Optimal. Leaf size=235 \[ -\frac {b}{4 a^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 b \log (x) (a+b x)}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b (a+b x) \log (a+b x)}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {4 b}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{a^5 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b}{2 a^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b}{3 a^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.10, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 44} \[ -\frac {3 b}{2 a^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b}{3 a^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b}{4 a^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {4 b}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{a^5 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 b \log (x) (a+b x)}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b (a+b x) \log (a+b x)}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 646
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {1}{a^5 b^5 x^2}-\frac {5}{a^6 b^4 x}+\frac {1}{a^2 b^3 (a+b x)^5}+\frac {2}{a^3 b^3 (a+b x)^4}+\frac {3}{a^4 b^3 (a+b x)^3}+\frac {4}{a^5 b^3 (a+b x)^2}+\frac {5}{a^6 b^3 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {4 b}{a^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b}{4 a^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 b}{3 a^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 b}{2 a^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{a^5 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 b (a+b x) \log (x)}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b (a+b x) \log (a+b x)}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 103, normalized size = 0.44 \[ \frac {-a \left (12 a^4+125 a^3 b x+260 a^2 b^2 x^2+210 a b^3 x^3+60 b^4 x^4\right )-60 b x \log (x) (a+b x)^4+60 b x (a+b x)^4 \log (a+b x)}{12 a^6 x (a+b x)^3 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 197, normalized size = 0.84 \[ -\frac {60 \, a b^{4} x^{4} + 210 \, a^{2} b^{3} x^{3} + 260 \, a^{3} b^{2} x^{2} + 125 \, a^{4} b x + 12 \, a^{5} - 60 \, {\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} \log \relax (x)}{12 \, {\left (a^{6} b^{4} x^{5} + 4 \, a^{7} b^{3} x^{4} + 6 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{2} + a^{10} x\right )}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 199, normalized size = 0.85 \[ \frac {\left (-60 b^{5} x^{5} \ln \relax (x )+60 b^{5} x^{5} \ln \left (b x +a \right )-240 a \,b^{4} x^{4} \ln \relax (x )+240 a \,b^{4} x^{4} \ln \left (b x +a \right )-360 a^{2} b^{3} x^{3} \ln \relax (x )+360 a^{2} b^{3} x^{3} \ln \left (b x +a \right )-60 a \,b^{4} x^{4}-240 a^{3} b^{2} x^{2} \ln \relax (x )+240 a^{3} b^{2} x^{2} \ln \left (b x +a \right )-210 a^{2} b^{3} x^{3}-60 a^{4} b x \ln \relax (x )+60 a^{4} b x \ln \left (b x +a \right )-260 a^{3} b^{2} x^{2}-125 a^{4} b x -12 a^{5}\right ) \left (b x +a \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} a^{6} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 148, normalized size = 0.63 \[ \frac {5 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} b \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{6}} - \frac {5 \, b}{3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3}} - \frac {5 \, b}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{5}} - \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} x} - \frac {5}{2 \, a^{4} b {\left (x + \frac {a}{b}\right )}^{2}} - \frac {1}{4 \, a^{2} b^{3} {\left (x + \frac {a}{b}\right )}^{4}} \]
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 \[ \int \frac {1}{x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{x^{2} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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