Optimal. Leaf size=205 \[ -\frac {f^2 (a+b x)^m (c+d x)^{-m} \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac {b (c+d x)}{b c-a d}\right )}{d^3 m}+\frac {(a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^2 (m+2) (b c-a d)}-\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1} (2 a d f (m+2)-b (c f (2 m+3)+d e))}{d^2 (m+1) (m+2) (b c-a d)^2} \]
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Rubi [A] time = 0.19, antiderivative size = 202, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {89, 79, 70, 69} \[ \frac {(a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m-2}}{d^2 (m+2) (b c-a d)}+\frac {(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-1} (-2 a d f (m+2)+b c f (2 m+3)+b d e)}{d^2 (m+1) (m+2) (b c-a d)^2}-\frac {f^2 (a+b x)^m (c+d x)^{-m} \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac {b (c+d x)}{b c-a d}\right )}{d^3 m} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 79
Rule 89
Rubi steps
\begin {align*} \int (a+b x)^m (c+d x)^{-3-m} (e+f x)^2 \, dx &=\frac {(d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^2 (b c-a d) (2+m)}-\frac {\int (a+b x)^m (c+d x)^{-2-m} \left (a d f (2 d e-c f) (2+m)-b \left (d^2 e^2+2 c d e f (1+m)-c^2 f^2 (1+m)\right )-d (b c-a d) f^2 (2+m) x\right ) \, dx}{d^2 (b c-a d) (2+m)}\\ &=\frac {(d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^2 (b c-a d) (2+m)}+\frac {(d e-c f) (b d e-2 a d f (2+m)+b c f (3+2 m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^2 (b c-a d)^2 (1+m) (2+m)}+\frac {f^2 \int (a+b x)^m (c+d x)^{-1-m} \, dx}{d^2}\\ &=\frac {(d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^2 (b c-a d) (2+m)}+\frac {(d e-c f) (b d e-2 a d f (2+m)+b c f (3+2 m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^2 (b c-a d)^2 (1+m) (2+m)}+\frac {\left (f^2 (a+b x)^m \left (\frac {d (a+b x)}{-b c+a d}\right )^{-m}\right ) \int (c+d x)^{-1-m} \left (-\frac {a d}{b c-a d}-\frac {b d x}{b c-a d}\right )^m \, dx}{d^2}\\ &=\frac {(d e-c f)^2 (a+b x)^{1+m} (c+d x)^{-2-m}}{d^2 (b c-a d) (2+m)}+\frac {(d e-c f) (b d e-2 a d f (2+m)+b c f (3+2 m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{d^2 (b c-a d)^2 (1+m) (2+m)}-\frac {f^2 (a+b x)^m \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} (c+d x)^{-m} \, _2F_1\left (-m,-m;1-m;\frac {b (c+d x)}{b c-a d}\right )}{d^3 m}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 184, normalized size = 0.90 \[ \frac {(a+b x)^m (c+d x)^{-m} \left (\frac {(a+b x) (d e-c f) (-2 a d f (m+2)+b c f (2 m+3)+b d e)}{(m+1) (c+d x) (b c-a d)}+\frac {(a+b x) (d e-c f)^2}{(c+d x)^2}+\frac {f^2 (m+2) (a+b x) \left (\frac {d (a+b x)}{a d-b c}\right )^{-m-1} \, _2F_1\left (-m,-m;1-m;\frac {b (c+d x)}{b c-a d}\right )}{m}\right )}{d^2 (m+2) (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 3}, x\right ) \]
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 \[ \int {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right )^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{-m -3}\, dx \]
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 \[ \int {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e+f\,x\right )}^2\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^{m+3}} \,d x \]
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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