left (d^2 x^2-3\right )+c^3 \left (-2 d^4 x^4-5 d^2 x^2+15\right )\right )-12 b^2 d^3 x \left (c \left (d^2 x^2-3\right )-2 a d^2\right )-8 b^3 d^3 \left (d^2 x^2-2\right )}{8 d^7 \sqrt{1-d^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.225, size = 755, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67893, size = 549, normalized size = 1.99 \begin{align*} -\frac{c^{3} x^{5}}{4 \, \sqrt{-d^{2} x^{2} + 1} d^{2}} - \frac{b c^{2} x^{4}}{\sqrt{-d^{2} x^{2} + 1} d^{2}} + \frac{a^{3} x}{\sqrt{-d^{2} x^{2} + 1}} - \frac{5 \, c^{3} x^{3}}{8 \, \sqrt{-d^{2} x^{2} + 1} d^{4}} - \frac{3 \,{\left (b^{2} c + a c^{2}\right )} x^{3}}{2 \, \sqrt{-d^{2} x^{2} + 1} d^{2}} + \frac{3 \, a^{2} b}{\sqrt{-d^{2} x^{2} + 1} d^{2}} - \frac{4 \, b c^{2} x^{2}}{\sqrt{-d^{2} x^{2} + 1} d^{4}} - \frac{{\left (b^{3} + 6 \, a b c\right )} x^{2}}{\sqrt{-d^{2} x^{2} + 1} d^{2}} + \frac{3 \,{\left (a b^{2} + a^{2} c\right )} x}{\sqrt{-d^{2} x^{2} + 1} d^{2}} - \frac{3 \,{\left (a b^{2} + a^{2} c\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}} d^{2}} + \frac{15 \, c^{3} x}{8 \, \sqrt{-d^{2} x^{2} + 1} d^{6}} + \frac{9 \,{\left (b^{2} c + a c^{2}\right )} x}{2 \, \sqrt{-d^{2} x^{2} + 1} d^{4}} - \frac{15 \, c^{3} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \sqrt{d^{2}} d^{6}} - \frac{9 \,{\left (b^{2} c + a c^{2}\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}} d^{4}} + \frac{8 \, b c^{2}}{\sqrt{-d^{2} x^{2} + 1} d^{6}} + \frac{2 \,{\left (b^{3} + 6 \, a b c\right )}}{\sqrt{-d^{2} x^{2} + 1} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71622, size = 802, normalized size = 2.91 \begin{align*} -\frac{24 \, a^{2} b d^{5} + 64 \, b c^{2} d + 16 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} - 8 \,{\left (3 \, a^{2} b d^{7} + 8 \, b c^{2} d^{3} + 2 \,{\left (b^{3} + 6 \, a b c\right )} d^{5}\right )} x^{2} -{\left (2 \, c^{3} d^{5} x^{5} + 8 \, b c^{2} d^{5} x^{4} - 24 \, a^{2} b d^{5} - 64 \, b c^{2} d - 16 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} +{\left (5 \, c^{3} d^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{5}\right )} x^{3} + 8 \,{\left (4 \, b c^{2} d^{3} +{\left (b^{3} + 6 \, a b c\right )} d^{5}\right )} x^{2} -{\left (8 \, a^{3} d^{7} + 24 \,{\left (a b^{2} + a^{2} c\right )} d^{5} + 15 \, c^{3} d + 36 \,{\left (b^{2} c + a c^{2}\right )} d^{3}\right )} x\right )} \sqrt{d x + 1} \sqrt{-d x + 1} + 6 \,{\left (8 \,{\left (a b^{2} + a^{2} c\right )} d^{4} + 5 \, c^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} -{\left (8 \,{\left (a b^{2} + a^{2} c\right )} d^{6} + 5 \, c^{3} d^{2} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{4}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{8 \,{\left (d^{9} x^{2} - d^{7}\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 [B] time = 1.57657, size = 961, normalized size = 3.48 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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