Optimal. Leaf size=75 \[ -\frac {2 b^4 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {b^4 \log (x)}{a^5}+\frac {2 b^3}{a^4 \sqrt {x}}-\frac {b^2}{a^3 x}+\frac {2 b}{3 a^2 x^{3/2}}-\frac {1}{2 a x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 44} \[ \frac {2 b^3}{a^4 \sqrt {x}}-\frac {b^2}{a^3 x}-\frac {2 b^4 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {b^4 \log (x)}{a^5}+\frac {2 b}{3 a^2 x^{3/2}}-\frac {1}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sqrt {x}\right ) x^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^5 (a+b x)} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {1}{a x^5}-\frac {b}{a^2 x^4}+\frac {b^2}{a^3 x^3}-\frac {b^3}{a^4 x^2}+\frac {b^4}{a^5 x}-\frac {b^5}{a^5 (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{2 a x^2}+\frac {2 b}{3 a^2 x^{3/2}}-\frac {b^2}{a^3 x}+\frac {2 b^3}{a^4 \sqrt {x}}-\frac {2 b^4 \log \left (a+b \sqrt {x}\right )}{a^5}+\frac {b^4 \log (x)}{a^5}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 69, normalized size = 0.92 \[ \frac {\frac {a \left (-3 a^3+4 a^2 b \sqrt {x}-6 a b^2 x+12 b^3 x^{3/2}\right )}{x^2}-12 b^4 \log \left (a+b \sqrt {x}\right )+6 b^4 \log (x)}{6 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 69, normalized size = 0.92 \[ -\frac {12 \, b^{4} x^{2} \log \left (b \sqrt {x} + a\right ) - 12 \, b^{4} x^{2} \log \left (\sqrt {x}\right ) + 6 \, a^{2} b^{2} x + 3 \, a^{4} - 4 \, {\left (3 \, a b^{3} x + a^{3} b\right )} \sqrt {x}}{6 \, a^{5} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 69, normalized size = 0.92 \[ -\frac {2 \, b^{4} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{5}} + \frac {b^{4} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {12 \, a b^{3} x^{\frac {3}{2}} - 6 \, a^{2} b^{2} x + 4 \, a^{3} b \sqrt {x} - 3 \, a^{4}}{6 \, a^{5} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 0.88 \[ \frac {b^{4} \ln \relax (x )}{a^{5}}-\frac {2 b^{4} \ln \left (b \sqrt {x}+a \right )}{a^{5}}+\frac {2 b^{3}}{a^{4} \sqrt {x}}-\frac {b^{2}}{a^{3} x}+\frac {2 b}{3 a^{2} x^{\frac {3}{2}}}-\frac {1}{2 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 64, normalized size = 0.85 \[ -\frac {2 \, b^{4} \log \left (b \sqrt {x} + a\right )}{a^{5}} + \frac {b^{4} \log \relax (x)}{a^{5}} + \frac {12 \, b^{3} x^{\frac {3}{2}} - 6 \, a b^{2} x + 4 \, a^{2} b \sqrt {x} - 3 \, a^{3}}{6 \, a^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 60, normalized size = 0.80 \[ -\frac {\frac {1}{2\,a}-\frac {2\,b\,\sqrt {x}}{3\,a^2}+\frac {b^2\,x}{a^3}-\frac {2\,b^3\,x^{3/2}}{a^4}}{x^2}-\frac {4\,b^4\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.53, size = 99, normalized size = 1.32 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{5 b x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {1}{2 a x^{2}} & \text {for}\: b = 0 \\- \frac {1}{2 a x^{2}} + \frac {2 b}{3 a^{2} x^{\frac {3}{2}}} - \frac {b^{2}}{a^{3} x} + \frac {2 b^{3}}{a^{4} \sqrt {x}} + \frac {b^{4} \log {\relax (x )}}{a^{5}} - \frac {2 b^{4} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{5}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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