Optimal. Leaf size=122 \[ \frac {b (b c-a d)^4 x}{d^5}-\frac {(b c-a d)^3 (a+b x)^2}{2 d^4}+\frac {(b c-a d)^2 (a+b x)^3}{3 d^3}-\frac {(b c-a d) (a+b x)^4}{4 d^2}+\frac {(a+b x)^5}{5 d}-\frac {(b c-a d)^5 \log (c+d x)}{d^6} \]
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Rubi [A]
time = 0.05, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 45}
\begin {gather*} -\frac {(b c-a d)^5 \log (c+d x)}{d^6}+\frac {b x (b c-a d)^4}{d^5}-\frac {(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac {(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac {(a+b x)^4 (b c-a d)}{4 d^2}+\frac {(a+b x)^5}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {(a+b x)^6}{a c+(b c+a d) x+b d x^2} \, dx &=\int \frac {(a+b x)^5}{c+d x} \, dx\\ &=\int \left (\frac {b (b c-a d)^4}{d^5}-\frac {b (b c-a d)^3 (a+b x)}{d^4}+\frac {b (b c-a d)^2 (a+b x)^2}{d^3}-\frac {b (b c-a d) (a+b x)^3}{d^2}+\frac {b (a+b x)^4}{d}+\frac {(-b c+a d)^5}{d^5 (c+d x)}\right ) \, dx\\ &=\frac {b (b c-a d)^4 x}{d^5}-\frac {(b c-a d)^3 (a+b x)^2}{2 d^4}+\frac {(b c-a d)^2 (a+b x)^3}{3 d^3}-\frac {(b c-a d) (a+b x)^4}{4 d^2}+\frac {(a+b x)^5}{5 d}-\frac {(b c-a d)^5 \log (c+d x)}{d^6}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 167, normalized size = 1.37 \begin {gather*} \frac {b d x \left (300 a^4 d^4+300 a^3 b d^3 (-2 c+d x)+100 a^2 b^2 d^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )+25 a b^3 d \left (-12 c^3+6 c^2 d x-4 c d^2 x^2+3 d^3 x^3\right )+b^4 \left (60 c^4-30 c^3 d x+20 c^2 d^2 x^2-15 c d^3 x^3+12 d^4 x^4\right )\right )-60 (b c-a d)^5 \log (c+d x)}{60 d^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(265\) vs.
\(2(114)=228\).
time = 0.67, size = 266, normalized size = 2.18
method | result | size |
norman | \(\frac {b \left (5 a^{4} d^{4}-10 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}-5 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x}{d^{5}}+\frac {b^{5} x^{5}}{5 d}+\frac {b^{2} \left (10 a^{3} d^{3}-10 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{2}}{2 d^{4}}+\frac {b^{3} \left (10 a^{2} d^{2}-5 a b c d +b^{2} c^{2}\right ) x^{3}}{3 d^{3}}+\frac {b^{4} \left (5 a d -b c \right ) x^{4}}{4 d^{2}}+\frac {\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \ln \left (d x +c \right )}{d^{6}}\) | \(244\) |
default | \(\frac {b \left (\frac {1}{5} b^{4} x^{5} d^{4}+\frac {5}{4} a \,b^{3} d^{4} x^{4}-\frac {1}{4} b^{4} c \,d^{3} x^{4}+\frac {10}{3} a^{2} b^{2} d^{4} x^{3}-\frac {5}{3} a \,b^{3} c \,d^{3} x^{3}+\frac {1}{3} b^{4} c^{2} d^{2} x^{3}+5 a^{3} b \,d^{4} x^{2}-5 a^{2} b^{2} c \,d^{3} x^{2}+\frac {5}{2} a \,b^{3} c^{2} d^{2} x^{2}-\frac {1}{2} b^{4} c^{3} d \,x^{2}+5 a^{4} d^{4} x -10 a^{3} b c \,d^{3} x +10 a^{2} b^{2} c^{2} d^{2} x -5 a \,b^{3} c^{3} d x +b^{4} c^{4} x \right )}{d^{5}}+\frac {\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \ln \left (d x +c \right )}{d^{6}}\) | \(266\) |
risch | \(\frac {b^{5} x^{5}}{5 d}+\frac {5 b^{4} a \,x^{4}}{4 d}-\frac {b^{5} c \,x^{4}}{4 d^{2}}+\frac {10 b^{3} a^{2} x^{3}}{3 d}-\frac {5 b^{4} a c \,x^{3}}{3 d^{2}}+\frac {b^{5} c^{2} x^{3}}{3 d^{3}}+\frac {5 b^{2} a^{3} x^{2}}{d}-\frac {5 b^{3} a^{2} c \,x^{2}}{d^{2}}+\frac {5 b^{4} a \,c^{2} x^{2}}{2 d^{3}}-\frac {b^{5} c^{3} x^{2}}{2 d^{4}}+\frac {5 b \,a^{4} x}{d}-\frac {10 b^{2} a^{3} c x}{d^{2}}+\frac {10 b^{3} a^{2} c^{2} x}{d^{3}}-\frac {5 b^{4} a \,c^{3} x}{d^{4}}+\frac {b^{5} c^{4} x}{d^{5}}+\frac {\ln \left (d x +c \right ) a^{5}}{d}-\frac {5 \ln \left (d x +c \right ) a^{4} b c}{d^{2}}+\frac {10 \ln \left (d x +c \right ) a^{3} b^{2} c^{2}}{d^{3}}-\frac {10 \ln \left (d x +c \right ) a^{2} b^{3} c^{3}}{d^{4}}+\frac {5 \ln \left (d x +c \right ) a \,b^{4} c^{4}}{d^{5}}-\frac {\ln \left (d x +c \right ) b^{5} c^{5}}{d^{6}}\) | \(302\) |
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. 258 vs.
\(2 (114) = 228\).
time = 0.27, size = 258, normalized size = 2.11 \begin {gather*} \frac {12 \, b^{5} d^{4} x^{5} - 15 \, {\left (b^{5} c d^{3} - 5 \, a b^{4} d^{4}\right )} x^{4} + 20 \, {\left (b^{5} c^{2} d^{2} - 5 \, a b^{4} c d^{3} + 10 \, a^{2} b^{3} d^{4}\right )} x^{3} - 30 \, {\left (b^{5} c^{3} d - 5 \, a b^{4} c^{2} d^{2} + 10 \, a^{2} b^{3} c d^{3} - 10 \, a^{3} b^{2} d^{4}\right )} x^{2} + 60 \, {\left (b^{5} c^{4} - 5 \, a b^{4} c^{3} d + 10 \, a^{2} b^{3} c^{2} d^{2} - 10 \, a^{3} b^{2} c d^{3} + 5 \, a^{4} b d^{4}\right )} x}{60 \, d^{5}} - \frac {{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left (d x + c\right )}{d^{6}} \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. 259 vs.
\(2 (114) = 228\).
time = 3.00, size = 259, normalized size = 2.12 \begin {gather*} \frac {12 \, b^{5} d^{5} x^{5} - 15 \, {\left (b^{5} c d^{4} - 5 \, a b^{4} d^{5}\right )} x^{4} + 20 \, {\left (b^{5} c^{2} d^{3} - 5 \, a b^{4} c d^{4} + 10 \, a^{2} b^{3} d^{5}\right )} x^{3} - 30 \, {\left (b^{5} c^{3} d^{2} - 5 \, a b^{4} c^{2} d^{3} + 10 \, a^{2} b^{3} c d^{4} - 10 \, a^{3} b^{2} d^{5}\right )} x^{2} + 60 \, {\left (b^{5} c^{4} d - 5 \, a b^{4} c^{3} d^{2} + 10 \, a^{2} b^{3} c^{2} d^{3} - 10 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x - 60 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left (d x + c\right )}{60 \, d^{6}} \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. 209 vs.
\(2 (104) = 208\).
time = 0.28, size = 209, normalized size = 1.71 \begin {gather*} \frac {b^{5} x^{5}}{5 d} + x^{4} \cdot \left (\frac {5 a b^{4}}{4 d} - \frac {b^{5} c}{4 d^{2}}\right ) + x^{3} \cdot \left (\frac {10 a^{2} b^{3}}{3 d} - \frac {5 a b^{4} c}{3 d^{2}} + \frac {b^{5} c^{2}}{3 d^{3}}\right ) + x^{2} \cdot \left (\frac {5 a^{3} b^{2}}{d} - \frac {5 a^{2} b^{3} c}{d^{2}} + \frac {5 a b^{4} c^{2}}{2 d^{3}} - \frac {b^{5} c^{3}}{2 d^{4}}\right ) + x \left (\frac {5 a^{4} b}{d} - \frac {10 a^{3} b^{2} c}{d^{2}} + \frac {10 a^{2} b^{3} c^{2}}{d^{3}} - \frac {5 a b^{4} c^{3}}{d^{4}} + \frac {b^{5} c^{4}}{d^{5}}\right ) + \frac {\left (a d - b c\right )^{5} \log {\left (c + d x \right )}}{d^{6}} \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. 273 vs.
\(2 (114) = 228\).
time = 1.01, size = 273, normalized size = 2.24 \begin {gather*} \frac {12 \, b^{5} d^{4} x^{5} - 15 \, b^{5} c d^{3} x^{4} + 75 \, a b^{4} d^{4} x^{4} + 20 \, b^{5} c^{2} d^{2} x^{3} - 100 \, a b^{4} c d^{3} x^{3} + 200 \, a^{2} b^{3} d^{4} x^{3} - 30 \, b^{5} c^{3} d x^{2} + 150 \, a b^{4} c^{2} d^{2} x^{2} - 300 \, a^{2} b^{3} c d^{3} x^{2} + 300 \, a^{3} b^{2} d^{4} x^{2} + 60 \, b^{5} c^{4} x - 300 \, a b^{4} c^{3} d x + 600 \, a^{2} b^{3} c^{2} d^{2} x - 600 \, a^{3} b^{2} c d^{3} x + 300 \, a^{4} b d^{4} x}{60 \, d^{5}} - \frac {{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.60, size = 280, normalized size = 2.30 \begin {gather*} x\,\left (\frac {5\,a^4\,b}{d}-\frac {c\,\left (\frac {10\,a^3\,b^2}{d}+\frac {c\,\left (\frac {c\,\left (\frac {5\,a\,b^4}{d}-\frac {b^5\,c}{d^2}\right )}{d}-\frac {10\,a^2\,b^3}{d}\right )}{d}\right )}{d}\right )+x^4\,\left (\frac {5\,a\,b^4}{4\,d}-\frac {b^5\,c}{4\,d^2}\right )+x^2\,\left (\frac {5\,a^3\,b^2}{d}+\frac {c\,\left (\frac {c\,\left (\frac {5\,a\,b^4}{d}-\frac {b^5\,c}{d^2}\right )}{d}-\frac {10\,a^2\,b^3}{d}\right )}{2\,d}\right )-x^3\,\left (\frac {c\,\left (\frac {5\,a\,b^4}{d}-\frac {b^5\,c}{d^2}\right )}{3\,d}-\frac {10\,a^2\,b^3}{3\,d}\right )+\frac {b^5\,x^5}{5\,d}+\frac {\ln \left (c+d\,x\right )\,\left (a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5\right )}{d^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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