Optimal. Leaf size=484 \[ -\frac {\sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \tan ^{-1}\left (\frac {b \sqrt {a^2-2 a c+b^2+c^2}-x \left ((a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right )}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2}}-\frac {\sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \tanh ^{-1}\left (\frac {x \left ((a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right )+b \sqrt {a^2-2 a c+b^2+c^2}}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2}} \]
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Rubi [A] time = 23.58, antiderivative size = 484, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1036, 1030, 208, 205} \[ -\frac {\sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \tan ^{-1}\left (\frac {b \sqrt {a^2-2 a c+b^2+c^2}-x \left ((a-c) \left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right )}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (2 c-\sqrt {a^2-2 a c+b^2+c^2}\right )+c \left (c-\sqrt {a^2-2 a c+b^2+c^2}\right )+a^2+b^2} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2}}-\frac {\sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \tanh ^{-1}\left (\frac {x \left ((a-c) \left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right )+b^2\right )+b \sqrt {a^2-2 a c+b^2+c^2}}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2} \sqrt {-a \left (\sqrt {a^2-2 a c+b^2+c^2}+2 c\right )+c \left (\sqrt {a^2-2 a c+b^2+c^2}+c\right )+a^2+b^2} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 a c+b^2+c^2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 1030
Rule 1036
Rubi steps
\begin {align*} \int \frac {a-c+b x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx &=-\frac {\int \frac {-b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx}{2 \sqrt {a^2+b^2-2 a c+c^2}}+\frac {\int \frac {-b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx}{2 \sqrt {a^2+b^2-2 a c+c^2}}\\ &=\left (b \left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-2 b \sqrt {a^2+b^2-2 a c+c^2} \left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac {b \sqrt {a^2+b^2-2 a c+c^2}+\left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) x}{\sqrt {a+b x+c x^2}}\right )+\left (b \left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 b \sqrt {a^2+b^2-2 a c+c^2} \left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac {-b \sqrt {a^2+b^2-2 a c+c^2}+\left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) x}{\sqrt {a+b x+c x^2}}\right )\\ &=-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \tan ^{-1}\left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) x}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2}}-\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \tanh ^{-1}\left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}+\left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) x}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 136, normalized size = 0.28 \[ \frac {1}{2} i \left (\sqrt {a+i b-c} \tanh ^{-1}\left (\frac {2 a+b (x+i)+2 i c x}{2 \sqrt {a+i b-c} \sqrt {a+x (b+c x)}}\right )-\sqrt {a-i b-c} \tanh ^{-1}\left (\frac {2 a+b (x-i)-2 i c x}{2 \sqrt {a-i b-c} \sqrt {a+x (b+c x)}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.52, size = 6871419, normalized size = 14197.15 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b x + a - c}{\sqrt {c x^{2} + b x + a} {\left (x^{2} + 1\right )}}\,{d x} \]
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 \[ \int \frac {a-c+b\,x}{\left (x^2+1\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b x - c}{\left (x^{2} + 1\right ) \sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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