Optimal. Leaf size=609 \[ -\frac{2 \left (c x \left (-a b e-2 a (c d-a f)+b^2 d\right )+a (a b f-2 a c e+b c d)\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac{f \left (2 d (c d-a f)-\left (e-\sqrt{e^2-4 d f}\right ) (b d-a e)\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} \sqrt{e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{f \left (2 d (c d-a f)-\left (\sqrt{e^2-4 d f}+e\right ) (b d-a e)\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} \sqrt{e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}} \]
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Rubi [A] time = 5.84168, antiderivative size = 609, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1061, 1032, 724, 206} \[ -\frac{2 \left (c x \left (-a b e-2 a (c d-a f)+b^2 d\right )+a (a b f-2 a c e+b c d)\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac{f \left (2 d (c d-a f)-\left (e-\sqrt{e^2-4 d f}\right ) (b d-a e)\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} \sqrt{e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{f \left (2 d (c d-a f)-\left (\sqrt{e^2-4 d f}+e\right ) (b d-a e)\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} \sqrt{e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}} \]
Antiderivative was successfully verified.
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Rule 1061
Rule 1032
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )} \, dx &=-\frac{2 \left (a (b c d-2 a c e+a b f)+c \left (b^2 d-a b e-2 a (c d-a f)\right ) x\right )}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{-\frac{1}{2} \left (b^2-4 a c\right ) d (c d-a f)-\frac{1}{2} \left (b^2-4 a c\right ) (b d-a e) f x}{\sqrt{a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\\ &=-\frac{2 \left (a (b c d-2 a c e+a b f)+c \left (b^2 d-a b e-2 a (c d-a f)\right ) x\right )}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt{a+b x+c x^2}}+\frac{\left (f \left (2 d (c d-a f)-(b d-a e) \left (e-\sqrt{e^2-4 d f}\right )\right )\right ) \int \frac{1}{\left (e-\sqrt{e^2-4 d f}+2 f x\right ) \sqrt{a+b x+c x^2}} \, dx}{\sqrt{e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}-\frac{\left (f \left (2 d (c d-a f)-(b d-a e) \left (e+\sqrt{e^2-4 d f}\right )\right )\right ) \int \frac{1}{\left (e+\sqrt{e^2-4 d f}+2 f x\right ) \sqrt{a+b x+c x^2}} \, dx}{\sqrt{e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\\ &=-\frac{2 \left (a (b c d-2 a c e+a b f)+c \left (b^2 d-a b e-2 a (c d-a f)\right ) x\right )}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt{a+b x+c x^2}}-\frac{\left (2 f \left (2 d (c d-a f)-(b d-a e) \left (e-\sqrt{e^2-4 d f}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{16 a f^2-8 b f \left (e-\sqrt{e^2-4 d f}\right )+4 c \left (e-\sqrt{e^2-4 d f}\right )^2-x^2} \, dx,x,\frac{4 a f-b \left (e-\sqrt{e^2-4 d f}\right )-\left (-2 b f+2 c \left (e-\sqrt{e^2-4 d f}\right )\right ) x}{\sqrt{a+b x+c x^2}}\right )}{\sqrt{e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}+\frac{\left (2 f \left (2 d (c d-a f)-(b d-a e) \left (e+\sqrt{e^2-4 d f}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{16 a f^2-8 b f \left (e+\sqrt{e^2-4 d f}\right )+4 c \left (e+\sqrt{e^2-4 d f}\right )^2-x^2} \, dx,x,\frac{4 a f-b \left (e+\sqrt{e^2-4 d f}\right )-\left (-2 b f+2 c \left (e+\sqrt{e^2-4 d f}\right )\right ) x}{\sqrt{a+b x+c x^2}}\right )}{\sqrt{e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right )}\\ &=-\frac{2 \left (a (b c d-2 a c e+a b f)+c \left (b^2 d-a b e-2 a (c d-a f)\right ) x\right )}{\left (b^2-4 a c\right ) \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt{a+b x+c x^2}}-\frac{f \left (2 d (c d-a f)-(b d-a e) \left (e-\sqrt{e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac{4 a f-b \left (e-\sqrt{e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right ) x}{2 \sqrt{2} \sqrt{c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt{e^2-4 d f}} \sqrt{a+b x+c x^2}}\right )}{\sqrt{2} \sqrt{e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt{c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt{e^2-4 d f}}}+\frac{f \left (2 d (c d-a f)-(b d-a e) \left (e+\sqrt{e^2-4 d f}\right )\right ) \tanh ^{-1}\left (\frac{4 a f-b \left (e+\sqrt{e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt{e^2-4 d f}\right )\right ) x}{2 \sqrt{2} \sqrt{c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt{e^2-4 d f}} \sqrt{a+b x+c x^2}}\right )}{\sqrt{2} \sqrt{e^2-4 d f} \left ((c d-a f)^2-(b d-a e) (c e-b f)\right ) \sqrt{c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt{e^2-4 d f}}}\\ \end{align*}
Mathematica [A] time = 6.61074, size = 1097, normalized size = 1.8 \[ \frac{16 \sqrt{2} f \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \left (e+\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}\right ) \tanh ^{-1}\left (\frac{4 a f-b \left (e-\sqrt{e^2-4 d f}\right )-\left (2 c \left (e-\sqrt{e^2-4 d f}\right )-2 b f\right ) x}{2 \sqrt{2} \sqrt{c e^2-b f e-c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f+b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right ) \left (c x^2+b x+a\right )^{3/2}}{\left (4 a f^2-2 b \left (e-\sqrt{e^2-4 d f}\right ) f+c \left (e-\sqrt{e^2-4 d f}\right )^2\right ) \left (16 a f^2-8 b \left (e-\sqrt{e^2-4 d f}\right ) f+4 c \left (e-\sqrt{e^2-4 d f}\right )^2\right ) (a+x (b+c x))^{3/2}}+\frac{16 \sqrt{2} f \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \left (e-\frac{2 d f-e^2}{\sqrt{e^2-4 d f}}\right ) \tanh ^{-1}\left (\frac{4 a f-b \left (e+\sqrt{e^2-4 d f}\right )-\left (2 c \left (e+\sqrt{e^2-4 d f}\right )-2 b f\right ) x}{2 \sqrt{2} \sqrt{c e^2-b f e+c \sqrt{e^2-4 d f} e+2 a f^2-2 c d f-b f \sqrt{e^2-4 d f}} \sqrt{c x^2+b x+a}}\right ) \left (c x^2+b x+a\right )^{3/2}}{\left (4 a f^2-2 b \left (e+\sqrt{e^2-4 d f}\right ) f+c \left (e+\sqrt{e^2-4 d f}\right )^2\right ) \left (16 a f^2-8 b \left (e+\sqrt{e^2-4 d f}\right ) f+4 c \left (e+\sqrt{e^2-4 d f}\right )^2\right ) (a+x (b+c x))^{3/2}}-\frac{2 \left (e-\frac{e^2-2 d f}{\sqrt{e^2-4 d f}}\right ) \left (2 f b^2-c \left (e-\sqrt{e^2-4 d f}\right ) b-4 a c f+2 c \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right ) x\right ) \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) f \left (4 a f^2-2 b \left (e-\sqrt{e^2-4 d f}\right ) f+c \left (e-\sqrt{e^2-4 d f}\right )^2\right ) (a+x (b+c x))^{3/2}}-\frac{2 \left (e+\frac{e^2-2 d f}{\sqrt{e^2-4 d f}}\right ) \left (2 f b^2-c \left (e+\sqrt{e^2-4 d f}\right ) b-4 a c f+2 c \left (b f-c \left (e+\sqrt{e^2-4 d f}\right )\right ) x\right ) \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) f \left (4 a f^2-2 b \left (e+\sqrt{e^2-4 d f}\right ) f+c \left (e+\sqrt{e^2-4 d f}\right )^2\right ) (a+x (b+c x))^{3/2}}+\frac{4 (b+2 c x) \left (-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}\right )^{3/2}}{c f (a+x (b+c x))^{3/2} \sqrt{1-\frac{(b+2 c x)^2}{b^2-4 a c}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.348, size = 11341, normalized size = 18.6 \begin{align*} \text{output 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 [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (d + e x + f x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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