Optimal. Leaf size=194 \[ \frac {(a+b x)^{n+1} (c+d x)^{1-n} (a d (n+2)+b c (2-n))}{6 a^2 c^2 x^2}+\frac {(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} \left (a^2 d^2 \left (n^2+3 n+2\right )+2 a b c d \left (1-n^2\right )+b^2 c^2 \left (n^2-3 n+2\right )\right ) \, _2F_1\left (2,n+1;n+2;\frac {c (a+b x)}{a (c+d x)}\right )}{6 a^4 c^2 (n+1)}-\frac {(a+b x)^{n+1} (c+d x)^{1-n}}{3 a c x^3} \]
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Rubi [A] time = 0.14, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {129, 151, 12, 131} \[ \frac {(b c-a d) (a+b x)^{n+1} (c+d x)^{-n-1} \left (a^2 d^2 \left (n^2+3 n+2\right )+2 a b c d \left (1-n^2\right )+b^2 c^2 \left (n^2-3 n+2\right )\right ) \, _2F_1\left (2,n+1;n+2;\frac {c (a+b x)}{a (c+d x)}\right )}{6 a^4 c^2 (n+1)}+\frac {(a+b x)^{n+1} (c+d x)^{1-n} (a d (n+2)+b c (2-n))}{6 a^2 c^2 x^2}-\frac {(a+b x)^{n+1} (c+d x)^{1-n}}{3 a c x^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 129
Rule 131
Rule 151
Rubi steps
\begin {align*} \int \frac {(a+b x)^n (c+d x)^{-n}}{x^4} \, dx &=-\frac {(a+b x)^{1+n} (c+d x)^{1-n}}{3 a c x^3}-\frac {\int \frac {(a+b x)^n (c+d x)^{-n} (b c (2-n)+a d (2+n)+b d x)}{x^3} \, dx}{3 a c}\\ &=-\frac {(a+b x)^{1+n} (c+d x)^{1-n}}{3 a c x^3}+\frac {(b c (2-n)+a d (2+n)) (a+b x)^{1+n} (c+d x)^{1-n}}{6 a^2 c^2 x^2}+\frac {\int \frac {\left (2 a b c d \left (1-n^2\right )+b^2 c^2 \left (2-3 n+n^2\right )+a^2 d^2 \left (2+3 n+n^2\right )\right ) (a+b x)^n (c+d x)^{-n}}{x^2} \, dx}{6 a^2 c^2}\\ &=-\frac {(a+b x)^{1+n} (c+d x)^{1-n}}{3 a c x^3}+\frac {(b c (2-n)+a d (2+n)) (a+b x)^{1+n} (c+d x)^{1-n}}{6 a^2 c^2 x^2}+\frac {\left (2 a b c d \left (1-n^2\right )+b^2 c^2 \left (2-3 n+n^2\right )+a^2 d^2 \left (2+3 n+n^2\right )\right ) \int \frac {(a+b x)^n (c+d x)^{-n}}{x^2} \, dx}{6 a^2 c^2}\\ &=-\frac {(a+b x)^{1+n} (c+d x)^{1-n}}{3 a c x^3}+\frac {(b c (2-n)+a d (2+n)) (a+b x)^{1+n} (c+d x)^{1-n}}{6 a^2 c^2 x^2}+\frac {(b c-a d) \left (2 a b c d \left (1-n^2\right )+b^2 c^2 \left (2-3 n+n^2\right )+a^2 d^2 \left (2+3 n+n^2\right )\right ) (a+b x)^{1+n} (c+d x)^{-1-n} \, _2F_1\left (2,1+n;2+n;\frac {c (a+b x)}{a (c+d x)}\right )}{6 a^4 c^2 (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 156, normalized size = 0.80 \[ \frac {(a+b x)^{n+1} (c+d x)^{-n-1} \left (-\frac {2 a^3 c (c+d x)^2}{x^3}+\frac {(b c-a d) \left (a^2 d^2 \left (n^2+3 n+2\right )-2 a b c d \left (n^2-1\right )+b^2 c^2 \left (n^2-3 n+2\right )\right ) \, _2F_1\left (2,n+1;n+2;\frac {c (a+b x)}{a (c+d x)}\right )}{n+1}+\frac {a^2 (c+d x)^2 (a d (n+2)-b c (n-2))}{x^2}\right )}{6 a^4 c^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{n} x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{n} x^{4}}\,{d x} \]
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 \[ \int \frac {\left (b x +a \right )^{n} \left (d x +c \right )^{-n}}{x^{4}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{n} x^{4}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^n}{x^4\,{\left (c+d\,x\right )}^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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