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 = 5.84, antiderivative size = 609, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 1032
Rule 1061
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*}
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Mathematica [A] time = 6.59, size = 1097, normalized size = 1.80 \begin {gather*} \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}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 1.40, size = 570, normalized size = 0.94 \begin {gather*} \frac {\text {RootSum}\left [\text {$\#$1}^4 f-2 \text {$\#$1}^3 \sqrt {c} e-2 \text {$\#$1}^2 a f+\text {$\#$1}^2 b e+4 \text {$\#$1}^2 c d+2 \text {$\#$1} a \sqrt {c} e-4 \text {$\#$1} b \sqrt {c} d+a^2 f-a b e+b^2 d\&,\frac {\text {$\#$1}^2 b d f \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )-\text {$\#$1}^2 a e f \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )+a^2 e f \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )-2 \text {$\#$1} c^{3/2} d^2 \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )+b c d^2 \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )-2 a b d f \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )+2 \text {$\#$1} a \sqrt {c} d f \log \left (-\text {$\#$1}+\sqrt {a+b x+c x^2}-\sqrt {c} x\right )}{-2 \text {$\#$1}^3 f+3 \text {$\#$1}^2 \sqrt {c} e+2 \text {$\#$1} a f-\text {$\#$1} b e-4 \text {$\#$1} c d-a \sqrt {c} e+2 b \sqrt {c} d}\&\right ]}{-a^2 f^2+a b e f+2 a c d f-a c e^2+b^2 (-d) f+b c d e-c^2 d^2}-\frac {2 \left (a^2 b f-2 a^2 c e+2 a^2 c f x+a b c d-a b c e x-2 a c^2 d x+b^2 c d x\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a^2 f^2-a b e f-2 a c d f+a c e^2+b^2 d f-b c d e+c^2 d^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 11341, normalized size = 18.62 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{{\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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