Optimal. Leaf size=103 \[ -\frac {2 b^6 \log \left (a+b \sqrt {x}\right )}{a^7}+\frac {b^6 \log (x)}{a^7}+\frac {2 b^5}{a^6 \sqrt {x}}-\frac {b^4}{a^5 x}+\frac {2 b^3}{3 a^4 x^{3/2}}-\frac {b^2}{2 a^3 x^2}+\frac {2 b}{5 a^2 x^{5/2}}-\frac {1}{3 a x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 103, 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} \[ \frac {2 b^3}{3 a^4 x^{3/2}}-\frac {b^2}{2 a^3 x^2}+\frac {2 b^5}{a^6 \sqrt {x}}-\frac {b^4}{a^5 x}-\frac {2 b^6 \log \left (a+b \sqrt {x}\right )}{a^7}+\frac {b^6 \log (x)}{a^7}+\frac {2 b}{5 a^2 x^{5/2}}-\frac {1}{3 a x^3} \]
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^4} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^7 (a+b x)} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {1}{a x^7}-\frac {b}{a^2 x^6}+\frac {b^2}{a^3 x^5}-\frac {b^3}{a^4 x^4}+\frac {b^4}{a^5 x^3}-\frac {b^5}{a^6 x^2}+\frac {b^6}{a^7 x}-\frac {b^7}{a^7 (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {1}{3 a x^3}+\frac {2 b}{5 a^2 x^{5/2}}-\frac {b^2}{2 a^3 x^2}+\frac {2 b^3}{3 a^4 x^{3/2}}-\frac {b^4}{a^5 x}+\frac {2 b^5}{a^6 \sqrt {x}}-\frac {2 b^6 \log \left (a+b \sqrt {x}\right )}{a^7}+\frac {b^6 \log (x)}{a^7}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 93, normalized size = 0.90 \[ \frac {\frac {a \left (-10 a^5+12 a^4 b \sqrt {x}-15 a^3 b^2 x+20 a^2 b^3 x^{3/2}-30 a b^4 x^2+60 b^5 x^{5/2}\right )}{x^3}-60 b^6 \log \left (a+b \sqrt {x}\right )+30 b^6 \log (x)}{30 a^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 92, normalized size = 0.89 \[ -\frac {60 \, b^{6} x^{3} \log \left (b \sqrt {x} + a\right ) - 60 \, b^{6} x^{3} \log \left (\sqrt {x}\right ) + 30 \, a^{2} b^{4} x^{2} + 15 \, a^{4} b^{2} x + 10 \, a^{6} - 4 \, {\left (15 \, a b^{5} x^{2} + 5 \, a^{3} b^{3} x + 3 \, a^{5} b\right )} \sqrt {x}}{30 \, a^{7} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 91, normalized size = 0.88 \[ -\frac {2 \, b^{6} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{7}} + \frac {b^{6} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac {60 \, a b^{5} x^{\frac {5}{2}} - 30 \, a^{2} b^{4} x^{2} + 20 \, a^{3} b^{3} x^{\frac {3}{2}} - 15 \, a^{4} b^{2} x + 12 \, a^{5} b \sqrt {x} - 10 \, a^{6}}{30 \, a^{7} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 88, normalized size = 0.85 \[ \frac {b^{6} \ln \relax (x )}{a^{7}}-\frac {2 b^{6} \ln \left (b \sqrt {x}+a \right )}{a^{7}}+\frac {2 b^{5}}{a^{6} \sqrt {x}}-\frac {b^{4}}{a^{5} x}+\frac {2 b^{3}}{3 a^{4} x^{\frac {3}{2}}}-\frac {b^{2}}{2 a^{3} x^{2}}+\frac {2 b}{5 a^{2} x^{\frac {5}{2}}}-\frac {1}{3 a \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 86, normalized size = 0.83 \[ -\frac {2 \, b^{6} \log \left (b \sqrt {x} + a\right )}{a^{7}} + \frac {b^{6} \log \relax (x)}{a^{7}} + \frac {60 \, b^{5} x^{\frac {5}{2}} - 30 \, a b^{4} x^{2} + 20 \, a^{2} b^{3} x^{\frac {3}{2}} - 15 \, a^{3} b^{2} x + 12 \, a^{4} b \sqrt {x} - 10 \, a^{5}}{30 \, a^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 82, normalized size = 0.80 \[ -\frac {\frac {1}{3\,a}-\frac {2\,b\,\sqrt {x}}{5\,a^2}+\frac {b^2\,x}{2\,a^3}+\frac {b^4\,x^2}{a^5}-\frac {2\,b^3\,x^{3/2}}{3\,a^4}-\frac {2\,b^5\,x^{5/2}}{a^6}}{x^3}-\frac {4\,b^6\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.19, size = 126, normalized size = 1.22 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{7 b x^{\frac {7}{2}}} & \text {for}\: a = 0 \\- \frac {1}{3 a x^{3}} & \text {for}\: b = 0 \\- \frac {1}{3 a x^{3}} + \frac {2 b}{5 a^{2} x^{\frac {5}{2}}} - \frac {b^{2}}{2 a^{3} x^{2}} + \frac {2 b^{3}}{3 a^{4} x^{\frac {3}{2}}} - \frac {b^{4}}{a^{5} x} + \frac {2 b^{5}}{a^{6} \sqrt {x}} + \frac {b^{6} \log {\relax (x )}}{a^{7}} - \frac {2 b^{6} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{7}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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