Optimal. Leaf size=103 \[ -\frac {d^2 x \left (a^2 d^2+6 b^2 c^2\right )}{b^4}-\frac {(a d+b c)^4 \log (a-b x)}{2 a b^5}+\frac {(b c-a d)^4 \log (a+b x)}{2 a b^5}-\frac {2 c d^3 x^2}{b^2}-\frac {d^4 x^3}{3 b^2} \]
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Rubi [A] time = 0.10, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {72} \[ -\frac {d^2 x \left (a^2 d^2+6 b^2 c^2\right )}{b^4}-\frac {(a d+b c)^4 \log (a-b x)}{2 a b^5}+\frac {(b c-a d)^4 \log (a+b x)}{2 a b^5}-\frac {2 c d^3 x^2}{b^2}-\frac {d^4 x^3}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin {align*} \int \frac {(c+d x)^4}{(a-b x) (a+b x)} \, dx &=\int \left (\frac {-6 b^2 c^2 d^2-a^2 d^4}{b^4}-\frac {4 c d^3 x}{b^2}-\frac {d^4 x^2}{b^2}+\frac {(b c+a d)^4}{2 a b^4 (a-b x)}+\frac {(-b c+a d)^4}{2 a b^4 (a+b x)}\right ) \, dx\\ &=-\frac {d^2 \left (6 b^2 c^2+a^2 d^2\right ) x}{b^4}-\frac {2 c d^3 x^2}{b^2}-\frac {d^4 x^3}{3 b^2}-\frac {(b c+a d)^4 \log (a-b x)}{2 a b^5}+\frac {(b c-a d)^4 \log (a+b x)}{2 a b^5}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 86, normalized size = 0.83 \[ \frac {-2 a b d^2 x \left (3 a^2 d^2+b^2 \left (18 c^2+6 c d x+d^2 x^2\right )\right )+3 (b c-a d)^4 \log (a+b x)-3 (a d+b c)^4 \log (a-b x)}{6 a b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 174, normalized size = 1.69 \[ -\frac {2 \, a b^{3} d^{4} x^{3} + 12 \, a b^{3} c d^{3} x^{2} + 6 \, {\left (6 \, a b^{3} c^{2} d^{2} + a^{3} b d^{4}\right )} x - 3 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x + a\right ) + 3 \, {\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x - a\right )}{6 \, a b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 183, normalized size = 1.78 \[ \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{2 \, a b^{5}} - \frac {{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, a b^{5}} - \frac {b^{4} d^{4} x^{3} + 6 \, b^{4} c d^{3} x^{2} + 18 \, b^{4} c^{2} d^{2} x + 3 \, a^{2} b^{2} d^{4} x}{3 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 229, normalized size = 2.22 \[ -\frac {d^{4} x^{3}}{3 b^{2}}-\frac {2 c \,d^{3} x^{2}}{b^{2}}-\frac {a^{3} d^{4} \ln \left (b x -a \right )}{2 b^{5}}+\frac {a^{3} d^{4} \ln \left (b x +a \right )}{2 b^{5}}-\frac {2 a^{2} c \,d^{3} \ln \left (b x -a \right )}{b^{4}}-\frac {2 a^{2} c \,d^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {a^{2} d^{4} x}{b^{4}}-\frac {3 a \,c^{2} d^{2} \ln \left (b x -a \right )}{b^{3}}+\frac {3 a \,c^{2} d^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {c^{4} \ln \left (b x -a \right )}{2 a b}+\frac {c^{4} \ln \left (b x +a \right )}{2 a b}-\frac {2 c^{3} d \ln \left (b x -a \right )}{b^{2}}-\frac {2 c^{3} d \ln \left (b x +a \right )}{b^{2}}-\frac {6 c^{2} d^{2} x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 179, normalized size = 1.74 \[ -\frac {b^{2} d^{4} x^{3} + 6 \, b^{2} c d^{3} x^{2} + 3 \, {\left (6 \, b^{2} c^{2} d^{2} + a^{2} d^{4}\right )} x}{3 \, b^{4}} + \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x + a\right )}{2 \, a b^{5}} - \frac {{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (b x - a\right )}{2 \, a b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 176, normalized size = 1.71 \[ \frac {\ln \left (a+b\,x\right )\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{2\,a\,b^5}-\frac {d^4\,x^3}{3\,b^2}-\frac {2\,c\,d^3\,x^2}{b^2}-x\,\left (\frac {a^2\,d^4}{b^4}+\frac {6\,c^2\,d^2}{b^2}\right )-\frac {\ln \left (a-b\,x\right )\,\left (a^4\,d^4+4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{2\,a\,b^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.33, size = 214, normalized size = 2.08 \[ - x \left (\frac {a^{2} d^{4}}{b^{4}} + \frac {6 c^{2} d^{2}}{b^{2}}\right ) - \frac {2 c d^{3} x^{2}}{b^{2}} - \frac {d^{4} x^{3}}{3 b^{2}} + \frac {\left (a d - b c\right )^{4} \log {\left (x + \frac {4 a^{4} c d^{3} + 4 a^{2} b^{2} c^{3} d + \frac {a \left (a d - b c\right )^{4}}{b}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} + b^{4} c^{4}} \right )}}{2 a b^{5}} - \frac {\left (a d + b c\right )^{4} \log {\left (x + \frac {4 a^{4} c d^{3} + 4 a^{2} b^{2} c^{3} d - \frac {a \left (a d + b c\right )^{4}}{b}}{a^{4} d^{4} + 6 a^{2} b^{2} c^{2} d^{2} + b^{4} c^{4}} \right )}}{2 a b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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