Optimal. Leaf size=190 \[ -\frac {d (b c-2 a d) (a+b x)^{1+n}}{a c^2 (b c-a d) (c+d x)}-\frac {(a+b x)^{1+n}}{a c x (c+d x)}-\frac {d^2 (2 a d-b c (2-n)) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{c^3 (b c-a d)^2 (1+n)}+\frac {(2 a d-b c n) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a^2 c^3 (1+n)} \]
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Rubi [A]
time = 0.14, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {105, 156, 162,
67, 70} \begin {gather*} \frac {(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 c^3 (n+1)}-\frac {d^2 (a+b x)^{n+1} (2 a d-b c (2-n)) \, _2F_1\left (1,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{c^3 (n+1) (b c-a d)^2}-\frac {d (b c-2 a d) (a+b x)^{n+1}}{a c^2 (c+d x) (b c-a d)}-\frac {(a+b x)^{n+1}}{a c x (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 70
Rule 105
Rule 156
Rule 162
Rubi steps
\begin {align*} \int \frac {(a+b x)^n}{x^2 (c+d x)^2} \, dx &=-\frac {(a+b x)^{1+n}}{a c x (c+d x)}-\frac {\int \frac {(a+b x)^n (2 a d-b c n+b d (1-n) x)}{x (c+d x)^2} \, dx}{a c}\\ &=-\frac {d (b c-2 a d) (a+b x)^{1+n}}{a c^2 (b c-a d) (c+d x)}-\frac {(a+b x)^{1+n}}{a c x (c+d x)}+\frac {\int \frac {(a+b x)^n (-(b c-a d) (2 a d-b c n)+b d (b c-2 a d) n x)}{x (c+d x)} \, dx}{a c^2 (b c-a d)}\\ &=-\frac {d (b c-2 a d) (a+b x)^{1+n}}{a c^2 (b c-a d) (c+d x)}-\frac {(a+b x)^{1+n}}{a c x (c+d x)}-\frac {\left (d^2 (2 a d-b c (2-n))\right ) \int \frac {(a+b x)^n}{c+d x} \, dx}{c^3 (b c-a d)}-\frac {(2 a d-b c n) \int \frac {(a+b x)^n}{x} \, dx}{a c^3}\\ &=-\frac {d (b c-2 a d) (a+b x)^{1+n}}{a c^2 (b c-a d) (c+d x)}-\frac {(a+b x)^{1+n}}{a c x (c+d x)}-\frac {d^2 (2 a d-b c (2-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^3 (b c-a d)^2 (1+n)}+\frac {(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 c^3 (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 176, normalized size = 0.93 \begin {gather*} -\frac {(a+b x)^{1+n} \left (a c^2 (b c-a d)^2 (1+n)+a c d (-b c+a d) (-b c+2 a d) (1+n) x-x (c+d x) \left (-a^2 d^2 (2 a d+b c (-2+n)) \, _2F_1\left (1,1+n;2+n;\frac {d (a+b x)}{-b c+a d}\right )+(b c-a d)^2 (2 a d-b c n) \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )\right )\right )}{a^2 c^3 (b c-a d)^2 (1+n) x (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{n}}{x^{2} \left (d x +c \right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{n}}{x^{2} \left (c + d x\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^n}{x^2\,{\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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