Optimal. Leaf size=139 \[ \frac{d (a+b x)^{n+1} (a d-b c (1-n)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{c^2 (n+1) (b c-a d)^2}-\frac{(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a c^2 (n+1)}-\frac{d (a+b x)^{n+1}}{c (c+d x) (b c-a d)} \]
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Rubi [A] time = 0.0814086, antiderivative size = 139, normalized size of antiderivative = 1., 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 = {103, 156, 65, 68} \[ \frac{d (a+b x)^{n+1} (a d-b c (1-n)) \, _2F_1\left (1,n+1;n+2;-\frac{d (a+b x)}{b c-a d}\right )}{c^2 (n+1) (b c-a d)^2}-\frac{(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a c^2 (n+1)}-\frac{d (a+b x)^{n+1}}{c (c+d x) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 103
Rule 156
Rule 65
Rule 68
Rubi steps
\begin{align*} \int \frac{(a+b x)^n}{x (c+d x)^2} \, dx &=-\frac{d (a+b x)^{1+n}}{c (b c-a d) (c+d x)}-\frac{\int \frac{(a+b x)^n (-b c+a d-b d n x)}{x (c+d x)} \, dx}{c (b c-a d)}\\ &=-\frac{d (a+b x)^{1+n}}{c (b c-a d) (c+d x)}+\frac{\int \frac{(a+b x)^n}{x} \, dx}{c^2}+\frac{(d (a d-b c (1-n))) \int \frac{(a+b x)^n}{c+d x} \, dx}{c^2 (b c-a d)}\\ &=-\frac{d (a+b x)^{1+n}}{c (b c-a d) (c+d x)}+\frac{d (a d-b c (1-n)) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac{d (a+b x)}{b c-a d}\right )}{c^2 (b c-a d)^2 (1+n)}-\frac{(a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac{b x}{a}\right )}{a c^2 (1+n)}\\ \end{align*}
Mathematica [A] time = 0.129849, size = 111, normalized size = 0.8 \[ \frac{(a+b x)^{n+1} \left (\frac{d \left (\frac{(a d+b c (n-1)) \, _2F_1\left (1,n+1;n+2;\frac{d (a+b x)}{a d-b c}\right )}{n+1}+\frac{c (a d-b c)}{c+d x}\right )}{(b c-a d)^2}-\frac{\, _2F_1\left (1,n+1;n+2;\frac{a+b x}{a}\right )}{a n+a}\right )}{c^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}}{x \left ( dx+c \right ) ^{2}}}\, 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 \frac{{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{2} x}\,{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 (\frac{{\left (b x + a\right )}^{n}}{d^{2} x^{3} + 2 \, c d x^{2} + c^{2} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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