Optimal. Leaf size=286 \[ \frac{\left (1-d^2 x^2\right )^{3/2} \left (8 \left (C \left (d^2 e^3-4 e f^2\right )-2 f \left (5 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-3 f x \left (5 f^2 \left (2 A d^2+C\right )-2 d^2 e (C e-2 B f)\right )\right )}{120 d^4 f}+\frac{x \sqrt{1-d^2 x^2} \left (2 d^2 \left (A \left (4 d^2 e^2+f^2\right )+2 B e f\right )+C \left (2 d^2 e^2+f^2\right )\right )}{16 d^4}+\frac{\sin ^{-1}(d x) \left (2 d^2 \left (A \left (4 d^2 e^2+f^2\right )+2 B e f\right )+C \left (2 d^2 e^2+f^2\right )\right )}{16 d^5}+\frac{\left (1-d^2 x^2\right )^{3/2} (e+f x)^2 (C e-2 B f)}{10 d^2 f}-\frac{C \left (1-d^2 x^2\right )^{3/2} (e+f x)^3}{6 d^2 f} \]
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Rubi [A] time = 0.563279, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {1609, 1654, 833, 780, 195, 216} \[ \frac{\left (1-d^2 x^2\right )^{3/2} \left (8 \left (C \left (d^2 e^3-4 e f^2\right )-2 f \left (5 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-3 f x \left (5 f^2 \left (2 A d^2+C\right )-2 d^2 e (C e-2 B f)\right )\right )}{120 d^4 f}+\frac{x \sqrt{1-d^2 x^2} \left (2 d^2 \left (A \left (4 d^2 e^2+f^2\right )+2 B e f\right )+C \left (2 d^2 e^2+f^2\right )\right )}{16 d^4}+\frac{\sin ^{-1}(d x) \left (2 d^2 \left (A \left (4 d^2 e^2+f^2\right )+2 B e f\right )+C \left (2 d^2 e^2+f^2\right )\right )}{16 d^5}+\frac{\left (1-d^2 x^2\right )^{3/2} (e+f x)^2 (C e-2 B f)}{10 d^2 f}-\frac{C \left (1-d^2 x^2\right )^{3/2} (e+f x)^3}{6 d^2 f} \]
Antiderivative was successfully verified.
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Rule 1609
Rule 1654
Rule 833
Rule 780
Rule 195
Rule 216
Rubi steps
\begin{align*} \int \sqrt{1-d x} \sqrt{1+d x} (e+f x)^2 \left (A+B x+C x^2\right ) \, dx &=\int (e+f x)^2 \left (A+B x+C x^2\right ) \sqrt{1-d^2 x^2} \, dx\\ &=-\frac{C (e+f x)^3 \left (1-d^2 x^2\right )^{3/2}}{6 d^2 f}-\frac{\int (e+f x)^2 \left (-3 \left (C+2 A d^2\right ) f^2+3 d^2 f (C e-2 B f) x\right ) \sqrt{1-d^2 x^2} \, dx}{6 d^2 f^2}\\ &=\frac{(C e-2 B f) (e+f x)^2 \left (1-d^2 x^2\right )^{3/2}}{10 d^2 f}-\frac{C (e+f x)^3 \left (1-d^2 x^2\right )^{3/2}}{6 d^2 f}+\frac{\int (e+f x) \left (3 d^2 f^2 \left (3 C e+10 A d^2 e+4 B f\right )+3 d^2 f \left (5 \left (C+2 A d^2\right ) f^2-2 d^2 e (C e-2 B f)\right ) x\right ) \sqrt{1-d^2 x^2} \, dx}{30 d^4 f^2}\\ &=\frac{(C e-2 B f) (e+f x)^2 \left (1-d^2 x^2\right )^{3/2}}{10 d^2 f}-\frac{C (e+f x)^3 \left (1-d^2 x^2\right )^{3/2}}{6 d^2 f}+\frac{\left (8 \left (C \left (d^2 e^3-4 e f^2\right )-2 f \left (5 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-3 f \left (5 \left (C+2 A d^2\right ) f^2-2 d^2 e (C e-2 B f)\right ) x\right ) \left (1-d^2 x^2\right )^{3/2}}{120 d^4 f}+\frac{\left (C \left (2 d^2 e^2+f^2\right )+2 d^2 \left (2 B e f+A \left (4 d^2 e^2+f^2\right )\right )\right ) \int \sqrt{1-d^2 x^2} \, dx}{8 d^4}\\ &=\frac{\left (C \left (2 d^2 e^2+f^2\right )+2 d^2 \left (2 B e f+A \left (4 d^2 e^2+f^2\right )\right )\right ) x \sqrt{1-d^2 x^2}}{16 d^4}+\frac{(C e-2 B f) (e+f x)^2 \left (1-d^2 x^2\right )^{3/2}}{10 d^2 f}-\frac{C (e+f x)^3 \left (1-d^2 x^2\right )^{3/2}}{6 d^2 f}+\frac{\left (8 \left (C \left (d^2 e^3-4 e f^2\right )-2 f \left (5 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-3 f \left (5 \left (C+2 A d^2\right ) f^2-2 d^2 e (C e-2 B f)\right ) x\right ) \left (1-d^2 x^2\right )^{3/2}}{120 d^4 f}+\frac{\left (C \left (2 d^2 e^2+f^2\right )+2 d^2 \left (2 B e f+A \left (4 d^2 e^2+f^2\right )\right )\right ) \int \frac{1}{\sqrt{1-d^2 x^2}} \, dx}{16 d^4}\\ &=\frac{\left (C \left (2 d^2 e^2+f^2\right )+2 d^2 \left (2 B e f+A \left (4 d^2 e^2+f^2\right )\right )\right ) x \sqrt{1-d^2 x^2}}{16 d^4}+\frac{(C e-2 B f) (e+f x)^2 \left (1-d^2 x^2\right )^{3/2}}{10 d^2 f}-\frac{C (e+f x)^3 \left (1-d^2 x^2\right )^{3/2}}{6 d^2 f}+\frac{\left (8 \left (C \left (d^2 e^3-4 e f^2\right )-2 f \left (5 A d^2 e f+B \left (d^2 e^2+f^2\right )\right )\right )-3 f \left (5 \left (C+2 A d^2\right ) f^2-2 d^2 e (C e-2 B f)\right ) x\right ) \left (1-d^2 x^2\right )^{3/2}}{120 d^4 f}+\frac{\left (C \left (2 d^2 e^2+f^2\right )+2 d^2 \left (2 B e f+A \left (4 d^2 e^2+f^2\right )\right )\right ) \sin ^{-1}(d x)}{16 d^5}\\ \end{align*}
Mathematica [A] time = 0.325953, size = 244, normalized size = 0.85 \[ \frac{d \sqrt{1-d^2 x^2} \left (10 A d^2 \left (12 d^2 e^2 x+16 e f \left (d^2 x^2-1\right )+3 f^2 x \left (2 d^2 x^2-1\right )\right )+4 B \left (2 d^4 x^2 \left (10 e^2+15 e f x+6 f^2 x^2\right )-d^2 \left (20 e^2+15 e f x+4 f^2 x^2\right )-8 f^2\right )+C \left (30 d^2 e^2 x \left (2 d^2 x^2-1\right )+32 e f \left (3 d^4 x^4-d^2 x^2-2\right )+5 f^2 x \left (8 d^4 x^4-2 d^2 x^2-3\right )\right )\right )+15 \sin ^{-1}(d x) \left (2 d^2 \left (A \left (4 d^2 e^2+f^2\right )+2 B e f\right )+C \left (2 d^2 e^2+f^2\right )\right )}{240 d^5} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 652, normalized size = 2.3 \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 = 3.75575, size = 459, normalized size = 1.6 \begin{align*} -\frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} C f^{2} x^{3}}{6 \, d^{2}} + \frac{1}{2} \, \sqrt{-d^{2} x^{2} + 1} A e^{2} x + \frac{A e^{2} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{2 \, \sqrt{d^{2}}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} B e^{2}}{3 \, d^{2}} - \frac{2 \,{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} A e f}{3 \, d^{2}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (2 \, C e f + B f^{2}\right )} x^{2}}{5 \, d^{2}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (C e^{2} + 2 \, B e f + A f^{2}\right )} x}{4 \, d^{2}} - \frac{{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}} C f^{2} x}{8 \, d^{4}} + \frac{\sqrt{-d^{2} x^{2} + 1}{\left (C e^{2} + 2 \, B e f + A f^{2}\right )} x}{8 \, d^{2}} + \frac{\sqrt{-d^{2} x^{2} + 1} C f^{2} x}{16 \, d^{4}} + \frac{{\left (C e^{2} + 2 \, B e f + A f^{2}\right )} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{8 \, \sqrt{d^{2}} d^{2}} + \frac{C f^{2} \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{16 \, \sqrt{d^{2}} d^{4}} - \frac{2 \,{\left (-d^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (2 \, C e f + B f^{2}\right )}}{15 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.10435, size = 617, normalized size = 2.16 \begin{align*} \frac{{\left (40 \, C d^{5} f^{2} x^{5} - 80 \, B d^{3} e^{2} + 48 \,{\left (2 \, C d^{5} e f + B d^{5} f^{2}\right )} x^{4} - 32 \, B d f^{2} + 10 \,{\left (6 \, C d^{5} e^{2} + 12 \, B d^{5} e f +{\left (6 \, A d^{5} - C d^{3}\right )} f^{2}\right )} x^{3} - 32 \,{\left (5 \, A d^{3} + 2 \, C d\right )} e f + 16 \,{\left (5 \, B d^{5} e^{2} - B d^{3} f^{2} + 2 \,{\left (5 \, A d^{5} - C d^{3}\right )} e f\right )} x^{2} - 15 \,{\left (4 \, B d^{3} e f - 2 \,{\left (4 \, A d^{5} - C d^{3}\right )} e^{2} +{\left (2 \, A d^{3} + C d\right )} f^{2}\right )} x\right )} \sqrt{d x + 1} \sqrt{-d x + 1} - 30 \,{\left (4 \, B d^{2} e f + 2 \,{\left (4 \, A d^{4} + C d^{2}\right )} e^{2} +{\left (2 \, A d^{2} + C\right )} f^{2}\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right )}{240 \, d^{5}} \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 = 1.65372, size = 772, 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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