Optimal. Leaf size=185 \[ -\frac {b \left (a+\frac {b}{x}\right )^{m+1} \left (6 a^2 c^2-6 a b c d (m+1)+b^2 d^2 \left (m^2+3 m+2\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{6 c^2 (m+1) (a c-b d)^4}+\frac {d^2 \left (a+\frac {b}{x}\right )^{m+1}}{3 c^2 \left (\frac {c}{x}+d\right )^3 (a c-b d)}-\frac {d \left (a+\frac {b}{x}\right )^{m+1} (6 a c-b d (m+4))}{6 c^2 \left (\frac {c}{x}+d\right )^2 (a c-b d)^2} \]
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Rubi [A] time = 0.18, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {434, 446, 89, 78, 68} \[ -\frac {b \left (a+\frac {b}{x}\right )^{m+1} \left (6 a^2 c^2-6 a b c d (m+1)+b^2 d^2 \left (m^2+3 m+2\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{6 c^2 (m+1) (a c-b d)^4}+\frac {d^2 \left (a+\frac {b}{x}\right )^{m+1}}{3 c^2 \left (\frac {c}{x}+d\right )^3 (a c-b d)}-\frac {d \left (a+\frac {b}{x}\right )^{m+1} (6 a c-b d (m+4))}{6 c^2 \left (\frac {c}{x}+d\right )^2 (a c-b d)^2} \]
Antiderivative was successfully verified.
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Rule 68
Rule 78
Rule 89
Rule 434
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^m}{(c+d x)^4} \, dx &=\int \frac {\left (a+\frac {b}{x}\right )^m}{\left (d+\frac {c}{x}\right )^4 x^4} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {x^2 (a+b x)^m}{(d+c x)^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {d^2 \left (a+\frac {b}{x}\right )^{1+m}}{3 c^2 (a c-b d) \left (d+\frac {c}{x}\right )^3}-\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^m (-d (3 a c-b d (1+m))+3 c (a c-b d) x)}{(d+c x)^3} \, dx,x,\frac {1}{x}\right )}{3 c^2 (a c-b d)}\\ &=\frac {d^2 \left (a+\frac {b}{x}\right )^{1+m}}{3 c^2 (a c-b d) \left (d+\frac {c}{x}\right )^3}-\frac {d (6 a c-b d (4+m)) \left (a+\frac {b}{x}\right )^{1+m}}{6 c^2 (a c-b d)^2 \left (d+\frac {c}{x}\right )^2}-\frac {\left (6 a^2 c^2-6 a b c d (1+m)+b^2 d^2 \left (2+3 m+m^2\right )\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^m}{(d+c x)^2} \, dx,x,\frac {1}{x}\right )}{6 c^2 (a c-b d)^2}\\ &=\frac {d^2 \left (a+\frac {b}{x}\right )^{1+m}}{3 c^2 (a c-b d) \left (d+\frac {c}{x}\right )^3}-\frac {d (6 a c-b d (4+m)) \left (a+\frac {b}{x}\right )^{1+m}}{6 c^2 (a c-b d)^2 \left (d+\frac {c}{x}\right )^2}-\frac {b \left (6 a^2 c^2-6 a b c d (1+m)+b^2 d^2 \left (2+3 m+m^2\right )\right ) \left (a+\frac {b}{x}\right )^{1+m} \, _2F_1\left (2,1+m;2+m;\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{6 c^2 (a c-b d)^4 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 155, normalized size = 0.84 \[ \frac {\left (a+\frac {b}{x}\right )^{m+1} \left (-\frac {b \left (6 a^2 c^2-6 a b c d (m+1)+b^2 d^2 \left (m^2+3 m+2\right )\right ) \, _2F_1\left (2,m+1;m+2;\frac {b c+a x c}{a c x-b d x}\right )}{(m+1) (a c-b d)^2}+\frac {2 d^2 x^3 (a c-b d)}{(c+d x)^3}+\frac {d x^2 (b d (m+4)-6 a c)}{(c+d x)^2}\right )}{6 c^2 (a c-b d)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a x + b}{x}\right )^{m}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}, x\right ) \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{m}}{{\left (d x + c\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +\frac {b}{x}\right )^{m}}{\left (d x +c \right )^{4}}\, dx \]
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 \[ \int \frac {{\left (a + \frac {b}{x}\right )}^{m}}{{\left (d x + c\right )}^{4}}\,{d x} \]
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 \[ \int \frac {{\left (a+\frac {b}{x}\right )}^m}{{\left (c+d\,x\right )}^4} \,d x \]
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 \[ \int \frac {\left (a + \frac {b}{x}\right )^{m}}{\left (c + d x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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