Optimal. Leaf size=88 \[ \frac {d (2 b c-a d) (a+b x)^{n+1}}{b^2 (n+1)}+\frac {d^2 (a+b x)^{n+2}}{b^2 (n+2)}-\frac {c^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a (n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {88, 65} \[ \frac {d (2 b c-a d) (a+b x)^{n+1}}{b^2 (n+1)}+\frac {d^2 (a+b x)^{n+2}}{b^2 (n+2)}-\frac {c^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a (n+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 88
Rubi steps
\begin {align*} \int \frac {(a+b x)^n (c+d x)^2}{x} \, dx &=\int \left (-\frac {d (-2 b c+a d) (a+b x)^n}{b}+\frac {c^2 (a+b x)^n}{x}+\frac {d^2 (a+b x)^{1+n}}{b}\right ) \, dx\\ &=\frac {d (2 b c-a d) (a+b x)^{1+n}}{b^2 (1+n)}+\frac {d^2 (a+b x)^{2+n}}{b^2 (2+n)}+c^2 \int \frac {(a+b x)^n}{x} \, dx\\ &=\frac {d (2 b c-a d) (a+b x)^{1+n}}{b^2 (1+n)}+\frac {d^2 (a+b x)^{2+n}}{b^2 (2+n)}-\frac {c^2 (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 77, normalized size = 0.88 \[ -\frac {(a+b x)^{n+1} \left (b^2 c^2 (n+2) \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )+a d (a d-b (2 c (n+2)+d (n+1) x))\right )}{a b^2 (n+1) (n+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} {\left (b x + a\right )}^{n}}{x}, x\right ) \]
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 \[ \int \frac {{\left (d x + c\right )}^{2} {\left (b x + a\right )}^{n}}{x}\,{d x} \]
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 \[ \int \frac {\left (d x +c \right )^{2} \left (b x +a \right )^{n}}{x}\, dx \]
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 \[ \frac {2 \, {\left (b x + a\right )}^{n + 1} c d}{b {\left (n + 1\right )}} + \int \frac {{\left (d^{2} x^{2} + c^{2}\right )} {\left (b x + a\right )}^{n}}{x}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.70, size = 386, normalized size = 4.39 \[ - \frac {b^{n} c^{2} n \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} - \frac {b^{n} c^{2} \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{\Gamma \left (n + 2\right )} + 2 c d \left (\begin {cases} a^{n} x & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (a + b x \right )} & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}\right ) + d^{2} \left (\begin {cases} \frac {a^{n} x^{2}}{2} & \text {for}\: b = 0 \\\frac {a \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac {a}{a b^{2} + b^{3} x} + \frac {b x \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text {for}\: n = -2 \\- \frac {a \log {\left (\frac {a}{b} + x \right )}}{b^{2}} + \frac {x}{b} & \text {for}\: n = -1 \\- \frac {a^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {a b n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {b^{2} x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} & \text {otherwise} \end {cases}\right ) - \frac {b b^{n} c^{2} n x \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b b^{n} c^{2} x \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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