Optimal. Leaf size=145 \[ -\frac {(b c+a d) \left (b^2 c^2+a^2 d^2\right ) x}{b^4 d^4}+\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) x^2}{2 b^3 d^3}-\frac {(b c+a d) x^3}{3 b^2 d^2}+\frac {x^4}{4 b d}-\frac {a^5 \log (a+b x)}{b^5 (b c-a d)}+\frac {c^5 \log (c+d x)}{d^5 (b c-a d)} \]
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Rubi [A]
time = 0.11, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {84}
\begin {gather*} -\frac {a^5 \log (a+b x)}{b^5 (b c-a d)}-\frac {x (a d+b c) \left (a^2 d^2+b^2 c^2\right )}{b^4 d^4}+\frac {x^2 \left (a^2 d^2+a b c d+b^2 c^2\right )}{2 b^3 d^3}-\frac {x^3 (a d+b c)}{3 b^2 d^2}+\frac {c^5 \log (c+d x)}{d^5 (b c-a d)}+\frac {x^4}{4 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rubi steps
\begin {align*} \int \frac {x^5}{(a+b x) (c+d x)} \, dx &=\int \left (\frac {(b c+a d) \left (-b^2 c^2-a^2 d^2\right )}{b^4 d^4}+\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) x}{b^3 d^3}-\frac {(b c+a d) x^2}{b^2 d^2}+\frac {x^3}{b d}-\frac {a^5}{b^4 (b c-a d) (a+b x)}-\frac {c^5}{d^4 (-b c+a d) (c+d x)}\right ) \, dx\\ &=-\frac {(b c+a d) \left (b^2 c^2+a^2 d^2\right ) x}{b^4 d^4}+\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) x^2}{2 b^3 d^3}-\frac {(b c+a d) x^3}{3 b^2 d^2}+\frac {x^4}{4 b d}-\frac {a^5 \log (a+b x)}{b^5 (b c-a d)}+\frac {c^5 \log (c+d x)}{d^5 (b c-a d)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 133, normalized size = 0.92 \begin {gather*} \frac {b d x \left (12 a^4 d^4-6 a^3 b d^4 x+4 a^2 b^2 d^4 x^2-3 a b^3 d^4 x^3+b^4 c \left (-12 c^3+6 c^2 d x-4 c d^2 x^2+3 d^3 x^3\right )\right )-12 a^5 d^5 \log (a+b x)+12 b^5 c^5 \log (c+d x)}{12 b^5 d^5 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 166, normalized size = 1.14
method | result | size |
norman | \(\frac {x^{4}}{4 b d}-\frac {\left (a d +b c \right ) x^{3}}{3 b^{2} d^{2}}+\frac {\left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x^{2}}{2 b^{3} d^{3}}-\frac {\left (a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x}{b^{4} d^{4}}+\frac {a^{5} \ln \left (b x +a \right )}{b^{5} \left (a d -b c \right )}-\frac {c^{5} \ln \left (d x +c \right )}{d^{5} \left (a d -b c \right )}\) | \(151\) |
default | \(-\frac {-\frac {1}{4} d^{3} x^{4} b^{3}+\frac {1}{3} a \,b^{2} d^{3} x^{3}+\frac {1}{3} b^{3} c \,d^{2} x^{3}-\frac {1}{2} a^{2} b \,d^{3} x^{2}-\frac {1}{2} a \,b^{2} c \,d^{2} x^{2}-\frac {1}{2} b^{3} c^{2} d \,x^{2}+a^{3} d^{3} x +a^{2} b c \,d^{2} x +a \,b^{2} c^{2} d x +b^{3} c^{3} x}{b^{4} d^{4}}+\frac {a^{5} \ln \left (b x +a \right )}{b^{5} \left (a d -b c \right )}-\frac {c^{5} \ln \left (d x +c \right )}{d^{5} \left (a d -b c \right )}\) | \(166\) |
risch | \(\frac {x^{4}}{4 b d}-\frac {a \,x^{3}}{3 b^{2} d}-\frac {c \,x^{3}}{3 b \,d^{2}}+\frac {a^{2} x^{2}}{2 b^{3} d}+\frac {a c \,x^{2}}{2 b^{2} d^{2}}+\frac {c^{2} x^{2}}{2 b \,d^{3}}-\frac {a^{3} x}{b^{4} d}-\frac {a^{2} c x}{b^{3} d^{2}}-\frac {a \,c^{2} x}{b^{2} d^{3}}-\frac {c^{3} x}{b \,d^{4}}+\frac {a^{5} \ln \left (-b x -a \right )}{b^{5} \left (a d -b c \right )}-\frac {c^{5} \ln \left (d x +c \right )}{d^{5} \left (a d -b c \right )}\) | \(178\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 161, normalized size = 1.11 \begin {gather*} -\frac {a^{5} \log \left (b x + a\right )}{b^{6} c - a b^{5} d} + \frac {c^{5} \log \left (d x + c\right )}{b c d^{5} - a d^{6}} + \frac {3 \, b^{3} d^{3} x^{4} - 4 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{3} + 6 \, {\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} - 12 \, {\left (b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} x}{12 \, b^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.11, size = 149, normalized size = 1.03 \begin {gather*} -\frac {12 \, a^{5} d^{5} \log \left (b x + a\right ) - 12 \, b^{5} c^{5} \log \left (d x + c\right ) - 3 \, {\left (b^{5} c d^{4} - a b^{4} d^{5}\right )} x^{4} + 4 \, {\left (b^{5} c^{2} d^{3} - a^{2} b^{3} d^{5}\right )} x^{3} - 6 \, {\left (b^{5} c^{3} d^{2} - a^{3} b^{2} d^{5}\right )} x^{2} + 12 \, {\left (b^{5} c^{4} d - a^{4} b d^{5}\right )} x}{12 \, {\left (b^{6} c d^{5} - a b^{5} d^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 306 vs.
\(2 (134) = 268\).
time = 1.40, size = 306, normalized size = 2.11 \begin {gather*} \frac {a^{5} \log {\left (x + \frac {\frac {a^{7} d^{6}}{b \left (a d - b c\right )} - \frac {2 a^{6} c d^{5}}{a d - b c} + \frac {a^{5} b c^{2} d^{4}}{a d - b c} + a^{5} c d^{4} + a b^{4} c^{5}}{a^{5} d^{5} + b^{5} c^{5}} \right )}}{b^{5} \left (a d - b c\right )} - \frac {c^{5} \log {\left (x + \frac {a^{5} c d^{4} - \frac {a^{2} b^{4} c^{5} d}{a d - b c} + \frac {2 a b^{5} c^{6}}{a d - b c} + a b^{4} c^{5} - \frac {b^{6} c^{7}}{d \left (a d - b c\right )}}{a^{5} d^{5} + b^{5} c^{5}} \right )}}{d^{5} \left (a d - b c\right )} + x^{3} \left (- \frac {a}{3 b^{2} d} - \frac {c}{3 b d^{2}}\right ) + x^{2} \left (\frac {a^{2}}{2 b^{3} d} + \frac {a c}{2 b^{2} d^{2}} + \frac {c^{2}}{2 b d^{3}}\right ) + x \left (- \frac {a^{3}}{b^{4} d} - \frac {a^{2} c}{b^{3} d^{2}} - \frac {a c^{2}}{b^{2} d^{3}} - \frac {c^{3}}{b d^{4}}\right ) + \frac {x^{4}}{4 b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.06, size = 175, normalized size = 1.21 \begin {gather*} -\frac {a^{5} \log \left ({\left | b x + a \right |}\right )}{b^{6} c - a b^{5} d} + \frac {c^{5} \log \left ({\left | d x + c \right |}\right )}{b c d^{5} - a d^{6}} + \frac {3 \, b^{3} d^{3} x^{4} - 4 \, b^{3} c d^{2} x^{3} - 4 \, a b^{2} d^{3} x^{3} + 6 \, b^{3} c^{2} d x^{2} + 6 \, a b^{2} c d^{2} x^{2} + 6 \, a^{2} b d^{3} x^{2} - 12 \, b^{3} c^{3} x - 12 \, a b^{2} c^{2} d x - 12 \, a^{2} b c d^{2} x - 12 \, a^{3} d^{3} x}{12 \, b^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.50, size = 174, normalized size = 1.20 \begin {gather*} x^2\,\left (\frac {{\left (a\,d+b\,c\right )}^2}{2\,b^3\,d^3}-\frac {a\,c}{2\,b^2\,d^2}\right )-x\,\left (\frac {\left (a\,d+b\,c\right )\,\left (\frac {{\left (a\,d+b\,c\right )}^2}{b^3\,d^3}-\frac {a\,c}{b^2\,d^2}\right )}{b\,d}-\frac {a\,c\,\left (a\,d+b\,c\right )}{b^3\,d^3}\right )-\frac {a^5\,\ln \left (a+b\,x\right )}{b^6\,c-a\,b^5\,d}+\frac {x^4}{4\,b\,d}-\frac {c^5\,\ln \left (c+d\,x\right )}{d^5\,\left (a\,d-b\,c\right )}-\frac {x^3\,\left (a\,d+b\,c\right )}{3\,b^2\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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