Optimal. Leaf size=87 \[ -\frac {c (a+b x)^{n+1} (2 a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a^2 (n+1)}-\frac {c^2 (a+b x)^{n+1}}{a x}+\frac {d^2 (a+b x)^{n+1}}{b (n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {89, 80, 65} \[ -\frac {c (a+b x)^{n+1} (2 a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a^2 (n+1)}-\frac {c^2 (a+b x)^{n+1}}{a x}+\frac {d^2 (a+b x)^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 65
Rule 80
Rule 89
Rubi steps
\begin {align*} \int \frac {(a+b x)^n (c+d x)^2}{x^2} \, dx &=-\frac {c^2 (a+b x)^{1+n}}{a x}+\frac {\int \frac {(a+b x)^n \left (c (2 a d+b c n)+a d^2 x\right )}{x} \, dx}{a}\\ &=\frac {d^2 (a+b x)^{1+n}}{b (1+n)}-\frac {c^2 (a+b x)^{1+n}}{a x}+\frac {(c (2 a d+b c n)) \int \frac {(a+b x)^n}{x} \, dx}{a}\\ &=\frac {d^2 (a+b x)^{1+n}}{b (1+n)}-\frac {c^2 (a+b x)^{1+n}}{a x}-\frac {c (2 a d+b c n) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a^2 (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 73, normalized size = 0.84 \[ \frac {(a+b x)^{n+1} \left (a \left (a d^2 x-b c^2 (n+1)\right )-b c x (2 a d+b c n) \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )\right )}{a^2 b (n+1) x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} {\left (b x + a\right )}^{n}}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (d x + c\right )}^{2} {\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{2} \left (b x +a \right )^{n}}{x^{2}}\, dx \]
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 \[ \frac {{\left (b x + a\right )}^{n + 1} d^{2}}{b {\left (n + 1\right )}} + \int \frac {{\left (2 \, c d x + c^{2}\right )} {\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^2}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.98, size = 554, normalized size = 6.37 \[ \frac {b^{n} c^{2} n^{2} \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{x \Gamma \left (n + 2\right )} + \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 )}{x \Gamma \left (n + 2\right )} - \frac {b^{n} c^{2} n \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{x \Gamma \left (n + 2\right )} - \frac {b^{n} c^{2} \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{x \Gamma \left (n + 2\right )} - \frac {2 b^{n} c d 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 {2 b^{n} c d \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 )} + d^{2} \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 ) + \frac {b b^{n} c^{2} n^{2} \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} n \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} n \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b b^{n} c^{2} \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {2 b b^{n} c d 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 {2 b b^{n} c d 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^{2} b^{n} c^{2} n^{2} \left (\frac {a}{b} + x\right )^{2} \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a^{2} x \Gamma \left (n + 2\right )} - \frac {b^{2} b^{n} c^{2} n \left (\frac {a}{b} + x\right )^{2} \left (\frac {a}{b} + x\right )^{n} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a^{2} x \Gamma \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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