Integrand size = 15, antiderivative size = 98 \[ \int \frac {(a+b x)^4}{c+d x} \, dx=-\frac {b (b c-a d)^3 x}{d^4}+\frac {(b c-a d)^2 (a+b x)^2}{2 d^3}-\frac {(b c-a d) (a+b x)^3}{3 d^2}+\frac {(a+b x)^4}{4 d}+\frac {(b c-a d)^4 \log (c+d x)}{d^5} \]
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Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {(a+b x)^4}{c+d x} \, dx=\frac {(b c-a d)^4 \log (c+d x)}{d^5}-\frac {b x (b c-a d)^3}{d^4}+\frac {(a+b x)^2 (b c-a d)^2}{2 d^3}-\frac {(a+b x)^3 (b c-a d)}{3 d^2}+\frac {(a+b x)^4}{4 d} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx \\ & = -\frac {b (b c-a d)^3 x}{d^4}+\frac {(b c-a d)^2 (a+b x)^2}{2 d^3}-\frac {(b c-a d) (a+b x)^3}{3 d^2}+\frac {(a+b x)^4}{4 d}+\frac {(b c-a d)^4 \log (c+d x)}{d^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^4}{c+d x} \, dx=\frac {b d x \left (48 a^3 d^3+36 a^2 b d^2 (-2 c+d x)+8 a b^2 d \left (6 c^2-3 c d x+2 d^2 x^2\right )+b^3 \left (-12 c^3+6 c^2 d x-4 c d^2 x^2+3 d^3 x^3\right )\right )+12 (b c-a d)^4 \log (c+d x)}{12 d^5} \]
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Time = 0.24 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.71
method | result | size |
norman | \(\frac {b \left (4 a^{3} d^{3}-6 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{d^{4}}+\frac {b^{4} x^{4}}{4 d}+\frac {b^{2} \left (6 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) x^{2}}{2 d^{3}}+\frac {b^{3} \left (4 a d -b c \right ) x^{3}}{3 d^{2}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (d x +c \right )}{d^{5}}\) | \(168\) |
default | \(\frac {b \left (\frac {d^{3} x^{4} b^{3}}{4}+\frac {\left (\left (2 a d -b c \right ) b^{2} d^{2}+2 a \,b^{2} d^{3}\right ) x^{3}}{3}+\frac {\left (2 \left (2 a d -b c \right ) a b \,d^{2}+b d \left (2 a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )\right ) x^{2}}{2}+\left (2 a d -b c \right ) \left (2 a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x \right )}{d^{4}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (d x +c \right )}{d^{5}}\) | \(189\) |
risch | \(\frac {b^{4} x^{4}}{4 d}+\frac {4 b^{3} x^{3} a}{3 d}-\frac {b^{4} x^{3} c}{3 d^{2}}+\frac {3 b^{2} x^{2} a^{2}}{d}-\frac {2 b^{3} x^{2} a c}{d^{2}}+\frac {b^{4} x^{2} c^{2}}{2 d^{3}}+\frac {4 b \,a^{3} x}{d}-\frac {6 b^{2} a^{2} c x}{d^{2}}+\frac {4 b^{3} a \,c^{2} x}{d^{3}}-\frac {b^{4} c^{3} x}{d^{4}}+\frac {\ln \left (d x +c \right ) a^{4}}{d}-\frac {4 \ln \left (d x +c \right ) a^{3} b c}{d^{2}}+\frac {6 \ln \left (d x +c \right ) a^{2} b^{2} c^{2}}{d^{3}}-\frac {4 \ln \left (d x +c \right ) a \,b^{3} c^{3}}{d^{4}}+\frac {\ln \left (d x +c \right ) b^{4} c^{4}}{d^{5}}\) | \(209\) |
parallelrisch | \(\frac {3 d^{4} x^{4} b^{4}+16 a \,b^{3} d^{4} x^{3}-4 b^{4} c \,d^{3} x^{3}+36 a^{2} b^{2} d^{4} x^{2}-24 a \,b^{3} c \,d^{3} x^{2}+6 b^{4} c^{2} d^{2} x^{2}+12 \ln \left (d x +c \right ) a^{4} d^{4}-48 \ln \left (d x +c \right ) a^{3} b c \,d^{3}+72 \ln \left (d x +c \right ) a^{2} b^{2} c^{2} d^{2}-48 \ln \left (d x +c \right ) a \,b^{3} c^{3} d +12 \ln \left (d x +c \right ) b^{4} c^{4}+48 a^{3} b \,d^{4} x -72 a^{2} b^{2} c \,d^{3} x +48 a \,b^{3} c^{2} d^{2} x -12 b^{4} c^{3} d x}{12 d^{5}}\) | \(209\) |
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Time = 0.23 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x)^4}{c+d x} \, dx=\frac {3 \, b^{4} d^{4} x^{4} - 4 \, {\left (b^{4} c d^{3} - 4 \, a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} d^{2} - 4 \, a b^{3} c d^{3} + 6 \, a^{2} b^{2} d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{3} d - 4 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} - 4 \, a^{3} b d^{4}\right )} x + 12 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (d x + c\right )}{12 \, d^{5}} \]
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Time = 0.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x)^4}{c+d x} \, dx=\frac {b^{4} x^{4}}{4 d} + x^{3} \cdot \left (\frac {4 a b^{3}}{3 d} - \frac {b^{4} c}{3 d^{2}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} b^{2}}{d} - \frac {2 a b^{3} c}{d^{2}} + \frac {b^{4} c^{2}}{2 d^{3}}\right ) + x \left (\frac {4 a^{3} b}{d} - \frac {6 a^{2} b^{2} c}{d^{2}} + \frac {4 a b^{3} c^{2}}{d^{3}} - \frac {b^{4} c^{3}}{d^{4}}\right ) + \frac {\left (a d - b c\right )^{4} \log {\left (c + d x \right )}}{d^{5}} \]
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Time = 0.21 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.81 \[ \int \frac {(a+b x)^4}{c+d x} \, dx=\frac {3 \, b^{4} d^{3} x^{4} - 4 \, {\left (b^{4} c d^{2} - 4 \, a b^{3} d^{3}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} d - 4 \, a b^{3} c d^{2} + 6 \, a^{2} b^{2} d^{3}\right )} x^{2} - 12 \, {\left (b^{4} c^{3} - 4 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x}{12 \, d^{4}} + \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (d x + c\right )}{d^{5}} \]
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Time = 0.34 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.88 \[ \int \frac {(a+b x)^4}{c+d x} \, dx=\frac {3 \, b^{4} d^{3} x^{4} - 4 \, b^{4} c d^{2} x^{3} + 16 \, a b^{3} d^{3} x^{3} + 6 \, b^{4} c^{2} d x^{2} - 24 \, a b^{3} c d^{2} x^{2} + 36 \, a^{2} b^{2} d^{3} x^{2} - 12 \, b^{4} c^{3} x + 48 \, a b^{3} c^{2} d x - 72 \, a^{2} b^{2} c d^{2} x + 48 \, a^{3} b d^{3} x}{12 \, d^{4}} + \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.93 \[ \int \frac {(a+b x)^4}{c+d x} \, dx=x^3\,\left (\frac {4\,a\,b^3}{3\,d}-\frac {b^4\,c}{3\,d^2}\right )+x\,\left (\frac {4\,a^3\,b}{d}+\frac {c\,\left (\frac {c\,\left (\frac {4\,a\,b^3}{d}-\frac {b^4\,c}{d^2}\right )}{d}-\frac {6\,a^2\,b^2}{d}\right )}{d}\right )-x^2\,\left (\frac {c\,\left (\frac {4\,a\,b^3}{d}-\frac {b^4\,c}{d^2}\right )}{2\,d}-\frac {3\,a^2\,b^2}{d}\right )+\frac {\ln \left (c+d\,x\right )\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{d^5}+\frac {b^4\,x^4}{4\,d} \]
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