Integrand size = 29, antiderivative size = 256 \[ \int \frac {A+B x}{x^5 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {A (a+b x)}{4 a x^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-a B) (a+b x)}{3 a^2 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (A b-a B) (a+b x)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (A b-a B) (a+b x)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 (A b-a B) (a+b x) \log (x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 (A b-a B) (a+b x) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.08 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {784, 78} \[ \int \frac {A+B x}{x^5 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {(a+b x) (A b-a B)}{3 a^2 x^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A (a+b x)}{4 a x^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^3 \log (x) (a+b x) (A b-a B)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^3 (a+b x) (A b-a B) \log (a+b x)}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (a+b x) (A b-a B)}{a^4 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (a+b x) (A b-a B)}{2 a^3 x^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 78
Rule 784
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.60 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x}{x^5 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {\frac {a^3 \left (-12 A b^3 x^3+6 a b^2 x^2 (A+2 B x)-2 a^2 b x (2 A+3 B x)+a^3 (3 A+4 B x)\right )}{\sqrt {a^2} x^4}-\frac {a \sqrt {(a+b x)^2} \left (-25 A b^3 x^3+a^3 (3 A+4 B x)-a^2 b x (7 A+10 B x)+a b^2 x^2 (13 A+22 B x)\right )}{x^4}+24 \sqrt {a^2} b^3 (-A b+a B) \log (x)-12 \left (-a+\sqrt {a^2}\right ) b^3 (-A b+a B) \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )-12 \left (a+\sqrt {a^2}\right ) b^3 (-A b+a B) \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )}{24 a^6} \]
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Time = 0.41 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.56
method | result | size |
default | \(\frac {\left (b x +a \right ) \left (12 A \ln \left (x \right ) x^{4} b^{4}-12 A \ln \left (b x +a \right ) x^{4} b^{4}-12 B \ln \left (x \right ) x^{4} a \,b^{3}+12 B \ln \left (b x +a \right ) x^{4} a \,b^{3}+12 a A \,b^{3} x^{3}-12 x^{3} B \,a^{2} b^{2}-6 a^{2} A \,b^{2} x^{2}+6 x^{2} B \,a^{3} b +4 a^{3} A b x -4 a^{4} B x -3 A \,a^{4}\right )}{12 \sqrt {\left (b x +a \right )^{2}}\, a^{5} x^{4}}\) | \(143\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {\left (A b -B a \right ) b^{2} x^{3}}{a^{4}}-\frac {\left (A b -B a \right ) b \,x^{2}}{2 a^{3}}+\frac {\left (A b -B a \right ) x}{3 a^{2}}-\frac {A}{4 a}\right )}{\left (b x +a \right ) x^{4}}-\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A b -B a \right ) b^{3} \ln \left (b x +a \right )}{\left (b x +a \right ) a^{5}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (A b -B a \right ) b^{3} \ln \left (-x \right )}{\left (b x +a \right ) a^{5}}\) | \(151\) |
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Time = 0.41 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.46 \[ \int \frac {A+B x}{x^5 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {12 \, {\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (b x + a\right ) - 12 \, {\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (x\right ) - 3 \, A a^{4} - 12 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} + 6 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} - 4 \, {\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \]
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\[ \int \frac {A+B x}{x^5 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {A + B x}{x^{5} \sqrt {\left (a + b x\right )^{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x}{x^5 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} B b^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{4}} - \frac {\left (-1\right )^{2 \, a b x + 2 \, a^{2}} A b^{4} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{5}} - \frac {11 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{2}}{6 \, a^{4} x} + \frac {25 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{3}}{12 \, a^{5} x} + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b}{6 \, a^{3} x^{2}} - \frac {13 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{2}}{12 \, a^{4} x^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B}{3 \, a^{2} x^{3}} + \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b}{12 \, a^{3} x^{3}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A}{4 \, a^{2} x^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.73 \[ \int \frac {A+B x}{x^5 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {{\left (B a b^{3} \mathrm {sgn}\left (b x + a\right ) - A b^{4} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {{\left (B a b^{4} \mathrm {sgn}\left (b x + a\right ) - A b^{5} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac {3 \, A a^{4} \mathrm {sgn}\left (b x + a\right ) + 12 \, {\left (B a^{2} b^{2} \mathrm {sgn}\left (b x + a\right ) - A a b^{3} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} - 6 \, {\left (B a^{3} b \mathrm {sgn}\left (b x + a\right ) - A a^{2} b^{2} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 4 \, {\left (B a^{4} \mathrm {sgn}\left (b x + a\right ) - A a^{3} b \mathrm {sgn}\left (b x + a\right )\right )} x}{12 \, a^{5} x^{4}} \]
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Timed out. \[ \int \frac {A+B x}{x^5 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {A+B\,x}{x^5\,\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
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