Optimal. Leaf size=256 \[ \frac {(a+b x) (A b-a B)}{3 a^2 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{4 a x^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 \log (x) (a+b x) (A b-a B)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 (a+b x) (A b-a B) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (a+b x) (A b-a B)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (a+b x) (A b-a B)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.12, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 77} \[ \frac {b^2 (a+b x) (A b-a B)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (a+b x) (A b-a B)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (A b-a B)}{3 a^2 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 \log (x) (a+b x) (A b-a B)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 (a+b x) (A b-a B) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{4 a x^4 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {A+B x}{x^5 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {A+B x}{x^5 \left (a b+b^2 x\right )} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {A}{a b x^5}+\frac {-A b+a B}{a^2 b x^4}+\frac {A b-a B}{a^3 x^3}+\frac {b (-A b+a B)}{a^4 x^2}-\frac {b^2 (-A b+a B)}{a^5 x}+\frac {b^3 (-A b+a B)}{a^5 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {A (a+b x)}{4 a x^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (a+b x)}{3 a^2 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (A b-a B) (a+b x)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (A b-a B) (a+b x)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 (A b-a B) (a+b x) \log (x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 (A b-a B) (a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 121, normalized size = 0.47 \[ -\frac {(a+b x) \left (a \left (a^3 (3 A+4 B x)-2 a^2 b x (2 A+3 B x)+6 a b^2 x^2 (A+2 B x)-12 A b^3 x^3\right )-12 b^3 x^4 \log (x) (A b-a B)+12 b^3 x^4 (A b-a B) \log (a+b x)\right )}{12 a^5 x^4 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 117, normalized size = 0.46 \[ \frac {12 \, {\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (b x + a\right ) - 12 \, {\left (B a b^{3} - A b^{4}\right )} x^{4} \log \relax (x) - 3 \, A a^{4} - 12 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} + 6 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} - 4 \, {\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 188, normalized size = 0.73 \[ -\frac {{\left (B a b^{3} \mathrm {sgn}\left (b x + a\right ) - A b^{4} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {{\left (B a b^{4} \mathrm {sgn}\left (b x + a\right ) - A b^{5} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac {3 \, A a^{4} \mathrm {sgn}\left (b x + a\right ) + 12 \, {\left (B a^{2} b^{2} \mathrm {sgn}\left (b x + a\right ) - A a b^{3} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} - 6 \, {\left (B a^{3} b \mathrm {sgn}\left (b x + a\right ) - A a^{2} b^{2} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 4 \, {\left (B a^{4} \mathrm {sgn}\left (b x + a\right ) - A a^{3} b \mathrm {sgn}\left (b x + a\right )\right )} x}{12 \, a^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 143, normalized size = 0.56 \[ -\frac {\left (b x +a \right ) \left (-12 A \,b^{4} x^{4} \ln \relax (x )+12 A \,b^{4} x^{4} \ln \left (b x +a \right )+12 B a \,b^{3} x^{4} \ln \relax (x )-12 B a \,b^{3} x^{4} \ln \left (b x +a \right )-12 A a \,b^{3} x^{3}+12 B \,a^{2} b^{2} x^{3}+6 A \,a^{2} b^{2} x^{2}-6 B \,a^{3} b \,x^{2}-4 A \,a^{3} b x +4 B \,a^{4} x +3 A \,a^{4}\right )}{12 \sqrt {\left (b x +a \right )^{2}}\, a^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 284, normalized size = 1.11 \[ \frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} B b^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{4}} - \frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} A b^{4} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{5}} - \frac {11 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{2}}{6 \, a^{4} x} + \frac {25 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{3}}{12 \, a^{5} x} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b}{6 \, a^{3} x^{2}} - \frac {13 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{2}}{12 \, a^{4} x^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B}{3 \, a^{2} x^{3}} + \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b}{12 \, a^{3} x^{3}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A}{4 \, a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x}{x^5\,\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 189, normalized size = 0.74 \[ \frac {- 3 A a^{3} + x^{3} \left (12 A b^{3} - 12 B a b^{2}\right ) + x^{2} \left (- 6 A a b^{2} + 6 B a^{2} b\right ) + x \left (4 A a^{2} b - 4 B a^{3}\right )}{12 a^{4} x^{4}} - \frac {b^{3} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{4} + B a^{2} b^{3} - a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} + \frac {b^{3} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{4} + B a^{2} b^{3} + a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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