Optimal. Leaf size=260 \[ \frac {(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \left (a^2 d^2 f^2 \left (m^2-5 m+6\right )-2 a b d f (2-m) (4 d e-c f (m+1))+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )-8 c d e f (m+1)+12 d^2 e^2\right )\right ) \, _2F_1\left (m-1,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{12 b^4 d^2 (m+1)}-\frac {f (a+b x)^{m+1} (c+d x)^{2-m} (a d f (3-m)-b (5 d e-c f (m+2)))}{12 b^2 d^2}+\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{2-m}}{4 b d} \]
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Rubi [A] time = 0.23, antiderivative size = 259, normalized size of antiderivative = 1.00, 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 = {90, 80, 70, 69} \[ \frac {(b c-a d) (a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \left (a^2 d^2 f^2 \left (m^2-5 m+6\right )-2 a b d f (2-m) (4 d e-c f (m+1))+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )-8 c d e f (m+1)+12 d^2 e^2\right )\right ) \, _2F_1\left (m-1,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{12 b^4 d^2 (m+1)}+\frac {f (a+b x)^{m+1} (c+d x)^{2-m} (-a d f (3-m)-b c f (m+2)+5 b d e)}{12 b^2 d^2}+\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{2-m}}{4 b d} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 80
Rule 90
Rubi steps
\begin {align*} \int (a+b x)^m (c+d x)^{1-m} (e+f x)^2 \, dx &=\frac {f (a+b x)^{1+m} (c+d x)^{2-m} (e+f x)}{4 b d}+\frac {\int (a+b x)^m (c+d x)^{1-m} (-a f (c f+d e (2-m))+b e (4 d e-c f (1+m))+f (5 b d e-a d f (3-m)-b c f (2+m)) x) \, dx}{4 b d}\\ &=\frac {f (5 b d e-a d f (3-m)-b c f (2+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{12 b^2 d^2}+\frac {f (a+b x)^{1+m} (c+d x)^{2-m} (e+f x)}{4 b d}+\frac {\left (a^2 d^2 f^2 \left (6-5 m+m^2\right )-2 a b d f (2-m) (4 d e-c f (1+m))+b^2 \left (12 d^2 e^2-8 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) \int (a+b x)^m (c+d x)^{1-m} \, dx}{12 b^2 d^2}\\ &=\frac {f (5 b d e-a d f (3-m)-b c f (2+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{12 b^2 d^2}+\frac {f (a+b x)^{1+m} (c+d x)^{2-m} (e+f x)}{4 b d}+\frac {\left ((b c-a d) \left (a^2 d^2 f^2 \left (6-5 m+m^2\right )-2 a b d f (2-m) (4 d e-c f (1+m))+b^2 \left (12 d^2 e^2-8 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{1-m} \, dx}{12 b^3 d^2}\\ &=\frac {f (5 b d e-a d f (3-m)-b c f (2+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{12 b^2 d^2}+\frac {f (a+b x)^{1+m} (c+d x)^{2-m} (e+f x)}{4 b d}+\frac {(b c-a d) \left (a^2 d^2 f^2 \left (6-5 m+m^2\right )-2 a b d f (2-m) (4 d e-c f (1+m))+b^2 \left (12 d^2 e^2-8 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (-1+m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{12 b^4 d^2 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 223, normalized size = 0.86 \[ \frac {(a+b x)^{m+1} (c+d x)^{-m} \left ((b c-a d) \left (\frac {b (c+d x)}{b c-a d}\right )^m \left (a^2 d^2 f^2 \left (m^2-5 m+6\right )-2 a b d f (m-2) (c f (m+1)-4 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )-8 c d e f (m+1)+12 d^2 e^2\right )\right ) \, _2F_1\left (m-1,m+1;m+2;\frac {d (a+b x)}{a d-b c}\right )+b^2 f (m+1) (c+d x)^2 (a d f (m-3)-b c f (m+2)+5 b d e)+3 b^3 d f (m+1) (c+d x)^2 (e+f x)\right )}{12 b^4 d^2 (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}, x\right ) \]
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 \[ \int {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}\,{d x} \]
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 \[ \int \left (f x +e \right )^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{-m +1}\, dx \]
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 \[ \int {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 1}\,{d x} \]
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 \[ \int {\left (e+f\,x\right )}^2\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{1-m} \,d x \]
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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