Optimal. Leaf size=72 \[ \frac {\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}-\frac {(d+e x) (a e-c d x)}{2 a c \left (a+c x^2\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {723, 205} \[ \frac {\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}-\frac {(d+e x) (a e-c d x)}{2 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 205
Rule 723
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx &=-\frac {(a e-c d x) (d+e x)}{2 a c \left (a+c x^2\right )}+\frac {\left (c d^2+a e^2\right ) \int \frac {1}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {(a e-c d x) (d+e x)}{2 a c \left (a+c x^2\right )}+\frac {\left (c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 77, normalized size = 1.07 \[ \frac {\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{3/2}}+\frac {-2 a d e-a e^2 x+c d^2 x}{2 a c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 223, normalized size = 3.10 \[ \left [-\frac {4 \, a^{2} c d e + {\left (a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} x}{4 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}, -\frac {2 \, a^{2} c d e - {\left (a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} x}{2 \, {\left (a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 69, normalized size = 0.96 \[ \frac {{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} + \frac {c d^{2} x - a x e^{2} - 2 \, a d e}{2 \, {\left (c x^{2} + a\right )} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 85, normalized size = 1.18 \[ \frac {d^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, a}+\frac {e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c}+\frac {-\frac {d e}{c}-\frac {\left (a \,e^{2}-c \,d^{2}\right ) x}{2 a c}}{c \,x^{2}+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 73, normalized size = 1.01 \[ -\frac {2 \, a d e - {\left (c d^{2} - a e^{2}\right )} x}{2 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} + \frac {{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 68, normalized size = 0.94 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (c\,d^2+a\,e^2\right )}{2\,a^{3/2}\,c^{3/2}}-\frac {\frac {d\,e}{c}+\frac {x\,\left (a\,e^2-c\,d^2\right )}{2\,a\,c}}{c\,x^2+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.56, size = 129, normalized size = 1.79 \[ - \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log {\left (- a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log {\left (a^{2} c \sqrt {- \frac {1}{a^{3} c^{3}}} + x \right )}}{4} + \frac {- 2 a d e + x \left (- a e^{2} + c d^{2}\right )}{2 a^{2} c + 2 a c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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