Optimal. Leaf size=65 \[ \frac {(a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (4,m+1;m+2;-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (m+1) (b c-a d)} \]
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Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {131} \[ \frac {(a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (4,m+1;m+2;-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (m+1) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 131
Rubi steps
\begin {align*} \int \frac {(a+b x)^m (c+d x)^{2-m}}{(b c+a d+2 b d x)^4} \, dx &=\frac {(a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (4,1+m;2+m;-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (b c-a d) (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 65, normalized size = 1.00 \[ \frac {(a+b x)^{m+1} (c+d x)^{-m-1} \, _2F_1\left (4,m+1;m+2;-\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 (m+1) (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2}}{16 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + a^{4} d^{4} + 32 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + 24 \, {\left (b^{4} c^{2} d^{2} + 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 8 \, {\left (b^{4} c^{3} d + 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2}}{{\left (2 \, b d x + b c + a d\right )}^{4}}\,{d x} \]
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 \[ \int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-m +2}}{\left (2 b d x +a d +b c \right )^{4}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 2}}{{\left (2 \, b d x + b c + a d\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{2-m}}{{\left (a\,d+b\,c+2\,b\,d\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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