Optimal. Leaf size=44 \[ -\frac {A (a+b x)^3}{4 a x^4}+\frac {(A b-4 a B) (a+b x)^3}{12 a^2 x^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {27, 79, 37}
\begin {gather*} \frac {(a+b x)^3 (A b-4 a B)}{12 a^2 x^3}-\frac {A (a+b x)^3}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rule 79
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^5} \, dx &=\int \frac {(a+b x)^2 (A+B x)}{x^5} \, dx\\ &=-\frac {A (a+b x)^3}{4 a x^4}+\frac {(-A b+4 a B) \int \frac {(a+b x)^2}{x^4} \, dx}{4 a}\\ &=-\frac {A (a+b x)^3}{4 a x^4}+\frac {(A b-4 a B) (a+b x)^3}{12 a^2 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 47, normalized size = 1.07 \begin {gather*} -\frac {6 b^2 x^2 (A+2 B x)+4 a b x (2 A+3 B x)+a^2 (3 A+4 B x)}{12 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 48, normalized size = 1.09
method | result | size |
default | \(-\frac {a^{2} A}{4 x^{4}}-\frac {b \left (A b +2 B a \right )}{2 x^{2}}-\frac {a \left (2 A b +B a \right )}{3 x^{3}}-\frac {b^{2} B}{x}\) | \(48\) |
norman | \(\frac {-b^{2} B \,x^{3}+\left (-\frac {1}{2} b^{2} A -a b B \right ) x^{2}+\left (-\frac {2}{3} a b A -\frac {1}{3} a^{2} B \right ) x -\frac {a^{2} A}{4}}{x^{4}}\) | \(51\) |
risch | \(\frac {-b^{2} B \,x^{3}+\left (-\frac {1}{2} b^{2} A -a b B \right ) x^{2}+\left (-\frac {2}{3} a b A -\frac {1}{3} a^{2} B \right ) x -\frac {a^{2} A}{4}}{x^{4}}\) | \(51\) |
gosper | \(-\frac {12 b^{2} B \,x^{3}+6 b^{2} A \,x^{2}+12 B a b \,x^{2}+8 a b A x +4 a^{2} B x +3 a^{2} A}{12 x^{4}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 51, normalized size = 1.16 \begin {gather*} -\frac {12 \, B b^{2} x^{3} + 3 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.61, size = 51, normalized size = 1.16 \begin {gather*} -\frac {12 \, B b^{2} x^{3} + 3 \, A a^{2} + 6 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} x}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.37, size = 56, normalized size = 1.27 \begin {gather*} \frac {- 3 A a^{2} - 12 B b^{2} x^{3} + x^{2} \left (- 6 A b^{2} - 12 B a b\right ) + x \left (- 8 A a b - 4 B a^{2}\right )}{12 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.04, size = 51, normalized size = 1.16 \begin {gather*} -\frac {12 \, B b^{2} x^{3} + 12 \, B a b x^{2} + 6 \, A b^{2} x^{2} + 4 \, B a^{2} x + 8 \, A a b x + 3 \, A a^{2}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 49, normalized size = 1.11 \begin {gather*} -\frac {x^2\,\left (\frac {A\,b^2}{2}+B\,a\,b\right )+\frac {A\,a^2}{4}+x\,\left (\frac {B\,a^2}{3}+\frac {2\,A\,b\,a}{3}\right )+B\,b^2\,x^3}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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