Optimal. Leaf size=259 \[ \frac {(a+b x)^{m+1} (c+d x)^{n+1} (f (a d (n+1)+b c (m+1)) (b d e (m+n+4)-f (a d (n+2)+b c (m+2)))+b d (m+n+2) (a f (c f+d e (n+1))+b e (c f (m+1)-d e (m+n+3)))) \, _2F_1\left (1,m+n+2;n+2;\frac {b (c+d x)}{b c-a d}\right )}{b^2 d^2 (n+1) (m+n+2) (m+n+3) (b c-a d)}+\frac {f (a+b x)^{m+1} (c+d x)^{n+1} (b d e (m+n+4)-f (a d (n+2)+b c (m+2)))}{b^2 d^2 (m+n+2) (m+n+3)}+\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+3)} \]
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Rubi [A] time = 0.34, antiderivative size = 272, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {90, 80, 70, 69} \[ \frac {(a+b x)^{m+1} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (b d (m+n+2) \left (b d e^2 (m+n+3)-f (a c f+a d e (n+1)+b c e (m+1))\right )-f (a d (n+1)+b c (m+1)) (b d e (m+n+4)-f (a d (n+2)+b c (m+2)))\right ) \, _2F_1\left (m+1,-n;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{b^3 d^2 (m+1) (m+n+2) (m+n+3)}+\frac {f (a+b x)^{m+1} (c+d x)^{n+1} (b d e (m+n+4)-f (a d (n+2)+b c (m+2)))}{b^2 d^2 (m+n+2) (m+n+3)}+\frac {f (e+f x) (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+3)} \]
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)^n (e+f x)^2 \, dx &=\frac {f (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\int (a+b x)^m (c+d x)^n \left (b d e^2 (3+m+n)-f (a c f+b c e (1+m)+a d e (1+n))+f (b d e (4+m+n)-f (b c (2+m)+a d (2+n))) x\right ) \, dx}{b d (3+m+n)}\\ &=\frac {f (b d e (4+m+n)-f (b c (2+m)+a d (2+n))) (a+b x)^{1+m} (c+d x)^{1+n}}{b^2 d^2 (2+m+n) (3+m+n)}+\frac {f (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\left (b d e^2 (3+m+n)-f (a c f+b c e (1+m)+a d e (1+n))+\frac {f (b c (1+m)+a d (1+n)) (b c f (2+m)+a d f (2+n)-b d e (4+m+n))}{b d (2+m+n)}\right ) \int (a+b x)^m (c+d x)^n \, dx}{b d (3+m+n)}\\ &=\frac {f (b d e (4+m+n)-f (b c (2+m)+a d (2+n))) (a+b x)^{1+m} (c+d x)^{1+n}}{b^2 d^2 (2+m+n) (3+m+n)}+\frac {f (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\left (\left (b d e^2 (3+m+n)-f (a c f+b c e (1+m)+a d e (1+n))+\frac {f (b c (1+m)+a d (1+n)) (b c f (2+m)+a d f (2+n)-b d e (4+m+n))}{b d (2+m+n)}\right ) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \, dx}{b d (3+m+n)}\\ &=\frac {f (b d e (4+m+n)-f (b c (2+m)+a d (2+n))) (a+b x)^{1+m} (c+d x)^{1+n}}{b^2 d^2 (2+m+n) (3+m+n)}+\frac {f (a+b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\left (b d e^2 (3+m+n)-f (a c f+b c e (1+m)+a d e (1+n))+\frac {f (b c (1+m)+a d (1+n)) (b c f (2+m)+a d f (2+n)-b d e (4+m+n))}{b d (2+m+n)}\right ) (a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{b^2 d (1+m) (3+m+n)}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 253, normalized size = 0.98 \[ \frac {(a+b x)^{m+1} (c+d x)^n \left (\frac {\left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (a^2 d^2 f^2 \left (n^2+3 n+2\right )-2 a b d f (n+1) (d e (m+n+3)-c f (m+1))+b^2 \left (c^2 f^2 \left (m^2+3 m+2\right )-2 c d e f (m+1) (m+n+3)+d^2 e^2 \left (m^2+m (2 n+5)+n^2+5 n+6\right )\right )\right ) \, _2F_1\left (m+1,-n;m+2;\frac {d (a+b x)}{a d-b c}\right )}{b^2 d (m+1) (m+n+2)}+\frac {f (c+d x) (b d e (m+n+4)-f (a d (n+2)+b c (m+2)))}{b d (m+n+2)}+f (c+d x) (e+f x)\right )}{b d (m+n+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (f^{2} x^{2} + 2 \, e f x + e^{2}\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n}, x\right ) \]
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 \[ \int {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n}\,{d x} \]
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 \[ \int \left (f x +e \right )^{2} \left (b x +a \right )^{m} \left (d x +c \right )^{n}\, dx \]
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 \[ \int {\left (f x + e\right )}^{2} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n}\,{d x} \]
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 \[ \int {\left (e+f\,x\right )}^2\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \]
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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