Optimal. Leaf size=206 \[ -\frac {(a+b x)^{n+1} (c+d x)^{p+1} \left (a^2 d^2 \left (p^2+3 p+2\right )+2 a b c d (n+1) (p+1)+b^2 c^2 \left (n^2+3 n+2\right )\right ) \, _2F_1\left (1,n+p+2;p+2;\frac {b (c+d x)}{b c-a d}\right )}{b^2 d^2 (p+1) (n+p+2) (n+p+3) (b c-a d)}-\frac {(a+b x)^{n+1} (c+d x)^{p+1} (a d (p+2)+b c (n+2))}{b^2 d^2 (n+p+2) (n+p+3)}+\frac {x (a+b x)^{n+1} (c+d x)^{p+1}}{b d (n+p+3)} \]
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Rubi [A] time = 0.19, antiderivative size = 216, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {90, 80, 70, 69} \[ \frac {(a+b x)^{n+1} (c+d x)^p \left (a^2 d^2 \left (p^2+3 p+2\right )+2 a b c d (n+1) (p+1)+b^2 c^2 \left (n^2+3 n+2\right )\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \, _2F_1\left (n+1,-p;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{b^3 d^2 (n+1) (n+p+2) (n+p+3)}-\frac {(a+b x)^{n+1} (c+d x)^{p+1} (a d (p+2)+b c (n+2))}{b^2 d^2 (n+p+2) (n+p+3)}+\frac {x (a+b x)^{n+1} (c+d x)^{p+1}}{b d (n+p+3)} \]
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)^p \, dx &=\frac {x (a+b x)^{1+n} (c+d x)^{1+p}}{b d (3+n+p)}+\frac {\int (a+b x)^n (c+d x)^p (-a c-(b c (2+n)+a d (2+p)) x) \, dx}{b d (3+n+p)}\\ &=-\frac {(b c (2+n)+a d (2+p)) (a+b x)^{1+n} (c+d x)^{1+p}}{b^2 d^2 (2+n+p) (3+n+p)}+\frac {x (a+b x)^{1+n} (c+d x)^{1+p}}{b d (3+n+p)}+\frac {\left (b^2 c^2 \left (2+3 n+n^2\right )+2 a b c d (1+n) (1+p)+a^2 d^2 \left (2+3 p+p^2\right )\right ) \int (a+b x)^n (c+d x)^p \, dx}{b^2 d^2 (2+n+p) (3+n+p)}\\ &=-\frac {(b c (2+n)+a d (2+p)) (a+b x)^{1+n} (c+d x)^{1+p}}{b^2 d^2 (2+n+p) (3+n+p)}+\frac {x (a+b x)^{1+n} (c+d x)^{1+p}}{b d (3+n+p)}+\frac {\left (\left (b^2 c^2 \left (2+3 n+n^2\right )+2 a b c d (1+n) (1+p)+a^2 d^2 \left (2+3 p+p^2\right )\right ) (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p}\right ) \int (a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^p \, dx}{b^2 d^2 (2+n+p) (3+n+p)}\\ &=-\frac {(b c (2+n)+a d (2+p)) (a+b x)^{1+n} (c+d x)^{1+p}}{b^2 d^2 (2+n+p) (3+n+p)}+\frac {x (a+b x)^{1+n} (c+d x)^{1+p}}{b d (3+n+p)}+\frac {\left (b^2 c^2 \left (2+3 n+n^2\right )+2 a b c d (1+n) (1+p)+a^2 d^2 \left (2+3 p+p^2\right )\right ) (a+b x)^{1+n} (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \, _2F_1\left (1+n,-p;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{b^3 d^2 (1+n) (2+n+p) (3+n+p)}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 178, normalized size = 0.86 \[ \frac {(a+b x)^{n+1} (c+d x)^p \left (\frac {\left (a^2 d^2 \left (p^2+3 p+2\right )+2 a b c d (n+1) (p+1)+b^2 c^2 \left (n^2+3 n+2\right )\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \, _2F_1\left (n+1,-p;n+2;\frac {d (a+b x)}{a d-b c}\right )}{b^2 d (n+1) (n+p+2)}-\frac {(c+d x) (a d (p+2)+b c (n+2))}{b d (n+p+2)}+x (c+d x)\right )}{b d (n+p+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{n} {\left (d x + c\right )}^{p} x^{2}, x\right ) \]
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 \[ \int {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{p} x^{2}\,{d x} \]
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 \[ \int x^{2} \left (b x +a \right )^{n} \left (d x +c \right )^{p}\, dx \]
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 \[ \int {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{p} x^{2}\,{d x} \]
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 \[ \int x^2\,{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^p \,d x \]
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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