Optimal. Leaf size=59 \[ -\frac {(b c-a d)^2}{2 b^3 (a+b x)^2}-\frac {2 d (b c-a d)}{b^3 (a+b x)}+\frac {d^2 \log (a+b x)}{b^3} \]
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Rubi [A]
time = 0.03, 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 = {640, 45}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
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 \begin {gather*} \frac {-\frac {(b c-a d) (3 a d+b (c+4 d x))}{(a+b x)^2}+2 d^2 \log (a+b x)}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 69, normalized size = 1.17
method | result | size |
risch | \(\frac {\frac {2 d \left (a d -b c \right ) x}{b^{2}}+\frac {3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}}{2 b^{3}}}{\left (b x +a \right )^{2}}+\frac {d^{2} \ln \left (b x +a \right )}{b^{3}}\) | \(67\) |
default | \(\frac {2 d \left (a d -b c \right )}{b^{3} \left (b x +a \right )}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 b^{3} \left (b x +a \right )^{2}}+\frac {d^{2} \ln \left (b x +a \right )}{b^{3}}\) | \(69\) |
norman | \(\frac {\frac {a \left (5 a^{2} b \,d^{2}-4 a \,b^{2} c d -b^{3} c^{2}\right ) x}{b^{3}}+\frac {a^{2} \left (3 a^{2} b \,d^{2}-2 a \,b^{2} c d -b^{3} c^{2}\right )}{2 b^{4}}+\frac {2 \left (a b \,d^{2}-b^{2} c d \right ) x^{3}}{b}+\frac {\left (11 a^{2} b \,d^{2}-10 a \,b^{2} c d -b^{3} c^{2}\right ) x^{2}}{2 b^{2}}}{\left (b x +a \right )^{4}}+\frac {d^{2} \ln \left (b x +a \right )}{b^{3}}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 79, normalized size = 1.34 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.79, size = 99, normalized size = 1.68 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 80, normalized size = 1.36 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.76, size = 110, normalized size = 1.86 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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