Optimal. Leaf size=110 \[ -\frac {(2 b c+a d) x}{b^2 d^3}+\frac {x^2}{2 b d^2}+\frac {c^4}{d^4 (b c-a d) (c+d x)}+\frac {a^4 \log (a+b x)}{b^3 (b c-a d)^2}+\frac {c^3 (3 b c-4 a d) \log (c+d x)}{d^4 (b c-a d)^2} \]
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Rubi [A]
time = 0.08, antiderivative size = 110, 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^4 \log (a+b x)}{b^3 (b c-a d)^2}-\frac {x (a d+2 b c)}{b^2 d^3}+\frac {c^4}{d^4 (c+d x) (b c-a d)}+\frac {c^3 (3 b c-4 a d) \log (c+d x)}{d^4 (b c-a d)^2}+\frac {x^2}{2 b d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rubi steps
\begin {align*} \int \frac {x^4}{(a+b x) (c+d x)^2} \, dx &=\int \left (\frac {-2 b c-a d}{b^2 d^3}+\frac {x}{b d^2}+\frac {a^4}{b^2 (b c-a d)^2 (a+b x)}+\frac {c^4}{d^3 (-b c+a d) (c+d x)^2}+\frac {c^3 (3 b c-4 a d)}{d^3 (-b c+a d)^2 (c+d x)}\right ) \, dx\\ &=-\frac {(2 b c+a d) x}{b^2 d^3}+\frac {x^2}{2 b d^2}+\frac {c^4}{d^4 (b c-a d) (c+d x)}+\frac {a^4 \log (a+b x)}{b^3 (b c-a d)^2}+\frac {c^3 (3 b c-4 a d) \log (c+d x)}{d^4 (b c-a d)^2}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 107, normalized size = 0.97 \begin {gather*} \frac {a^4 \log (a+b x)}{b^3 (b c-a d)^2}+\frac {-\frac {2 a d^2 x}{b^2}+\frac {d x (-4 c+d x)}{b}+\frac {2 c^4}{(b c-a d) (c+d x)}+\frac {2 c^3 (3 b c-4 a d) \log (c+d x)}{(b c-a d)^2}}{2 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 108, normalized size = 0.98
method | result | size |
default | \(-\frac {-\frac {1}{2} b d \,x^{2}+a d x +2 b c x}{b^{2} d^{3}}+\frac {a^{4} \ln \left (b x +a \right )}{b^{3} \left (a d -b c \right )^{2}}-\frac {c^{4}}{d^{4} \left (a d -b c \right ) \left (d x +c \right )}-\frac {c^{3} \left (4 a d -3 b c \right ) \ln \left (d x +c \right )}{d^{4} \left (a d -b c \right )^{2}}\) | \(108\) |
norman | \(\frac {\frac {\left (a^{2} d^{2} c +b \,c^{2} d a -3 b^{2} c^{3}\right ) c}{d^{4} b^{2} \left (a d -b c \right )}+\frac {x^{3}}{2 b d}-\frac {\left (2 a d +3 b c \right ) x^{2}}{2 b^{2} d^{2}}}{d x +c}+\frac {a^{4} \ln \left (b x +a \right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{3}}-\frac {c^{3} \left (4 a d -3 b c \right ) \ln \left (d x +c \right )}{d^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(166\) |
risch | \(\frac {x^{2}}{2 b \,d^{2}}-\frac {a x}{b^{2} d^{2}}-\frac {2 c x}{b \,d^{3}}-\frac {c^{4}}{d^{4} \left (a d -b c \right ) \left (d x +c \right )}-\frac {4 c^{3} \ln \left (d x +c \right ) a}{d^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {3 c^{4} \ln \left (d x +c \right ) b}{d^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {a^{4} \ln \left (-b x -a \right )}{b^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(173\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 151, normalized size = 1.37 \begin {gather*} \frac {a^{4} \log \left (b x + a\right )}{b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}} + \frac {c^{4}}{b c^{2} d^{4} - a c d^{5} + {\left (b c d^{5} - a d^{6}\right )} x} + \frac {{\left (3 \, b c^{4} - 4 \, a c^{3} d\right )} \log \left (d x + c\right )}{b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}} + \frac {b d x^{2} - 2 \, {\left (2 \, b c + a d\right )} x}{2 \, b^{2} d^{3}} \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. 285 vs.
\(2 (108) = 216\).
time = 1.42, size = 285, normalized size = 2.59 \begin {gather*} \frac {2 \, b^{4} c^{5} - 2 \, a b^{3} c^{4} d + {\left (b^{4} c^{2} d^{3} - 2 \, a b^{3} c d^{4} + a^{2} b^{2} d^{5}\right )} x^{3} - {\left (3 \, b^{4} c^{3} d^{2} - 4 \, a b^{3} c^{2} d^{3} - a^{2} b^{2} c d^{4} + 2 \, a^{3} b d^{5}\right )} x^{2} - 2 \, {\left (2 \, b^{4} c^{4} d - 3 \, a b^{3} c^{3} d^{2} + a^{3} b c d^{4}\right )} x + 2 \, {\left (a^{4} d^{5} x + a^{4} c d^{4}\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, b^{4} c^{5} - 4 \, a b^{3} c^{4} d + {\left (3 \, b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (b^{5} c^{3} d^{4} - 2 \, a b^{4} c^{2} d^{5} + a^{2} b^{3} c d^{6} + {\left (b^{5} c^{2} d^{5} - 2 \, a b^{4} c d^{6} + a^{2} b^{3} d^{7}\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. 428 vs.
\(2 (100) = 200\).
time = 4.74, size = 428, normalized size = 3.89 \begin {gather*} \frac {a^{4} \log {\left (x + \frac {\frac {a^{7} d^{6}}{b \left (a d - b c\right )^{2}} - \frac {3 a^{6} c d^{5}}{\left (a d - b c\right )^{2}} + \frac {3 a^{5} b c^{2} d^{4}}{\left (a d - b c\right )^{2}} - \frac {a^{4} b^{2} c^{3} d^{3}}{\left (a d - b c\right )^{2}} + a^{4} c d^{3} + 4 a^{2} b^{2} c^{3} d - 3 a b^{3} c^{4}}{a^{4} d^{4} + 4 a b^{3} c^{3} d - 3 b^{4} c^{4}} \right )}}{b^{3} \left (a d - b c\right )^{2}} - \frac {c^{4}}{a c d^{5} - b c^{2} d^{4} + x \left (a d^{6} - b c d^{5}\right )} - \frac {c^{3} \cdot \left (4 a d - 3 b c\right ) \log {\left (x + \frac {a^{4} c d^{3} - \frac {a^{3} b^{2} c^{3} d^{2} \cdot \left (4 a d - 3 b c\right )}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{3} c^{4} d \left (4 a d - 3 b c\right )}{\left (a d - b c\right )^{2}} + 4 a^{2} b^{2} c^{3} d - \frac {3 a b^{4} c^{5} \cdot \left (4 a d - 3 b c\right )}{\left (a d - b c\right )^{2}} - 3 a b^{3} c^{4} + \frac {b^{5} c^{6} \cdot \left (4 a d - 3 b c\right )}{d \left (a d - b c\right )^{2}}}{a^{4} d^{4} + 4 a b^{3} c^{3} d - 3 b^{4} c^{4}} \right )}}{d^{4} \left (a d - b c\right )^{2}} + x \left (- \frac {a}{b^{2} d^{2}} - \frac {2 c}{b d^{3}}\right ) + \frac {x^{2}}{2 b d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.56, size = 185, normalized size = 1.68 \begin {gather*} \frac {c^{4} d^{3}}{{\left (b c d^{7} - a d^{8}\right )} {\left (d x + c\right )}} + \frac {a^{4} d \log \left ({\left | b - \frac {b c}{d x + c} + \frac {a d}{d x + c} \right |}\right )}{b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}} + \frac {{\left (b^{2} - \frac {2 \, {\left (3 \, b^{2} c d + a b d^{2}\right )}}{{\left (d x + c\right )} d}\right )} {\left (d x + c\right )}^{2}}{2 \, b^{3} d^{4}} - \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{b^{3} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.55, size = 144, normalized size = 1.31 \begin {gather*} \frac {\ln \left (c+d\,x\right )\,\left (3\,b\,c^4-4\,a\,c^3\,d\right )}{a^2\,d^6-2\,a\,b\,c\,d^5+b^2\,c^2\,d^4}+\frac {x^2}{2\,b\,d^2}+\frac {a^4\,\ln \left (a+b\,x\right )}{b^3\,{\left (a\,d-b\,c\right )}^2}-\frac {x\,\left (a\,d^2+2\,b\,c\,d\right )}{b^2\,d^4}-\frac {b^2\,c^4}{d\,\left (x\,b^2\,d^4+c\,b^2\,d^3\right )\,\left (a\,d-b\,c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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