Integrand size = 18, antiderivative size = 143 \[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=\frac {d (a+b x)^{1+n} (c+d x)^2}{b (2+n) x}-\frac {(a+b x)^{1+n} \left (b c^2 (1+n) (a d+b c (2+n))+a d^2 (a d-b c (4+n)) x\right )}{a b^2 (1+n) (2+n) x}-\frac {c^2 (3 a d+b c n) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a^2 (1+n)} \]
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Time = 0.09 (sec) , antiderivative size = 143, 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 = {102, 151, 67} \[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=-\frac {c^2 (a+b x)^{n+1} (3 a d+b c n) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a^2 (n+1)}-\frac {(a+b x)^{n+1} \left (b c^2 (n+1) (a d+b c (n+2))+a d^2 x (a d-b c (n+4))\right )}{a b^2 (n+1) (n+2) x}+\frac {d (c+d x)^2 (a+b x)^{n+1}}{b (n+2) x} \]
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Rule 67
Rule 102
Rule 151
Rubi steps \begin{align*} \text {integral}& = \frac {d (a+b x)^{1+n} (c+d x)^2}{b (2+n) x}+\frac {\int \frac {(a+b x)^n (c+d x) (c (a d+b c (2+n))-d (a d-b c (4+n)) x)}{x^2} \, dx}{b (2+n)} \\ & = \frac {d (a+b x)^{1+n} (c+d x)^2}{b (2+n) x}-\frac {(a+b x)^{1+n} \left (b c^2 (1+n) (a d+b c (2+n))+a d^2 (a d-b c (4+n)) x\right )}{a b^2 (1+n) (2+n) x}+\frac {\left (c^2 (3 a d+b c n)\right ) \int \frac {(a+b x)^n}{x} \, dx}{a} \\ & = \frac {d (a+b x)^{1+n} (c+d x)^2}{b (2+n) x}-\frac {(a+b x)^{1+n} \left (b c^2 (1+n) (a d+b c (2+n))+a d^2 (a d-b c (4+n)) x\right )}{a b^2 (1+n) (2+n) x}-\frac {c^2 (3 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 = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=-\frac {(a+b x)^{1+n} \left (a \left (b^2 c^3 \left (2+3 n+n^2\right )+a^2 d^3 x-a b d^2 x (3 c (2+n)+d (1+n) x)\right )+b^2 c^2 (2+n) (3 a d+b c n) x \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )}{a^2 b^2 (1+n) (2+n) x} \]
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\[\int \frac {\left (b x +a \right )^{n} \left (d x +c \right )^{3}}{x^{2}}d x\]
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\[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3} {\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]
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Time = 2.94 (sec) , antiderivative size = 478, normalized size of antiderivative = 3.34 \[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=3 c 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 ) + d^{3} \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 {3 b^{n + 1} c^{2} 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 {3 b^{n + 1} c^{2} 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^{3} 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^{3} \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^{3} 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^{3} 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 )} \]
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\[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3} {\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]
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\[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3} {\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^3}{x^2} \,d x \]
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