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 = 0.187068, 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 90
Rule 80
Rule 70
Rule 69
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*}
Mathematica [A] time = 0.224318, 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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Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( bx+a \right ) ^{n} \left ( dx+c \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{p} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x + a\right )}^{n}{\left (d x + c\right )}^{p} x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b x\right )^{n} \left (c + d x\right )^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{n}{\left (d x + c\right )}^{p} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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