Integrand size = 18, antiderivative size = 87 \[ \int \frac {(a+b x)^n (c+d x)^2}{x^2} \, dx=\frac {d^2 (a+b x)^{1+n}}{b (1+n)}-\frac {c^2 (a+b x)^{1+n}}{a x}-\frac {c (2 a d+b c n) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a^2 (1+n)} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {91, 81, 67} \[ \int \frac {(a+b x)^n (c+d x)^2}{x^2} \, dx=-\frac {c (a+b x)^{n+1} (2 a d+b c n) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a^2 (n+1)}-\frac {c^2 (a+b x)^{n+1}}{a x}+\frac {d^2 (a+b x)^{n+1}}{b (n+1)} \]
[In]
[Out]
Rule 67
Rule 81
Rule 91
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x)^n (c+d x)^2}{x^2} \, dx=\frac {(a+b x)^{1+n} \left (a \left (-b c^2 (1+n)+a d^2 x\right )-b c (2 a d+b c n) x \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )}{a^2 b (1+n) x} \]
[In]
[Out]
\[\int \frac {\left (b x +a \right )^{n} \left (d x +c \right )^{2}}{x^{2}}d x\]
[In]
[Out]
\[ \int \frac {(a+b x)^n (c+d x)^2}{x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2} {\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]
[In]
[Out]
Time = 2.53 (sec) , antiderivative size = 265, normalized size of antiderivative = 3.05 \[ \int \frac {(a+b x)^n (c+d x)^2}{x^2} \, dx=d^{2} \left (\begin {cases} a^{n} x & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (a + b x \right )} & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}\right ) - \frac {2 b^{n + 1} c d n \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {2 b^{n + 1} c d \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b^{n + 2} c^{2} n \left (\frac {a}{b} + x\right )^{n + 1} \Gamma \left (n + 1\right )}{a b x \Gamma \left (n + 2\right )} - \frac {b^{n + 2} c^{2} \left (\frac {a}{b} + x\right )^{n + 1} \Gamma \left (n + 1\right )}{a b x \Gamma \left (n + 2\right )} - \frac {b^{n + 2} c^{2} n^{2} \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a^{2} \Gamma \left (n + 2\right )} - \frac {b^{n + 2} c^{2} n \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a^{2} \Gamma \left (n + 2\right )} \]
[In]
[Out]
\[ \int \frac {(a+b x)^n (c+d x)^2}{x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2} {\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a+b x)^n (c+d x)^2}{x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2} {\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b x)^n (c+d x)^2}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^2}{x^2} \,d x \]
[In]
[Out]