Optimal. Leaf size=59 \[ -\frac {2 d (b c-a d)}{b^3 (a+b x)}-\frac {(b c-a d)^2}{2 b^3 (a+b x)^2}+\frac {d^2 \log (a+b x)}{b^3} \]
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Rubi [A] time = 0.04, antiderivative size = 59, 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} \[ -\frac {2 d (b c-a d)}{b^3 (a+b x)}-\frac {(b c-a d)^2}{2 b^3 (a+b x)^2}+\frac {d^2 \log (a+b x)}{b^3} \]
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)^5} \, dx &=\int \frac {(c+d x)^2}{(a+b x)^3} \, dx\\ &=\int \left (\frac {(b c-a d)^2}{b^2 (a+b x)^3}+\frac {2 d (b c-a d)}{b^2 (a+b x)^2}+\frac {d^2}{b^2 (a+b x)}\right ) \, dx\\ &=-\frac {(b c-a d)^2}{2 b^3 (a+b x)^2}-\frac {2 d (b c-a d)}{b^3 (a+b x)}+\frac {d^2 \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.83 \[ \frac {2 d^2 \log (a+b x)-\frac {(b c-a d) (3 a d+b (c+4 d x))}{(a+b x)^2}}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 99, normalized size = 1.68 \[ -\frac {b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 4 \, {\left (b^{2} c d - a b d^{2}\right )} x - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 110, normalized size = 1.86 \[ -\frac {d^{2} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{3}} - \frac {\frac {b^{5} c^{2}}{{\left (b x + a\right )}^{2}} + \frac {4 \, b^{4} c d}{b x + a} - \frac {2 \, a b^{4} c d}{{\left (b x + a\right )}^{2}} - \frac {4 \, a b^{3} d^{2}}{b x + a} + \frac {a^{2} b^{3} d^{2}}{{\left (b x + a\right )}^{2}}}{2 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 92, normalized size = 1.56 \[ -\frac {a^{2} d^{2}}{2 \left (b x +a \right )^{2} b^{3}}+\frac {a c d}{\left (b x +a \right )^{2} b^{2}}-\frac {c^{2}}{2 \left (b x +a \right )^{2} b}+\frac {2 a \,d^{2}}{\left (b x +a \right ) b^{3}}-\frac {2 c d}{\left (b x +a \right ) b^{2}}+\frac {d^{2} \ln \left (b x +a \right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 79, normalized size = 1.34 \[ -\frac {b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 4 \, {\left (b^{2} c d - a b d^{2}\right )} x}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac {d^{2} \log \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 77, normalized size = 1.31 \[ \frac {d^2\,\ln \left (a+b\,x\right )}{b^3}-\frac {\frac {-3\,a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2}{2\,b^3}-\frac {2\,d\,x\,\left (a\,d-b\,c\right )}{b^2}}{a^2+2\,a\,b\,x+b^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.54, size = 80, normalized size = 1.36 \[ \frac {3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2} + x \left (4 a b d^{2} - 4 b^{2} c d\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {d^{2} \log {\left (a + b x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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