Optimal. Leaf size=144 \[ \frac {(b c-a d) (a+b x)^m (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (m-1,m;m+1;-\frac {d (a+b x)}{b c-a d}\right )}{4 b^3 d m}-\frac {(b c-a d) (a+b x)^m (c+d x)^{-m} \, _2F_1\left (2,m;m+1;-\frac {d (a+b x)}{b (c+d x)}\right )}{4 b^3 d m} \]
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Rubi [C] time = 0.06, antiderivative size = 93, normalized size of antiderivative = 0.65, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {137, 136} \[ \frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m F_1\left (m+1;m-2,2;m+2;-\frac {d (a+b x)}{b c-a d},-\frac {2 d (a+b x)}{b c-a d}\right )}{b^3 (m+1)} \]
Warning: Unable to verify antiderivative.
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Rule 136
Rule 137
Rubi steps
\begin {align*} \int \frac {(a+b x)^m (c+d x)^{2-m}}{(b c+a d+2 b d x)^2} \, dx &=\frac {\left ((b c-a d)^2 (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m\right ) \int \frac {(a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{2-m}}{(b c+a d+2 b d x)^2} \, dx}{b^2}\\ &=\frac {(a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m F_1\left (1+m;-2+m,2;2+m;-\frac {d (a+b x)}{b c-a d},-\frac {2 d (a+b x)}{b c-a d}\right )}{b^3 (1+m)}\\ \end {align*}
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Mathematica [C] time = 1.39, size = 308, normalized size = 2.14 \[ \frac {(a+b x)^m (c+d x)^{-m} \left (\frac {\left (\frac {d (a+b x)}{a d-b c}\right )^{-m} \left (4 b (m+1) (c+d x) F_1\left (1-m;-m,1;2-m;\frac {b (c+d x)}{b c-a d},\frac {2 b (c+d x)}{b c-a d}\right )+2 d (m-1) (a+b x) \left (-\frac {b d (a+b x) (c+d x)}{(b c-a d)^2}\right )^m \, _2F_1\left (m,m+1;m+2;\frac {d (a+b x)}{a d-b c}\right )\right )}{d (m-1) (m+1)}-\frac {(b c-a d)^2 \left (\frac {d (a+b x)}{a d+b (c+2 d x)}\right )^{1-m} \left (\frac {b (c+d x)}{a d+b (c+2 d x)}\right )^m F_1\left (1;m,-m;2;\frac {a d-b c}{a d+b (c+2 d x)},\frac {b c-a d}{b c+a d+2 b d x}\right )}{d^2 (a+b x)}\right )}{8 b^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2}}{4 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \, {\left (b^{2} c d + a b d^{2}\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2}}{{\left (2 \, b d x + b c + a d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-m +2}}{\left (2 b d x +a d +b c \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2}}{{\left (2 \, b d x + b c + a d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{2-m}}{{\left (a\,d+b\,c+2\,b\,d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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