Optimal. Leaf size=124 \[ \frac {(a+b x)^{n+1} (a d-b c n) \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a^2 c^2 (n+1)}+\frac {d^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{c^2 (n+1) (b c-a d)}-\frac {(a+b x)^{n+1}}{a c x} \]
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Rubi [A] time = 0.07, antiderivative size = 124, normalized size of antiderivative = 1.00, 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} \[ \frac {(a+b x)^{n+1} (a d-b c n) \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a^2 c^2 (n+1)}+\frac {d^2 (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{c^2 (n+1) (b c-a d)}-\frac {(a+b x)^{n+1}}{a c x} \]
Antiderivative was successfully verified.
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Rule 65
Rule 68
Rule 103
Rule 156
Rubi steps
\begin {align*} \int \frac {(a+b x)^n}{x^2 (c+d x)} \, dx &=-\frac {(a+b x)^{1+n}}{a c x}-\frac {\int \frac {(a+b x)^n (a d-b c n-b d n x)}{x (c+d x)} \, dx}{a c}\\ &=-\frac {(a+b x)^{1+n}}{a c x}+\frac {d^2 \int \frac {(a+b x)^n}{c+d x} \, dx}{c^2}-\frac {(a d-b c n) \int \frac {(a+b x)^n}{x} \, dx}{a c^2}\\ &=-\frac {(a+b x)^{1+n}}{a c x}+\frac {d^2 (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) (1+n)}+\frac {(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 c^2 (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 113, normalized size = 0.91 \[ -\frac {(a+b x)^{n+1} \left (a^2 d^2 x \, _2F_1\left (1,n+1;n+2;\frac {d (a+b x)}{a d-b c}\right )+(a d-b c) \left (\, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right ) (b c n x-a d x)+a c (n+1)\right )\right )}{a^2 c^2 (n+1) x (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{n}}{d x^{3} + c x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {\left (b x +a \right )^{n}}{\left (d x +c \right ) x^{2}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^n}{x^2\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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