Optimal. Leaf size=59 \[ \frac {\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {d e \log \left (a+c x^2\right )}{c}+\frac {e^2 x}{c} \]
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Rubi [A] time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {702, 635, 205, 260} \begin {gather*} \frac {\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {d e \log \left (a+c x^2\right )}{c}+\frac {e^2 x}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 702
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{a+c x^2} \, dx &=\int \left (\frac {e^2}{c}+\frac {c d^2-a e^2+2 c d e x}{c \left (a+c x^2\right )}\right ) \, dx\\ &=\frac {e^2 x}{c}+\frac {\int \frac {c d^2-a e^2+2 c d e x}{a+c x^2} \, dx}{c}\\ &=\frac {e^2 x}{c}+(2 d e) \int \frac {x}{a+c x^2} \, dx+\frac {\left (c d^2-a e^2\right ) \int \frac {1}{a+c x^2} \, dx}{c}\\ &=\frac {e^2 x}{c}+\frac {\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {d e \log \left (a+c x^2\right )}{c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 0.95 \begin {gather*} \frac {\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{3/2}}+\frac {e \left (d \log \left (a+c x^2\right )+e x\right )}{c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^2}{a+c x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 137, normalized size = 2.32 \begin {gather*} \left [\frac {2 \, a c e^{2} x + 2 \, a c d e \log \left (c x^{2} + a\right ) + {\left (c d^{2} - a e^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right )}{2 \, a c^{2}}, \frac {a c e^{2} x + a c d e \log \left (c x^{2} + a\right ) + {\left (c d^{2} - a e^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right )}{a c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 52, normalized size = 0.88 \begin {gather*} \frac {d e \log \left (c x^{2} + a\right )}{c} + \frac {x e^{2}}{c} + \frac {{\left (c d^{2} - a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 65, normalized size = 1.10 \begin {gather*} -\frac {a \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, c}+\frac {d^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}+\frac {d e \ln \left (c \,x^{2}+a \right )}{c}+\frac {e^{2} x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 53, normalized size = 0.90 \begin {gather*} \frac {e^{2} x}{c} + \frac {d e \log \left (c x^{2} + a\right )}{c} + \frac {{\left (c d^{2} - a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 62, normalized size = 1.05 \begin {gather*} \frac {e^2\,x}{c}+\frac {d^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {c}}-\frac {\sqrt {a}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{c^{3/2}}+\frac {d\,e\,\ln \left (c\,x^2+a\right )}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.48, size = 185, normalized size = 3.14 \begin {gather*} \left (\frac {d e}{c} - \frac {\sqrt {- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) \log {\left (x + \frac {- 2 a c \left (\frac {d e}{c} - \frac {\sqrt {- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) + 2 a d e}{a e^{2} - c d^{2}} \right )} + \left (\frac {d e}{c} + \frac {\sqrt {- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) \log {\left (x + \frac {- 2 a c \left (\frac {d e}{c} + \frac {\sqrt {- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) + 2 a d e}{a e^{2} - c d^{2}} \right )} + \frac {e^{2} x}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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