Optimal. Leaf size=278 \[ \frac {15 b^2 \log (x) (a+b x)}{a^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {10 b^2}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b (a+b x)}{a^6 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b^2}{a^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{2 a^5 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2}{a^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2}{4 a^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 278, 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} \[ \frac {3 b^2}{a^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2}{a^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2}{4 a^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {10 b^2}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b (a+b x)}{a^6 x \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{2 a^5 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 b^2 \log (x) (a+b x)}{a^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 646
Rubi steps
\begin {align*} \int \frac {1}{x^3 \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^3 \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^3}-\frac {5}{a^6 b^4 x^2}+\frac {15}{a^7 b^3 x}-\frac {1}{a^3 b^2 (a+b x)^5}-\frac {3}{a^4 b^2 (a+b x)^4}-\frac {6}{a^5 b^2 (a+b x)^3}-\frac {10}{a^6 b^2 (a+b x)^2}-\frac {15}{a^7 b^2 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {10 b^2}{a^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2}{4 a^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2}{a^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b^2}{a^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a+b x}{2 a^5 x^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b (a+b x)}{a^6 x \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 b^2 (a+b x) \log (x)}{a^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 b^2 (a+b x) \log (a+b x)}{a^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 121, normalized size = 0.44 \[ \frac {a \left (-2 a^5+12 a^4 b x+125 a^3 b^2 x^2+260 a^2 b^3 x^3+210 a b^4 x^4+60 b^5 x^5\right )+60 b^2 x^2 \log (x) (a+b x)^4-60 b^2 x^2 (a+b x)^4 \log (a+b x)}{4 a^7 x^2 (a+b x)^3 \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 218, normalized size = 0.78 \[ \frac {60 \, a b^{5} x^{5} + 210 \, a^{2} b^{4} x^{4} + 260 \, a^{3} b^{3} x^{3} + 125 \, a^{4} b^{2} x^{2} + 12 \, a^{5} b x - 2 \, a^{6} - 60 \, {\left (b^{6} x^{6} + 4 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{6} x^{6} + 4 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x^{2}\right )} \log \relax (x)}{4 \, {\left (a^{7} b^{4} x^{6} + 4 \, a^{8} b^{3} x^{5} + 6 \, a^{9} b^{2} x^{4} + 4 \, a^{10} b x^{3} + a^{11} x^{2}\right )}} \]
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 = 0.07, size = 218, normalized size = 0.78 \[ -\frac {\left (-60 b^{6} x^{6} \ln \relax (x )+60 b^{6} x^{6} \ln \left (b x +a \right )-240 a \,b^{5} x^{5} \ln \relax (x )+240 a \,b^{5} x^{5} \ln \left (b x +a \right )-360 a^{2} b^{4} x^{4} \ln \relax (x )+360 a^{2} b^{4} x^{4} \ln \left (b x +a \right )-60 a \,b^{5} x^{5}-240 a^{3} b^{3} x^{3} \ln \relax (x )+240 a^{3} b^{3} x^{3} \ln \left (b x +a \right )-210 a^{2} b^{4} x^{4}-60 a^{4} b^{2} x^{2} \ln \relax (x )+60 a^{4} b^{2} x^{2} \ln \left (b x +a \right )-260 a^{3} b^{3} x^{3}-125 a^{4} b^{2} x^{2}-12 a^{5} b x +2 a^{6}\right ) \left (b x +a \right )}{4 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} a^{7} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 178, normalized size = 0.64 \[ -\frac {15 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} b^{2} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right )}{a^{7}} + \frac {5 \, b^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{4}} + \frac {15 \, b^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{6}} + \frac {7 \, b}{2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{3} x} - \frac {1}{2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2} x^{2}} + \frac {15}{2 \, a^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {1}{4 \, a^{3} b^{2} {\left (x + \frac {a}{b}\right )}^{4}} \]
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 \[ \int \frac {1}{x^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{x^{3} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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