Optimal. Leaf size=209 \[ -\frac {f (c f (2+m)-d e (4+m+n)) (b x)^{1+m} (c+d x)^{1+n}}{b d^2 (2+m+n) (3+m+n)}+\frac {f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f (1+m) (3+m+n)+d^2 e^2 \left (6+m^2+5 n+n^2+m (5+2 n)\right )\right ) (b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {d x}{c}\right )}{b d^2 (1+m) (2+m+n) (3+m+n)} \]
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Rubi [A]
time = 0.12, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {92, 81, 68, 66}
\begin {gather*} \frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \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 ) \, _2F_1\left (m+1,-n;m+2;-\frac {d x}{c}\right )}{b d^2 (m+1) (m+n+2) (m+n+3)}-\frac {f (b x)^{m+1} (c+d x)^{n+1} (c f (m+2)-d e (m+n+4))}{b d^2 (m+n+2) (m+n+3)}+\frac {f (b x)^{m+1} (e+f x) (c+d x)^{n+1}}{b d (m+n+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 68
Rule 81
Rule 92
Rubi steps
\begin {align*} \int (b x)^m (c+d x)^n (e+f x)^2 \, dx &=\frac {f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\int (b x)^m (c+d x)^n (-b e (c f (1+m)-d e (3+m+n))-b f (c f (2+m)-d e (4+m+n)) x) \, dx}{b d (3+m+n)}\\ &=-\frac {f (c f (2+m)-d e (4+m+n)) (b x)^{1+m} (c+d x)^{1+n}}{b d^2 (2+m+n) (3+m+n)}+\frac {f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f (1+m) (3+m+n)+d^2 e^2 \left (6+m^2+5 n+n^2+m (5+2 n)\right )\right ) \int (b x)^m (c+d x)^n \, dx}{d^2 (2+m+n) (3+m+n)}\\ &=-\frac {f (c f (2+m)-d e (4+m+n)) (b x)^{1+m} (c+d x)^{1+n}}{b d^2 (2+m+n) (3+m+n)}+\frac {f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\left (\left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f (1+m) (3+m+n)+d^2 e^2 \left (6+m^2+5 n+n^2+m (5+2 n)\right )\right ) (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n}\right ) \int (b x)^m \left (1+\frac {d x}{c}\right )^n \, dx}{d^2 (2+m+n) (3+m+n)}\\ &=-\frac {f (c f (2+m)-d e (4+m+n)) (b x)^{1+m} (c+d x)^{1+n}}{b d^2 (2+m+n) (3+m+n)}+\frac {f (b x)^{1+m} (c+d x)^{1+n} (e+f x)}{b d (3+m+n)}+\frac {\left (c^2 f^2 \left (2+3 m+m^2\right )-2 c d e f (1+m) (3+m+n)+d^2 e^2 \left (6+m^2+5 n+n^2+m (5+2 n)\right )\right ) (b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {d x}{c}\right )}{b d^2 (1+m) (2+m+n) (3+m+n)}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 153, normalized size = 0.73 \begin {gather*} \frac {x (b x)^m (c+d x)^n \left (f (c+d x) (e+f x)-\frac {f (1+m) (c f (2+m)-d e (4+m+n)) (c+d x)+(d e (2+m+n) (c f (1+m)-d e (3+m+n))-c f (1+m) (c f (2+m)-d e (4+m+n))) \left (1+\frac {d x}{c}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {d x}{c}\right )}{d (1+m) (2+m+n)}\right )}{d (3+m+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (b x \right )^{m} \left (d x +c \right )^{n} \left (f x +e \right )^{2}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 11.01, size = 131, normalized size = 0.63 \begin {gather*} \frac {b^{m} c^{n} e^{2} x x^{m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 2\right )} + \frac {2 b^{m} c^{n} e f x^{2} x^{m} \Gamma \left (m + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 2 \\ m + 3 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 3\right )} + \frac {b^{m} c^{n} f^{2} x^{3} x^{m} \Gamma \left (m + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 3 \\ m + 4 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 4\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (e+f\,x\right )}^2\,{\left (b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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