Integrand size = 51, antiderivative size = 97 \[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=-\frac {(a+b x)^{-\frac {b c}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{b c}+\frac {(a+b x)^{-\frac {a d}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{a b c} \]
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Time = 0.01 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {47, 37} \[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\frac {(a+b x)^{-\frac {a d}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{a b c}-\frac {(a+b x)^{-\frac {b c}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{b c} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{-\frac {b c}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{b c}-\frac {d \int (a+b x)^{\frac {b c}{-b c+a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx}{b c} \\ & = -\frac {(a+b x)^{-\frac {b c}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{b c}+\frac {(a+b x)^{-\frac {a d}{b c-a d}} (c+d x)^{\frac {a d}{b c-a d}}}{a b c} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.47 \[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\frac {x (a+b x)^{\frac {b c}{-b c+a d}} (c+d x)^{\frac {a d}{b c-a d}}}{a c} \]
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Time = 0.61 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.68
method | result | size |
gosper | \(\frac {x \left (b x +a \right )^{1-\frac {a d -2 b c}{a d -b c}} \left (d x +c \right )^{1-\frac {2 a d -b c}{a d -b c}}}{a c}\) | \(66\) |
parallelrisch | \(\frac {x^{3} \left (b x +a \right )^{-\frac {a d -2 b c}{a d -b c}} \left (d x +c \right )^{-\frac {2 a d -b c}{a d -b c}} b^{2} d^{2}+x^{2} \left (b x +a \right )^{-\frac {a d -2 b c}{a d -b c}} \left (d x +c \right )^{-\frac {2 a d -b c}{a d -b c}} a b \,d^{2}+x^{2} \left (b x +a \right )^{-\frac {a d -2 b c}{a d -b c}} \left (d x +c \right )^{-\frac {2 a d -b c}{a d -b c}} b^{2} c d +x \left (b x +a \right )^{-\frac {a d -2 b c}{a d -b c}} \left (d x +c \right )^{-\frac {2 a d -b c}{a d -b c}} a b c d}{a b c d}\) | \(261\) |
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Time = 0.24 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\frac {b d x^{3} + a c x + {\left (b c + a d\right )} x^{2}}{{\left (b x + a\right )}^{\frac {2 \, b c - a d}{b c - a d}} {\left (d x + c\right )}^{\frac {b c - 2 \, a d}{b c - a d}} a c} \]
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Timed out. \[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\text {Timed out} \]
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\[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {2 \, b c - a d}{b c - a d}} {\left (d x + c\right )}^{\frac {b c - 2 \, a d}{b c - a d}}} \,d x } \]
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\[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {2 \, b c - a d}{b c - a d}} {\left (d x + c\right )}^{\frac {b c - 2 \, a d}{b c - a d}}} \,d x } \]
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Time = 0.90 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.46 \[ \int (a+b x)^{\frac {-2 b c+a d}{b c-a d}} (c+d x)^{\frac {b c-2 a d}{-b c+a d}} \, dx=\frac {\frac {x}{{\left (a+b\,x\right )}^{\frac {a\,d-2\,b\,c}{a\,d-b\,c}}}+\frac {x^2\,\left (a\,d+b\,c\right )}{a\,c\,{\left (a+b\,x\right )}^{\frac {a\,d-2\,b\,c}{a\,d-b\,c}}}+\frac {b\,d\,x^3}{a\,c\,{\left (a+b\,x\right )}^{\frac {a\,d-2\,b\,c}{a\,d-b\,c}}}}{{\left (c+d\,x\right )}^{\frac {2\,a\,d-b\,c}{a\,d-b\,c}}} \]
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