Optimal. Leaf size=50 \[ \frac {\left (a^2-c\right ) \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{b^3 \sqrt {c}}-\frac {a \log \left ((a+b x)^2+c\right )}{b^3}+\frac {x}{b^2} \]
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Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {371, 702, 635, 203, 260} \[ \frac {\left (a^2-c\right ) \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{b^3 \sqrt {c}}-\frac {a \log \left ((a+b x)^2+c\right )}{b^3}+\frac {x}{b^2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 371
Rule 635
Rule 702
Rubi steps
\begin {align*} \int \frac {x^2}{c+(a+b x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-a+x)^2}{c+x^2} \, dx,x,a+b x\right )}{b^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {a^2-c-2 a x}{c+x^2}\right ) \, dx,x,a+b x\right )}{b^3}\\ &=\frac {x}{b^2}+\frac {\operatorname {Subst}\left (\int \frac {a^2-c-2 a x}{c+x^2} \, dx,x,a+b x\right )}{b^3}\\ &=\frac {x}{b^2}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {x}{c+x^2} \, dx,x,a+b x\right )}{b^3}+\frac {\left (a^2-c\right ) \operatorname {Subst}\left (\int \frac {1}{c+x^2} \, dx,x,a+b x\right )}{b^3}\\ &=\frac {x}{b^2}+\frac {\left (a^2-c\right ) \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{b^3 \sqrt {c}}-\frac {a \log \left (c+(a+b x)^2\right )}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 54, normalized size = 1.08 \[ \frac {-a \log \left (a^2+2 a b x+b^2 x^2+c\right )+\frac {\left (a^2-c\right ) \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c}}+b x}{b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 157, normalized size = 3.14 \[ \left [\frac {2 \, b c x - 2 \, a c \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) + {\left (a^{2} - c\right )} \sqrt {-c} \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 2 \, {\left (b x + a\right )} \sqrt {-c} - c}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right )}{2 \, b^{3} c}, \frac {b c x - a c \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) + {\left (a^{2} - c\right )} \sqrt {c} \arctan \left (\frac {b x + a}{\sqrt {c}}\right )}{b^{3} c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 54, normalized size = 1.08 \[ \frac {x}{b^{2}} - \frac {a \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{b^{3}} + \frac {{\left (a^{2} - c\right )} \arctan \left (\frac {b x + a}{\sqrt {c}}\right )}{b^{3} \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 89, normalized size = 1.78 \[ \frac {a^{2} \arctan \left (\frac {2 b^{2} x +2 b a}{2 b \sqrt {c}}\right )}{b^{3} \sqrt {c}}-\frac {a \ln \left (b^{2} x^{2}+2 a b x +a^{2}+c \right )}{b^{3}}+\frac {x}{b^{2}}-\frac {\sqrt {c}\, \arctan \left (\frac {2 b^{2} x +2 b a}{2 b \sqrt {c}}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.65, size = 61, normalized size = 1.22 \[ \frac {x}{b^{2}} - \frac {a \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{b^{3}} + \frac {{\left (a^{2} - c\right )} \arctan \left (\frac {b^{2} x + a b}{b \sqrt {c}}\right )}{b^{3} \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 206, normalized size = 4.12 \[ \frac {x}{b^2}-\frac {a\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+c\right )}{b^3}+\frac {\sqrt {c}\,\mathrm {atan}\left (\frac {a^3}{\sqrt {c}\,\left (c-a^2\right )}-\frac {\sqrt {c}\,x}{\frac {c}{b}-\frac {a^2}{b}}-\frac {a\,\sqrt {c}}{c-a^2}+\frac {a^2\,x}{\sqrt {c}\,\left (\frac {c}{b}-\frac {a^2}{b}\right )}\right )}{b^3}-\frac {a^2\,\mathrm {atan}\left (\frac {a^3}{\sqrt {c}\,\left (c-a^2\right )}-\frac {\sqrt {c}\,x}{\frac {c}{b}-\frac {a^2}{b}}-\frac {a\,\sqrt {c}}{c-a^2}+\frac {a^2\,x}{\sqrt {c}\,\left (\frac {c}{b}-\frac {a^2}{b}\right )}\right )}{b^3\,\sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.46, size = 153, normalized size = 3.06 \[ \left (- \frac {a}{b^{3}} - \frac {\sqrt {- c} \left (a^{2} - c\right )}{2 b^{3} c}\right ) \log {\left (x + \frac {a^{3} + a c + 2 b^{3} c \left (- \frac {a}{b^{3}} - \frac {\sqrt {- c} \left (a^{2} - c\right )}{2 b^{3} c}\right )}{a^{2} b - b c} \right )} + \left (- \frac {a}{b^{3}} + \frac {\sqrt {- c} \left (a^{2} - c\right )}{2 b^{3} c}\right ) \log {\left (x + \frac {a^{3} + a c + 2 b^{3} c \left (- \frac {a}{b^{3}} + \frac {\sqrt {- c} \left (a^{2} - c\right )}{2 b^{3} c}\right )}{a^{2} b - b c} \right )} + \frac {x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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