Optimal. Leaf size=199 \[ \frac {(a+b x)^{n+1} (c+d x)^{-n} \left (a^2 d^2 \left (n^2-3 n+2\right )+2 a b c d \left (1-n^2\right )+b^2 c^2 \left (n^2+3 n+2\right )\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{6 b^3 d^2 (n+1)}-\frac {(a+b x)^{n+1} (c+d x)^{1-n} (a d (2-n)+b c (n+2))}{6 b^2 d^2}+\frac {x (a+b x)^{n+1} (c+d x)^{1-n}}{3 b d} \]
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Rubi [A] time = 0.14, antiderivative size = 199, 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 = {90, 80, 70, 69} \[ \frac {(a+b x)^{n+1} (c+d x)^{-n} \left (a^2 d^2 \left (n^2-3 n+2\right )+2 a b c d \left (1-n^2\right )+b^2 c^2 \left (n^2+3 n+2\right )\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{6 b^3 d^2 (n+1)}-\frac {(a+b x)^{n+1} (c+d x)^{1-n} (a d (2-n)+b c (n+2))}{6 b^2 d^2}+\frac {x (a+b x)^{n+1} (c+d x)^{1-n}}{3 b d} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 80
Rule 90
Rubi steps
\begin {align*} \int x^2 (a+b x)^n (c+d x)^{-n} \, dx &=\frac {x (a+b x)^{1+n} (c+d x)^{1-n}}{3 b d}+\frac {\int (a+b x)^n (c+d x)^{-n} (-a c-(a d (2-n)+b c (2+n)) x) \, dx}{3 b d}\\ &=-\frac {(a d (2-n)+b c (2+n)) (a+b x)^{1+n} (c+d x)^{1-n}}{6 b^2 d^2}+\frac {x (a+b x)^{1+n} (c+d x)^{1-n}}{3 b d}+\frac {\left (2 a b c d \left (1-n^2\right )+a^2 d^2 \left (2-3 n+n^2\right )+b^2 c^2 \left (2+3 n+n^2\right )\right ) \int (a+b x)^n (c+d x)^{-n} \, dx}{6 b^2 d^2}\\ &=-\frac {(a d (2-n)+b c (2+n)) (a+b x)^{1+n} (c+d x)^{1-n}}{6 b^2 d^2}+\frac {x (a+b x)^{1+n} (c+d x)^{1-n}}{3 b d}+\frac {\left (\left (2 a b c d \left (1-n^2\right )+a^2 d^2 \left (2-3 n+n^2\right )+b^2 c^2 \left (2+3 n+n^2\right )\right ) (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n\right ) \int (a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-n} \, dx}{6 b^2 d^2}\\ &=-\frac {(a d (2-n)+b c (2+n)) (a+b x)^{1+n} (c+d x)^{1-n}}{6 b^2 d^2}+\frac {x (a+b x)^{1+n} (c+d x)^{1-n}}{3 b d}+\frac {\left (2 a b c d \left (1-n^2\right )+a^2 d^2 \left (2-3 n+n^2\right )+b^2 c^2 \left (2+3 n+n^2\right )\right ) (a+b x)^{1+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{6 b^3 d^2 (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 154, normalized size = 0.77 \[ \frac {(a+b x)^{n+1} (c+d x)^{-n} \left (\frac {\left (a^2 d^2 \left (n^2-3 n+2\right )-2 a b c d \left (n^2-1\right )+b^2 c^2 \left (n^2+3 n+2\right )\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,n+1;n+2;\frac {d (a+b x)}{a d-b c}\right )}{n+1}+b (c+d x) (a d (n-2)-b c (n+2))+2 b^2 d x (c+d x)\right )}{6 b^3 d^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{n} x^{2}}{{\left (d x + c\right )}^{n}}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )}^{n} x^{2}}{{\left (d x + c\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int x^{2} \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{n} x^{2}}{{\left (d x + c\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n} \,d x \]
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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