Optimal. Leaf size=179 \[ \frac {\left (3 b^2 c^2+4 a b c d+3 a^2 d^2\right ) x}{b^4 d^4}-\frac {(b c+a d) x^2}{b^3 d^3}+\frac {x^3}{3 b^2 d^2}-\frac {a^6}{b^5 (b c-a d)^2 (a+b x)}-\frac {c^6}{d^5 (b c-a d)^2 (c+d x)}-\frac {2 a^5 (3 b c-2 a d) \log (a+b x)}{b^5 (b c-a d)^3}-\frac {2 c^5 (2 b c-3 a d) \log (c+d x)}{d^5 (b c-a d)^3} \]
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Rubi [A]
time = 0.17, antiderivative size = 179, 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 = {90}
\begin {gather*} -\frac {a^6}{b^5 (a+b x) (b c-a d)^2}-\frac {2 a^5 (3 b c-2 a d) \log (a+b x)}{b^5 (b c-a d)^3}+\frac {x \left (3 a^2 d^2+4 a b c d+3 b^2 c^2\right )}{b^4 d^4}-\frac {x^2 (a d+b c)}{b^3 d^3}-\frac {c^6}{d^5 (c+d x) (b c-a d)^2}-\frac {2 c^5 (2 b c-3 a d) \log (c+d x)}{d^5 (b c-a d)^3}+\frac {x^3}{3 b^2 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rubi steps
\begin {align*} \int \frac {x^6}{(a+b x)^2 (c+d x)^2} \, dx &=\int \left (\frac {3 b^2 c^2+4 a b c d+3 a^2 d^2}{b^4 d^4}-\frac {2 (b c+a d) x}{b^3 d^3}+\frac {x^2}{b^2 d^2}+\frac {a^6}{b^4 (b c-a d)^2 (a+b x)^2}+\frac {2 a^5 (-3 b c+2 a d)}{b^4 (b c-a d)^3 (a+b x)}+\frac {c^6}{d^4 (-b c+a d)^2 (c+d x)^2}+\frac {2 c^5 (2 b c-3 a d)}{d^4 (-b c+a d)^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (3 b^2 c^2+4 a b c d+3 a^2 d^2\right ) x}{b^4 d^4}-\frac {(b c+a d) x^2}{b^3 d^3}+\frac {x^3}{3 b^2 d^2}-\frac {a^6}{b^5 (b c-a d)^2 (a+b x)}-\frac {c^6}{d^5 (b c-a d)^2 (c+d x)}-\frac {2 a^5 (3 b c-2 a d) \log (a+b x)}{b^5 (b c-a d)^3}-\frac {2 c^5 (2 b c-3 a d) \log (c+d x)}{d^5 (b c-a d)^3}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 179, normalized size = 1.00 \begin {gather*} \frac {\left (3 b^2 c^2+4 a b c d+3 a^2 d^2\right ) x}{b^4 d^4}-\frac {(b c+a d) x^2}{b^3 d^3}+\frac {x^3}{3 b^2 d^2}-\frac {a^6}{b^5 (b c-a d)^2 (a+b x)}-\frac {c^6}{d^5 (b c-a d)^2 (c+d x)}+\frac {2 a^5 (-3 b c+2 a d) \log (a+b x)}{b^5 (b c-a d)^3}+\frac {2 c^5 (2 b c-3 a d) \log (c+d x)}{d^5 (-b c+a d)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 182, normalized size = 1.02
method | result | size |
default | \(\frac {\frac {1}{3} d^{2} x^{3} b^{2}-a b \,d^{2} x^{2}-b^{2} c d \,x^{2}+3 a^{2} d^{2} x +4 a b c d x +3 b^{2} c^{2} x}{b^{4} d^{4}}-\frac {a^{6}}{b^{5} \left (a d -b c \right )^{2} \left (b x +a \right )}-\frac {2 a^{5} \left (2 a d -3 b c \right ) \ln \left (b x +a \right )}{b^{5} \left (a d -b c \right )^{3}}-\frac {c^{6}}{d^{5} \left (a d -b c \right )^{2} \left (d x +c \right )}-\frac {2 c^{5} \left (3 a d -2 b c \right ) \ln \left (d x +c \right )}{d^{5} \left (a d -b c \right )^{3}}\) | \(182\) |
norman | \(\frac {\frac {x^{5}}{3 b d}-\frac {2 \left (a d +b c \right ) x^{4}}{3 b^{2} d^{2}}+\frac {\left (6 a^{2} d^{2}+7 a b c d +6 b^{2} c^{2}\right ) x^{3}}{3 b^{3} d^{3}}-\frac {\left (4 a^{6} d^{6}-a^{4} b^{2} c^{2} d^{4}-4 a^{3} b^{3} c^{3} d^{3}-a^{2} b^{4} c^{4} d^{2}+4 b^{6} c^{6}\right ) x}{d^{5} b^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (4 a^{5} d^{5}-3 a^{3} b^{2} c^{2} d^{3}-3 a^{2} b^{3} c^{3} d^{2}+4 b^{5} c^{5}\right ) a c}{d^{5} b^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}-\frac {2 a^{5} \left (2 a d -3 b c \right ) \ln \left (b x +a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{5}}-\frac {2 c^{5} \left (3 a d -2 b c \right ) \ln \left (d x +c \right )}{d^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(372\) |
risch | \(\frac {x^{3}}{3 b^{2} d^{2}}-\frac {a \,x^{2}}{b^{3} d^{2}}-\frac {c \,x^{2}}{b^{2} d^{3}}+\frac {3 a^{2} x}{b^{4} d^{2}}+\frac {4 a c x}{b^{3} d^{3}}+\frac {3 c^{2} x}{b^{2} d^{4}}+\frac {-\frac {\left (a^{6} d^{6}+b^{6} c^{6}\right ) x}{b d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {a c \left (a^{5} d^{5}+b^{5} c^{5}\right )}{b d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{b^{4} d^{4} \left (b x +a \right ) \left (d x +c \right )}-\frac {6 c^{5} \ln \left (-d x -c \right ) a}{d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {4 c^{6} \ln \left (-d x -c \right ) b}{d^{5} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {4 a^{6} \ln \left (b x +a \right ) d}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{5}}+\frac {6 a^{5} \ln \left (b x +a \right ) c}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{4}}\) | \(407\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 351, normalized size = 1.96 \begin {gather*} -\frac {2 \, {\left (3 \, a^{5} b c - 2 \, a^{6} d\right )} \log \left (b x + a\right )}{b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}} - \frac {2 \, {\left (2 \, b c^{6} - 3 \, a c^{5} d\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}} - \frac {a b^{5} c^{6} + a^{6} c d^{5} + {\left (b^{6} c^{6} + a^{6} d^{6}\right )} x}{a b^{7} c^{3} d^{5} - 2 \, a^{2} b^{6} c^{2} d^{6} + a^{3} b^{5} c d^{7} + {\left (b^{8} c^{2} d^{6} - 2 \, a b^{7} c d^{7} + a^{2} b^{6} d^{8}\right )} x^{2} + {\left (b^{8} c^{3} d^{5} - a b^{7} c^{2} d^{6} - a^{2} b^{6} c d^{7} + a^{3} b^{5} d^{8}\right )} x} + \frac {b^{2} d^{2} x^{3} - 3 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 3 \, {\left (3 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x}{3 \, b^{4} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 700 vs.
\(2 (177) = 354\).
time = 1.70, size = 700, normalized size = 3.91 \begin {gather*} -\frac {3 \, a b^{6} c^{7} - 3 \, a^{2} b^{5} c^{6} d + 3 \, a^{6} b c^{2} d^{5} - 3 \, a^{7} c d^{6} - {\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{5} + 2 \, {\left (b^{7} c^{4} d^{3} - 2 \, a b^{6} c^{3} d^{4} + 2 \, a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x^{4} - {\left (6 \, b^{7} c^{5} d^{2} - 11 \, a b^{6} c^{4} d^{3} + 3 \, a^{2} b^{5} c^{3} d^{4} - 3 \, a^{3} b^{4} c^{2} d^{5} + 11 \, a^{4} b^{3} c d^{6} - 6 \, a^{5} b^{2} d^{7}\right )} x^{3} - 9 \, {\left (b^{7} c^{6} d - a b^{6} c^{5} d^{2} - a^{2} b^{5} c^{4} d^{3} + a^{4} b^{3} c^{2} d^{5} + a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x^{2} + 3 \, {\left (b^{7} c^{7} - 4 \, a b^{6} c^{6} d + 5 \, a^{2} b^{5} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} - a^{7} d^{7}\right )} x + 6 \, {\left (3 \, a^{6} b c^{2} d^{5} - 2 \, a^{7} c d^{6} + {\left (3 \, a^{5} b^{2} c d^{6} - 2 \, a^{6} b d^{7}\right )} x^{2} + {\left (3 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - 2 \, a^{7} d^{7}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (2 \, a b^{6} c^{7} - 3 \, a^{2} b^{5} c^{6} d + {\left (2 \, b^{7} c^{6} d - 3 \, a b^{6} c^{5} d^{2}\right )} x^{2} + {\left (2 \, b^{7} c^{7} - a b^{6} c^{6} d - 3 \, a^{2} b^{5} c^{5} d^{2}\right )} x\right )} \log \left (d x + c\right )}{3 \, {\left (a b^{8} c^{4} d^{5} - 3 \, a^{2} b^{7} c^{3} d^{6} + 3 \, a^{3} b^{6} c^{2} d^{7} - a^{4} b^{5} c d^{8} + {\left (b^{9} c^{3} d^{6} - 3 \, a b^{8} c^{2} d^{7} + 3 \, a^{2} b^{7} c d^{8} - a^{3} b^{6} d^{9}\right )} x^{2} + {\left (b^{9} c^{4} d^{5} - 2 \, a b^{8} c^{3} d^{6} + 2 \, a^{3} b^{6} c d^{8} - a^{4} b^{5} d^{9}\right )} x\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. 779 vs.
\(2 (170) = 340\).
time = 226.46, size = 779, normalized size = 4.35 \begin {gather*} - \frac {2 a^{5} \cdot \left (2 a d - 3 b c\right ) \log {\left (x + \frac {\frac {2 a^{9} d^{8} \cdot \left (2 a d - 3 b c\right )}{b \left (a d - b c\right )^{3}} - \frac {8 a^{8} c d^{7} \cdot \left (2 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + \frac {12 a^{7} b c^{2} d^{6} \cdot \left (2 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - \frac {8 a^{6} b^{2} c^{3} d^{5} \cdot \left (2 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + 4 a^{6} c d^{5} + \frac {2 a^{5} b^{3} c^{4} d^{4} \cdot \left (2 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - 6 a^{5} b c^{2} d^{4} - 6 a^{2} b^{4} c^{5} d + 4 a b^{5} c^{6}}{4 a^{6} d^{6} - 6 a^{5} b c d^{5} - 6 a b^{5} c^{5} d + 4 b^{6} c^{6}} \right )}}{b^{5} \left (a d - b c\right )^{3}} - \frac {2 c^{5} \cdot \left (3 a d - 2 b c\right ) \log {\left (x + \frac {4 a^{6} c d^{5} - 6 a^{5} b c^{2} d^{4} + \frac {2 a^{4} b^{4} c^{5} d^{3} \cdot \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{3}} - \frac {8 a^{3} b^{5} c^{6} d^{2} \cdot \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{3}} + \frac {12 a^{2} b^{6} c^{7} d \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{3}} - 6 a^{2} b^{4} c^{5} d - \frac {8 a b^{7} c^{8} \cdot \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{3}} + 4 a b^{5} c^{6} + \frac {2 b^{8} c^{9} \cdot \left (3 a d - 2 b c\right )}{d \left (a d - b c\right )^{3}}}{4 a^{6} d^{6} - 6 a^{5} b c d^{5} - 6 a b^{5} c^{5} d + 4 b^{6} c^{6}} \right )}}{d^{5} \left (a d - b c\right )^{3}} + x^{2} \left (- \frac {a}{b^{3} d^{2}} - \frac {c}{b^{2} d^{3}}\right ) + x \left (\frac {3 a^{2}}{b^{4} d^{2}} + \frac {4 a c}{b^{3} d^{3}} + \frac {3 c^{2}}{b^{2} d^{4}}\right ) + \frac {- a^{6} c d^{5} - a b^{5} c^{6} + x \left (- a^{6} d^{6} - b^{6} c^{6}\right )}{a^{3} b^{5} c d^{7} - 2 a^{2} b^{6} c^{2} d^{6} + a b^{7} c^{3} d^{5} + x^{2} \left (a^{2} b^{6} d^{8} - 2 a b^{7} c d^{7} + b^{8} c^{2} d^{6}\right ) + x \left (a^{3} b^{5} d^{8} - a^{2} b^{6} c d^{7} - a b^{7} c^{2} d^{6} + b^{8} c^{3} d^{5}\right )} + \frac {x^{3}}{3 b^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 517 vs.
\(2 (177) = 354\).
time = 1.40, size = 517, normalized size = 2.89 \begin {gather*} -\frac {a^{6} b^{5}}{{\left (b^{12} c^{2} - 2 \, a b^{11} c d + a^{2} b^{10} d^{2}\right )} {\left (b x + a\right )}} - \frac {2 \, {\left (2 \, b^{2} c^{6} - 3 \, a b c^{5} d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} d^{5} - 3 \, a b^{3} c^{2} d^{6} + 3 \, a^{2} b^{2} c d^{7} - a^{3} b d^{8}} + \frac {2 \, {\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{5} d^{5}} + \frac {{\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7} - \frac {2 \, b^{5} c^{4} d^{3} + a b^{4} c^{3} d^{4} - 15 \, a^{2} b^{3} c^{2} d^{5} + 19 \, a^{3} b^{2} c d^{6} - 7 \, a^{4} b d^{7}}{{\left (b x + a\right )} b} + \frac {3 \, {\left (2 \, b^{7} c^{5} d^{2} - a b^{6} c^{4} d^{3} - a^{2} b^{5} c^{3} d^{4} - 11 \, a^{3} b^{4} c^{2} d^{5} + 19 \, a^{4} b^{3} c d^{6} - 8 \, a^{5} b^{2} d^{7}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {3 \, {\left (4 \, b^{9} c^{6} d - 6 \, a b^{8} c^{5} d^{2} + 15 \, a^{4} b^{5} c^{2} d^{5} - 18 \, a^{5} b^{4} c d^{6} + 6 \, a^{6} b^{3} d^{7}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )} {\left (b x + a\right )}^{3}}{3 \, {\left (b c - a d\right )}^{3} b^{5} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 344, normalized size = 1.92 \begin {gather*} x\,\left (\frac {4\,{\left (a\,d+b\,c\right )}^2}{b^4\,d^4}-\frac {a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2}{b^4\,d^4}\right )-\frac {\frac {a^6\,c\,d^5+a\,b^5\,c^6}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (a^6\,d^6+b^6\,c^6\right )}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x\,\left (c\,b^5\,d^4+a\,b^4\,d^5\right )+b^5\,d^5\,x^2+a\,b^4\,c\,d^4}+\frac {\ln \left (a+b\,x\right )\,\left (4\,a^6\,d-6\,a^5\,b\,c\right )}{-a^3\,b^5\,d^3+3\,a^2\,b^6\,c\,d^2-3\,a\,b^7\,c^2\,d+b^8\,c^3}+\frac {\ln \left (c+d\,x\right )\,\left (4\,b\,c^6-6\,a\,c^5\,d\right )}{a^3\,d^8-3\,a^2\,b\,c\,d^7+3\,a\,b^2\,c^2\,d^6-b^3\,c^3\,d^5}+\frac {x^3}{3\,b^2\,d^2}-\frac {x^2\,\left (a\,d+b\,c\right )}{b^3\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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