Optimal. Leaf size=250 \[ \frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \left (a^2 d^2 f^2 \left (m^2-3 m+2\right )-2 a b d f (1-m) (3 d e-c f (m+1))+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )-6 c d e f (m+1)+6 d^2 e^2\right )\right ) \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{6 b^3 d^2 (m+1)}-\frac {f (a+b x)^{m+1} (c+d x)^{1-m} (a d f (2-m)-b (4 d e-c f (m+2)))}{6 b^2 d^2}+\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{1-m}}{3 b d} \]
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Rubi [A] time = 0.21, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 80, 70, 69} \[ \frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \left (a^2 d^2 f^2 \left (m^2-3 m+2\right )-2 a b d f (1-m) (3 d e-c f (m+1))+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )-6 c d e f (m+1)+6 d^2 e^2\right )\right ) \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{6 b^3 d^2 (m+1)}+\frac {f (a+b x)^{m+1} (c+d x)^{1-m} (-a d f (2-m)-b c f (m+2)+4 b d e)}{6 b^2 d^2}+\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{1-m}}{3 b d} \]
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)^{-m} (e+f x)^2 \, dx &=\frac {f (a+b x)^{1+m} (c+d x)^{1-m} (e+f x)}{3 b d}+\frac {\int (a+b x)^m (c+d x)^{-m} \left (3 b d e^2-f (a c f+a d e (1-m)+b c e (1+m))+f (4 b d e-a d f (2-m)-b c f (2+m)) x\right ) \, dx}{3 b d}\\ &=\frac {f (4 b d e-a d f (2-m)-b c f (2+m)) (a+b x)^{1+m} (c+d x)^{1-m}}{6 b^2 d^2}+\frac {f (a+b x)^{1+m} (c+d x)^{1-m} (e+f x)}{3 b d}+\frac {\left (a^2 d^2 f^2 \left (2-3 m+m^2\right )-2 a b d f (1-m) (3 d e-c f (1+m))+b^2 \left (6 d^2 e^2-6 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)^{-m} \, dx}{6 b^2 d^2}\\ &=\frac {f (4 b d e-a d f (2-m)-b c f (2+m)) (a+b x)^{1+m} (c+d x)^{1-m}}{6 b^2 d^2}+\frac {f (a+b x)^{1+m} (c+d x)^{1-m} (e+f x)}{3 b d}+\frac {\left (\left (a^2 d^2 f^2 \left (2-3 m+m^2\right )-2 a b d f (1-m) (3 d e-c f (1+m))+b^2 \left (6 d^2 e^2-6 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 )^{-m} \, dx}{6 b^2 d^2}\\ &=\frac {f (4 b d e-a d f (2-m)-b c f (2+m)) (a+b x)^{1+m} (c+d x)^{1-m}}{6 b^2 d^2}+\frac {f (a+b x)^{1+m} (c+d x)^{1-m} (e+f x)}{3 b d}+\frac {\left (a^2 d^2 f^2 \left (2-3 m+m^2\right )-2 a b d f (1-m) (3 d e-c f (1+m))+b^2 \left (6 d^2 e^2-6 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 (m,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{6 b^3 d^2 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 201, normalized size = 0.80 \[ \frac {(a+b x)^{m+1} (c+d x)^{-m} \left (\frac {\left (\frac {b (c+d x)}{b c-a d}\right )^m \left (a^2 d^2 f^2 \left (m^2-3 m+2\right )-2 a b d f (m-1) (c f (m+1)-3 d e)+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )-6 c d e f (m+1)+6 d^2 e^2\right )\right ) \, _2F_1\left (m,m+1;m+2;\frac {d (a+b x)}{a d-b c}\right )}{m+1}+b f (c+d x) (a d f (m-2)-b c f (m+2)+4 b d e)+2 b^2 d f (c+d x) (e+f x)\right )}{6 b^3 d^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )} {\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{m}}, x\right ) \]
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 \[ \int \frac {{\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{m}}\,{d x} \]
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 \[ \int \left (f x +e \right )^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{-m}\, dx \]
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 \[ \int \frac {{\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m}}{{\left (d x + c\right )}^{m}}\,{d x} \]
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 \[ \int \frac {{\left (e+f\,x\right )}^2\,{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^m} \,d x \]
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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