Optimal. Leaf size=198 \[ -\frac {(3 b c+2 a d) x}{b^3 d^4}+\frac {x^2}{2 b^2 d^3}-\frac {a^6}{b^4 (b c-a d)^3 (a+b x)}-\frac {c^6}{2 d^5 (b c-a d)^2 (c+d x)^2}+\frac {2 c^5 (2 b c-3 a d)}{d^5 (b c-a d)^3 (c+d x)}-\frac {3 a^5 (2 b c-a d) \log (a+b x)}{b^4 (b c-a d)^4}+\frac {3 c^4 \left (2 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \log (c+d x)}{d^5 (b c-a d)^4} \]
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Rubi [A]
time = 0.18, antiderivative size = 198, 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^4 (a+b x) (b c-a d)^3}-\frac {3 a^5 (2 b c-a d) \log (a+b x)}{b^4 (b c-a d)^4}+\frac {3 c^4 \left (5 a^2 d^2-6 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^4}-\frac {x (2 a d+3 b c)}{b^3 d^4}-\frac {c^6}{2 d^5 (c+d x)^2 (b c-a d)^2}+\frac {2 c^5 (2 b c-3 a d)}{d^5 (c+d x) (b c-a d)^3}+\frac {x^2}{2 b^2 d^3} \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)^3} \, dx &=\int \left (\frac {-3 b c-2 a d}{b^3 d^4}+\frac {x}{b^2 d^3}+\frac {a^6}{b^3 (b c-a d)^3 (a+b x)^2}+\frac {3 a^5 (-2 b c+a d)}{b^3 (b c-a d)^4 (a+b x)}+\frac {c^6}{d^4 (-b c+a d)^2 (c+d x)^3}+\frac {2 c^5 (2 b c-3 a d)}{d^4 (-b c+a d)^3 (c+d x)^2}+\frac {3 c^4 \left (2 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{d^4 (-b c+a d)^4 (c+d x)}\right ) \, dx\\ &=-\frac {(3 b c+2 a d) x}{b^3 d^4}+\frac {x^2}{2 b^2 d^3}-\frac {a^6}{b^4 (b c-a d)^3 (a+b x)}-\frac {c^6}{2 d^5 (b c-a d)^2 (c+d x)^2}+\frac {2 c^5 (2 b c-3 a d)}{d^5 (b c-a d)^3 (c+d x)}-\frac {3 a^5 (2 b c-a d) \log (a+b x)}{b^4 (b c-a d)^4}+\frac {3 c^4 \left (2 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \log (c+d x)}{d^5 (b c-a d)^4}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 198, normalized size = 1.00 \begin {gather*} -\frac {(3 b c+2 a d) x}{b^3 d^4}+\frac {x^2}{2 b^2 d^3}-\frac {a^6}{b^4 (b c-a d)^3 (a+b x)}-\frac {c^6}{2 d^5 (b c-a d)^2 (c+d x)^2}+\frac {-4 b c^6+6 a c^5 d}{d^5 (-b c+a d)^3 (c+d x)}+\frac {3 a^5 (-2 b c+a d) \log (a+b x)}{b^4 (b c-a d)^4}+\frac {3 c^4 \left (2 b^2 c^2-6 a b c d+5 a^2 d^2\right ) \log (c+d x)}{d^5 (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 190, normalized size = 0.96
method | result | size |
default | \(-\frac {-\frac {1}{2} b d \,x^{2}+2 a d x +3 b c x}{b^{3} d^{4}}+\frac {a^{6}}{b^{4} \left (a d -b c \right )^{3} \left (b x +a \right )}+\frac {3 a^{5} \left (a d -2 b c \right ) \ln \left (b x +a \right )}{b^{4} \left (a d -b c \right )^{4}}-\frac {c^{6}}{2 d^{5} \left (a d -b c \right )^{2} \left (d x +c \right )^{2}}+\frac {3 c^{4} \left (5 a^{2} d^{2}-6 a b c d +2 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5} \left (a d -b c \right )^{4}}+\frac {2 c^{5} \left (3 a d -2 b c \right )}{d^{5} \left (a d -b c \right )^{3} \left (d x +c \right )}\) | \(190\) |
norman | \(\frac {\frac {\left (3 a^{6} d^{6}-2 a^{4} b^{2} c^{2} d^{4}-6 a^{3} b^{3} c^{3} d^{3}+20 a \,b^{5} c^{5} d -12 b^{6} c^{6}\right ) x^{2}}{d^{4} b^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {x^{5}}{2 b d}-\frac {\left (3 a d +4 b c \right ) x^{4}}{2 b^{2} d^{2}}+\frac {c \left (12 a^{6} d^{6}-20 a^{4} b^{2} c^{2} d^{4}-9 a^{3} b^{3} c^{3} d^{3}+39 a^{2} b^{4} c^{4} d^{2}+8 a \,b^{5} c^{5} d -18 b^{6} c^{6}\right ) x}{2 d^{5} b^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c^{2} a \left (6 a^{5} d^{5}-13 a^{3} b^{2} c^{2} d^{3}-a^{2} b^{3} c^{3} d^{2}+32 a \,b^{4} c^{4} d -18 b^{5} c^{5}\right )}{2 d^{5} b^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {3 a^{5} \left (a d -2 b c \right ) \ln \left (b x +a \right )}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{4}}+\frac {3 c^{4} \left (5 a^{2} d^{2}-6 a b c d +2 b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(535\) |
risch | \(\frac {x^{2}}{2 b^{2} d^{3}}-\frac {2 a x}{b^{3} d^{3}}-\frac {3 c x}{b^{2} d^{4}}+\frac {\frac {\left (a^{6} d^{6}+6 a \,b^{5} c^{5} d -4 b^{6} c^{6}\right ) x^{2}}{b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c \left (4 a^{6} d^{6}+12 a^{2} b^{4} c^{4} d^{2}+3 a \,b^{5} c^{5} d -7 b^{6} c^{6}\right ) x}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b d}+\frac {a \,c^{2} \left (2 a^{5} d^{5}+11 a \,b^{4} c^{4} d -7 b^{5} c^{5}\right )}{2 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d}}{b^{3} d^{4} \left (b x +a \right ) \left (d x +c \right )^{2}}+\frac {15 c^{4} \ln \left (d x +c \right ) a^{2}}{d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {18 c^{5} \ln \left (d x +c \right ) a b}{d^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {6 c^{6} \ln \left (d x +c \right ) b^{2}}{d^{5} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {3 a^{6} \ln \left (-b x -a \right ) d}{b^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}-\frac {6 a^{5} \ln \left (-b x -a \right ) c}{b^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(633\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 547 vs.
\(2 (194) = 388\).
time = 0.31, size = 547, normalized size = 2.76 \begin {gather*} -\frac {3 \, {\left (2 \, a^{5} b c - a^{6} d\right )} \log \left (b x + a\right )}{b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}} + \frac {3 \, {\left (2 \, b^{2} c^{6} - 6 \, a b c^{5} d + 5 \, a^{2} c^{4} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{5} - 4 \, a b^{3} c^{3} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}} + \frac {7 \, a b^{5} c^{7} - 11 \, a^{2} b^{4} c^{6} d - 2 \, a^{6} c^{2} d^{5} + 2 \, {\left (4 \, b^{6} c^{6} d - 6 \, a b^{5} c^{5} d^{2} - a^{6} d^{7}\right )} x^{2} + {\left (7 \, b^{6} c^{7} - 3 \, a b^{5} c^{6} d - 12 \, a^{2} b^{4} c^{5} d^{2} - 4 \, a^{6} c d^{6}\right )} x}{2 \, {\left (a b^{7} c^{5} d^{5} - 3 \, a^{2} b^{6} c^{4} d^{6} + 3 \, a^{3} b^{5} c^{3} d^{7} - a^{4} b^{4} c^{2} d^{8} + {\left (b^{8} c^{3} d^{7} - 3 \, a b^{7} c^{2} d^{8} + 3 \, a^{2} b^{6} c d^{9} - a^{3} b^{5} d^{10}\right )} x^{3} + {\left (2 \, b^{8} c^{4} d^{6} - 5 \, a b^{7} c^{3} d^{7} + 3 \, a^{2} b^{6} c^{2} d^{8} + a^{3} b^{5} c d^{9} - a^{4} b^{4} d^{10}\right )} x^{2} + {\left (b^{8} c^{5} d^{5} - a b^{7} c^{4} d^{6} - 3 \, a^{2} b^{6} c^{3} d^{7} + 5 \, a^{3} b^{5} c^{2} d^{8} - 2 \, a^{4} b^{4} c d^{9}\right )} x\right )}} + \frac {b d x^{2} - 2 \, {\left (3 \, b c + 2 \, a d\right )} x}{2 \, b^{3} 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. 1081 vs.
\(2 (194) = 388\).
time = 0.96, size = 1081, normalized size = 5.46 \begin {gather*} \frac {7 \, a b^{6} c^{8} - 18 \, a^{2} b^{5} c^{7} d + 11 \, a^{3} b^{4} c^{6} d^{2} - 2 \, a^{6} b c^{3} d^{5} + 2 \, a^{7} c^{2} d^{6} + {\left (b^{7} c^{4} d^{4} - 4 \, a b^{6} c^{3} d^{5} + 6 \, a^{2} b^{5} c^{2} d^{6} - 4 \, a^{3} b^{4} c d^{7} + a^{4} b^{3} d^{8}\right )} x^{5} - {\left (4 \, b^{7} c^{5} d^{3} - 13 \, a b^{6} c^{4} d^{4} + 12 \, a^{2} b^{5} c^{3} d^{5} + 2 \, a^{3} b^{4} c^{2} d^{6} - 8 \, a^{4} b^{3} c d^{7} + 3 \, a^{5} b^{2} d^{8}\right )} x^{4} - {\left (11 \, b^{7} c^{6} d^{2} - 32 \, a b^{6} c^{5} d^{3} + 22 \, a^{2} b^{5} c^{4} d^{4} + 12 \, a^{3} b^{4} c^{3} d^{5} - 13 \, a^{4} b^{3} c^{2} d^{6} - 4 \, a^{5} b^{2} c d^{7} + 4 \, a^{6} b d^{8}\right )} x^{3} + {\left (2 \, b^{7} c^{7} d - 11 \, a b^{6} c^{6} d^{2} + 28 \, a^{2} b^{5} c^{5} d^{3} - 34 \, a^{3} b^{4} c^{4} d^{4} + 6 \, a^{4} b^{3} c^{3} d^{5} + 17 \, a^{5} b^{2} c^{2} d^{6} - 10 \, a^{6} b c d^{7} + 2 \, a^{7} d^{8}\right )} x^{2} + {\left (7 \, b^{7} c^{8} - 16 \, a b^{6} c^{7} d + 11 \, a^{2} b^{5} c^{6} d^{2} - 8 \, a^{3} b^{4} c^{5} d^{3} + 10 \, a^{5} b^{2} c^{3} d^{5} - 8 \, a^{6} b c^{2} d^{6} + 4 \, a^{7} c d^{7}\right )} x - 6 \, {\left (2 \, a^{6} b c^{3} d^{5} - a^{7} c^{2} d^{6} + {\left (2 \, a^{5} b^{2} c d^{7} - a^{6} b d^{8}\right )} x^{3} + {\left (4 \, a^{5} b^{2} c^{2} d^{6} - a^{7} d^{8}\right )} x^{2} + {\left (2 \, a^{5} b^{2} c^{3} d^{5} + 3 \, a^{6} b c^{2} d^{6} - 2 \, a^{7} c d^{7}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (2 \, a b^{6} c^{8} - 6 \, a^{2} b^{5} c^{7} d + 5 \, a^{3} b^{4} c^{6} d^{2} + {\left (2 \, b^{7} c^{6} d^{2} - 6 \, a b^{6} c^{5} d^{3} + 5 \, a^{2} b^{5} c^{4} d^{4}\right )} x^{3} + {\left (4 \, b^{7} c^{7} d - 10 \, a b^{6} c^{6} d^{2} + 4 \, a^{2} b^{5} c^{5} d^{3} + 5 \, a^{3} b^{4} c^{4} d^{4}\right )} x^{2} + {\left (2 \, b^{7} c^{8} - 2 \, a b^{6} c^{7} d - 7 \, a^{2} b^{5} c^{6} d^{2} + 10 \, a^{3} b^{4} c^{5} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{8} c^{6} d^{5} - 4 \, a^{2} b^{7} c^{5} d^{6} + 6 \, a^{3} b^{6} c^{4} d^{7} - 4 \, a^{4} b^{5} c^{3} d^{8} + a^{5} b^{4} c^{2} d^{9} + {\left (b^{9} c^{4} d^{7} - 4 \, a b^{8} c^{3} d^{8} + 6 \, a^{2} b^{7} c^{2} d^{9} - 4 \, a^{3} b^{6} c d^{10} + a^{4} b^{5} d^{11}\right )} x^{3} + {\left (2 \, b^{9} c^{5} d^{6} - 7 \, a b^{8} c^{4} d^{7} + 8 \, a^{2} b^{7} c^{3} d^{8} - 2 \, a^{3} b^{6} c^{2} d^{9} - 2 \, a^{4} b^{5} c d^{10} + a^{5} b^{4} d^{11}\right )} x^{2} + {\left (b^{9} c^{6} d^{5} - 2 \, a b^{8} c^{5} d^{6} - 2 \, a^{2} b^{7} c^{4} d^{7} + 8 \, a^{3} b^{6} c^{3} d^{8} - 7 \, a^{4} b^{5} c^{2} d^{9} + 2 \, a^{5} b^{4} c d^{10}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 621 vs.
\(2 (194) = 388\).
time = 1.98, size = 621, normalized size = 3.14 \begin {gather*} -\frac {a^{6} b^{5}}{{\left (b^{12} c^{3} - 3 \, a b^{11} c^{2} d + 3 \, a^{2} b^{10} c d^{2} - a^{3} b^{9} d^{3}\right )} {\left (b x + a\right )}} + \frac {3 \, {\left (2 \, b^{3} c^{6} - 6 \, a b^{2} c^{5} d + 5 \, a^{2} b c^{4} d^{2}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} d^{5} - 4 \, a b^{4} c^{3} d^{6} + 6 \, a^{2} b^{3} c^{2} d^{7} - 4 \, a^{3} b^{2} c d^{8} + a^{4} b d^{9}} - \frac {3 \, {\left (2 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4} d^{5}} + \frac {{\left (b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7} - \frac {4 \, {\left (b^{6} c^{5} d^{2} - 2 \, a b^{5} c^{4} d^{3} - 2 \, a^{2} b^{4} c^{3} d^{4} + 8 \, a^{3} b^{3} c^{2} d^{5} - 7 \, a^{4} b^{2} c d^{6} + 2 \, a^{5} b d^{7}\right )}}{{\left (b x + a\right )} b} - \frac {18 \, b^{8} c^{6} d - 54 \, a b^{7} c^{5} d^{2} + 45 \, a^{2} b^{6} c^{4} d^{3} + 20 \, a^{3} b^{5} c^{3} d^{4} - 75 \, a^{4} b^{4} c^{2} d^{5} + 54 \, a^{5} b^{3} c d^{6} - 13 \, a^{6} b^{2} d^{7}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {6 \, {\left (2 \, b^{10} c^{7} - 8 \, a b^{9} c^{6} d + 11 \, a^{2} b^{8} c^{5} d^{2} - 5 \, a^{3} b^{7} c^{4} d^{3} - 5 \, a^{4} b^{6} c^{3} d^{4} + 9 \, a^{5} b^{5} c^{2} d^{5} - 5 \, a^{6} b^{4} c d^{6} + a^{7} b^{3} d^{7}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )} {\left (b x + a\right )}^{2}}{2 \, {\left (b c - a d\right )}^{4} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.83, size = 508, normalized size = 2.57 \begin {gather*} \frac {\frac {2\,a^6\,c^2\,d^5+11\,a^2\,b^4\,c^6\,d-7\,a\,b^5\,c^7}{2\,b\,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 )}+\frac {x^2\,\left (a^6\,d^6+6\,a\,b^5\,c^5\,d-4\,b^6\,c^6\right )}{b\,\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 {x\,\left (4\,a^6\,c\,d^6+12\,a^2\,b^4\,c^5\,d^2+3\,a\,b^5\,c^6\,d-7\,b^6\,c^7\right )}{2\,b\,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 )}}{x^2\,\left (2\,c\,b^4\,d^5+a\,b^3\,d^6\right )+x\,\left (b^4\,c^2\,d^4+2\,a\,b^3\,c\,d^5\right )+b^4\,d^6\,x^3+a\,b^3\,c^2\,d^4}+\frac {\ln \left (c+d\,x\right )\,\left (15\,a^2\,c^4\,d^2-18\,a\,b\,c^5\,d+6\,b^2\,c^6\right )}{a^4\,d^9-4\,a^3\,b\,c\,d^8+6\,a^2\,b^2\,c^2\,d^7-4\,a\,b^3\,c^3\,d^6+b^4\,c^4\,d^5}+\frac {x^2}{2\,b^2\,d^3}+\frac {\ln \left (a+b\,x\right )\,\left (3\,a^6\,d-6\,a^5\,b\,c\right )}{a^4\,b^4\,d^4-4\,a^3\,b^5\,c\,d^3+6\,a^2\,b^6\,c^2\,d^2-4\,a\,b^7\,c^3\,d+b^8\,c^4}-\frac {x\,\left (2\,a\,d+3\,b\,c\right )}{b^3\,d^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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