Optimal. Leaf size=125 \[ \frac {b^2 x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {b x}{a}\right )}{a (m+1) (b c-a d)^2}-\frac {d x^{m+1} (a d m+b c (1-m)) \, _2F_1\left (1,m+1;m+2;-\frac {d x}{c}\right )}{c^2 (m+1) (b c-a d)^2}-\frac {d x^{m+1}}{c (c+d x) (b c-a d)} \]
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Rubi [A] time = 0.10, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {103, 156, 64} \[ \frac {b^2 x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {b x}{a}\right )}{a (m+1) (b c-a d)^2}-\frac {d x^{m+1} (a d m+b c (1-m)) \, _2F_1\left (1,m+1;m+2;-\frac {d x}{c}\right )}{c^2 (m+1) (b c-a d)^2}-\frac {d x^{m+1}}{c (c+d x) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 64
Rule 103
Rule 156
Rubi steps
\begin {align*} \int \frac {x^m}{(a+b x) (c+d x)^2} \, dx &=-\frac {d x^{1+m}}{c (b c-a d) (c+d x)}-\frac {\int \frac {x^m (-b c-a d m-b d m x)}{(a+b x) (c+d x)} \, dx}{c (b c-a d)}\\ &=-\frac {d x^{1+m}}{c (b c-a d) (c+d x)}+\frac {b^2 \int \frac {x^m}{a+b x} \, dx}{(b c-a d)^2}-\frac {(d (a d m+b (c-c m))) \int \frac {x^m}{c+d x} \, dx}{c (b c-a d)^2}\\ &=-\frac {d x^{1+m}}{c (b c-a d) (c+d x)}+\frac {b^2 x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a (b c-a d)^2 (1+m)}-\frac {d (a d m+b (c-c m)) x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {d x}{c}\right )}{c^2 (b c-a d)^2 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 113, normalized size = 0.90 \[ \frac {x^{m+1} \left (b^2 c^2 (c+d x) \, _2F_1\left (1,m+1;m+2;-\frac {b x}{a}\right )+a d \left ((c+d x) (b c (m-1)-a d m) \, _2F_1\left (1,m+1;m+2;-\frac {d x}{c}\right )-c (m+1) (b c-a d)\right )\right )}{a c^2 (m+1) (c+d x) (b c-a d)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{b d^{2} x^{3} + a c^{2} + {\left (2 \, b c d + a d^{2}\right )} x^{2} + {\left (b c^{2} + 2 \, a c d\right )} x}, x\right ) \]
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 \[ \int \frac {x^{m}}{{\left (b x + a\right )} {\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (b x +a \right ) \left (d x +c \right )^{2}}\, dx \]
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 \[ \int \frac {x^{m}}{{\left (b x + a\right )} {\left (d x + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {x^m}{\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]
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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