Optimal. Leaf size=51 \[ -\frac {(b c-a d)^2}{b^3 (a+b x)}+\frac {2 d (b c-a d) \log (a+b x)}{b^3}+\frac {d^2 x}{b^2} \]
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Rubi [A] time = 0.04, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {626, 43} \[ -\frac {(b c-a d)^2}{b^3 (a+b x)}+\frac {2 d (b c-a d) \log (a+b x)}{b^3}+\frac {d^2 x}{b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 626
Rubi steps
\begin {align*} \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^4} \, dx &=\int \frac {(c+d x)^2}{(a+b x)^2} \, dx\\ &=\int \left (\frac {d^2}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)^2}+\frac {2 d (b c-a d)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {d^2 x}{b^2}-\frac {(b c-a d)^2}{b^3 (a+b x)}+\frac {2 d (b c-a d) \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 0.92 \[ \frac {-\frac {(b c-a d)^2}{a+b x}+2 d (b c-a d) \log (a+b x)+b d^2 x}{b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 92, normalized size = 1.80 \[ \frac {b^{2} d^{2} x^{2} + a b d^{2} x - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} + 2 \, {\left (a b c d - a^{2} d^{2} + {\left (b^{2} c d - a b d^{2}\right )} x\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 65, normalized size = 1.27 \[ \frac {d^{2} x}{b^{2}} + \frac {2 \, {\left (b c d - a d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} - \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{{\left (b x + a\right )} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 86, normalized size = 1.69 \[ -\frac {a^{2} d^{2}}{\left (b x +a \right ) b^{3}}+\frac {2 a c d}{\left (b x +a \right ) b^{2}}-\frac {2 a \,d^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {c^{2}}{\left (b x +a \right ) b}+\frac {2 c d \ln \left (b x +a \right )}{b^{2}}+\frac {d^{2} x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 67, normalized size = 1.31 \[ \frac {d^{2} x}{b^{2}} - \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{b^{4} x + a b^{3}} + \frac {2 \, {\left (b c d - a d^{2}\right )} \log \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 71, normalized size = 1.39 \[ \frac {d^2\,x}{b^2}-\frac {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}{b\,\left (x\,b^3+a\,b^2\right )}-\frac {\ln \left (a+b\,x\right )\,\left (2\,a\,d^2-2\,b\,c\,d\right )}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 60, normalized size = 1.18 \[ \frac {- a^{2} d^{2} + 2 a b c d - b^{2} c^{2}}{a b^{3} + b^{4} x} + \frac {d^{2} x}{b^{2}} - \frac {2 d \left (a d - b c\right ) \log {\left (a + b x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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