Optimal. Leaf size=309 \[ \frac{2 \left (-x \left (c^2 \left (2 a^2 f^2+6 a b e f+b^2 \left (2 d f+e^2\right )\right )-2 b^2 c f (2 a f+b e)-2 c^3 \left (a \left (2 d f+e^2\right )+b d e\right )+b^4 f^2+2 c^4 d^2\right )-b c \left (-3 a^2 f^2+a c \left (2 d f+e^2\right )+c^2 d^2\right )+2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)\right )}{c^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-12 c f (a f+2 b e)+15 b^2 f^2+8 c^2 \left (2 d f+e^2\right )\right )}{8 c^{7/2}}+\frac{f \sqrt{a+b x+c x^2} (8 c e-7 b f)}{4 c^3}+\frac{f^2 x \sqrt{a+b x+c x^2}}{2 c^2} \]
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Rubi [A] time = 0.446654, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1660, 1661, 640, 621, 206} \[ \frac{2 \left (-x \left (c^2 \left (2 a^2 f^2+6 a b e f+b^2 \left (2 d f+e^2\right )\right )-2 b^2 c f (2 a f+b e)-2 c^3 \left (a \left (2 d f+e^2\right )+b d e\right )+b^4 f^2+2 c^4 d^2\right )-b c \left (-3 a^2 f^2+a c \left (2 d f+e^2\right )+c^2 d^2\right )+2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)\right )}{c^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-12 c f (a f+2 b e)+15 b^2 f^2+8 c^2 \left (2 d f+e^2\right )\right )}{8 c^{7/2}}+\frac{f \sqrt{a+b x+c x^2} (8 c e-7 b f)}{4 c^3}+\frac{f^2 x \sqrt{a+b x+c x^2}}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 1661
Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (d+e x+f x^2\right )^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac{2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{-\frac{\left (b^2-4 a c\right ) \left (b^2 f^2-c f (2 b e+a f)+c^2 \left (e^2+2 d f\right )\right )}{2 c^3}-\frac{\left (b^2-4 a c\right ) f (2 c e-b f) x}{2 c^2}-\frac{\left (b^2-4 a c\right ) f^2 x^2}{2 c}}{\sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=\frac{2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f^2 x \sqrt{a+b x+c x^2}}{2 c^2}-\frac{\int \frac{-\frac{\left (b^2-4 a c\right ) \left (2 b^2 f^2-c f (4 b e+3 a f)+2 c^2 \left (e^2+2 d f\right )\right )}{2 c^2}-\frac{\left (b^2-4 a c\right ) f (8 c e-7 b f) x}{4 c}}{\sqrt{a+b x+c x^2}} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac{2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f (8 c e-7 b f) \sqrt{a+b x+c x^2}}{4 c^3}+\frac{f^2 x \sqrt{a+b x+c x^2}}{2 c^2}+\frac{\left (15 b^2 f^2-12 c f (2 b e+a f)+8 c^2 \left (e^2+2 d f\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{8 c^3}\\ &=\frac{2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f (8 c e-7 b f) \sqrt{a+b x+c x^2}}{4 c^3}+\frac{f^2 x \sqrt{a+b x+c x^2}}{2 c^2}+\frac{\left (15 b^2 f^2-12 c f (2 b e+a f)+8 c^2 \left (e^2+2 d f\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{4 c^3}\\ &=\frac{2 \left (2 a b^2 c e f-a b^3 f^2+4 a c^2 e (c d-a f)-b c \left (c^2 d^2-3 a^2 f^2+a c \left (e^2+2 d f\right )\right )-\left (2 c^4 d^2+b^4 f^2-2 b^2 c f (b e+2 a f)-2 c^3 \left (b d e+a \left (e^2+2 d f\right )\right )+c^2 \left (6 a b e f+2 a^2 f^2+b^2 \left (e^2+2 d f\right )\right )\right ) x\right )}{c^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f (8 c e-7 b f) \sqrt{a+b x+c x^2}}{4 c^3}+\frac{f^2 x \sqrt{a+b x+c x^2}}{2 c^2}+\frac{\left (15 b^2 f^2-12 c f (2 b e+a f)+8 c^2 \left (e^2+2 d f\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.756993, size = 288, normalized size = 0.93 \[ \frac{4 b c \left (-13 a^2 f^2+a c \left (4 d f+2 e^2+20 e f x-5 f^2 x^2\right )+2 c^2 d (d-2 e x)\right )+8 c^2 \left (a^2 f (8 e+3 f x)+a c \left (x \left (-2 e^2+4 e f x+f^2 x^2\right )-4 d (e+f x)\right )+2 c^2 d^2 x\right )-2 b^2 c \left (a f (12 e+31 f x)+c x \left (-8 d f-4 e^2+4 e f x+f^2 x^2\right )\right )+b^3 f (15 a f+c x (5 f x-24 e))+15 b^4 f^2 x}{4 c^3 \left (4 a c-b^2\right ) \sqrt{a+x (b+c x)}}+\frac{\log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (-12 c f (a f+2 b e)+15 b^2 f^2+8 c^2 \left (2 d f+e^2\right )\right )}{8 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.067, size = 1011, normalized size = 3.3 \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 [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 [B] time = 9.59512, size = 2750, normalized size = 8.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x + f x^{2}\right )^{2}}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.65767, size = 549, normalized size = 1.78 \begin{align*} \frac{{\left ({\left (\frac{2 \,{\left (b^{2} c^{2} f^{2} - 4 \, a c^{3} f^{2}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} - \frac{5 \, b^{3} c f^{2} - 20 \, a b c^{2} f^{2} - 8 \, b^{2} c^{2} f e + 32 \, a c^{3} f e}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac{16 \, c^{4} d^{2} + 16 \, b^{2} c^{2} d f - 32 \, a c^{3} d f + 15 \, b^{4} f^{2} - 62 \, a b^{2} c f^{2} + 24 \, a^{2} c^{2} f^{2} - 16 \, b c^{3} d e - 24 \, b^{3} c f e + 80 \, a b c^{2} f e + 8 \, b^{2} c^{2} e^{2} - 16 \, a c^{3} e^{2}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac{8 \, b c^{3} d^{2} + 16 \, a b c^{2} d f + 15 \, a b^{3} f^{2} - 52 \, a^{2} b c f^{2} - 32 \, a c^{3} d e - 24 \, a b^{2} c f e + 64 \, a^{2} c^{2} f e + 8 \, a b c^{2} e^{2}}{b^{2} c^{3} - 4 \, a c^{4}}}{4 \, \sqrt{c x^{2} + b x + a}} - \frac{{\left (16 \, c^{2} d f + 15 \, b^{2} f^{2} - 12 \, a c f^{2} - 24 \, b c f e + 8 \, c^{2} e^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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