Optimal. Leaf size=123 \[ -\frac {14 b^6 \log \left (a+b \sqrt {x}\right )}{a^8}+\frac {7 b^6 \log (x)}{a^8}+\frac {2 b^6}{a^7 \left (a+b \sqrt {x}\right )}+\frac {12 b^5}{a^7 \sqrt {x}}-\frac {5 b^4}{a^6 x}+\frac {8 b^3}{3 a^5 x^{3/2}}-\frac {3 b^2}{2 a^4 x^2}+\frac {4 b}{5 a^3 x^{5/2}}-\frac {1}{3 a^2 x^3} \]
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Rubi [A] time = 0.08, antiderivative size = 123, 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} \[ \frac {8 b^3}{3 a^5 x^{3/2}}-\frac {3 b^2}{2 a^4 x^2}+\frac {2 b^6}{a^7 \left (a+b \sqrt {x}\right )}+\frac {12 b^5}{a^7 \sqrt {x}}-\frac {5 b^4}{a^6 x}-\frac {14 b^6 \log \left (a+b \sqrt {x}\right )}{a^8}+\frac {7 b^6 \log (x)}{a^8}+\frac {4 b}{5 a^3 x^{5/2}}-\frac {1}{3 a^2 x^3} \]
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 )^2 x^4} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^7 (a+b x)^2} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {1}{a^2 x^7}-\frac {2 b}{a^3 x^6}+\frac {3 b^2}{a^4 x^5}-\frac {4 b^3}{a^5 x^4}+\frac {5 b^4}{a^6 x^3}-\frac {6 b^5}{a^7 x^2}+\frac {7 b^6}{a^8 x}-\frac {b^7}{a^7 (a+b x)^2}-\frac {7 b^7}{a^8 (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 b^6}{a^7 \left (a+b \sqrt {x}\right )}-\frac {1}{3 a^2 x^3}+\frac {4 b}{5 a^3 x^{5/2}}-\frac {3 b^2}{2 a^4 x^2}+\frac {8 b^3}{3 a^5 x^{3/2}}-\frac {5 b^4}{a^6 x}+\frac {12 b^5}{a^7 \sqrt {x}}-\frac {14 b^6 \log \left (a+b \sqrt {x}\right )}{a^8}+\frac {7 b^6 \log (x)}{a^8}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 115, normalized size = 0.93 \[ \frac {\frac {a \left (-10 a^6+14 a^5 b \sqrt {x}-21 a^4 b^2 x+35 a^3 b^3 x^{3/2}-70 a^2 b^4 x^2+210 a b^5 x^{5/2}+420 b^6 x^3\right )}{x^3 \left (a+b \sqrt {x}\right )}-420 b^6 \log \left (a+b \sqrt {x}\right )+210 b^6 \log (x)}{30 a^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 155, normalized size = 1.26 \[ -\frac {210 \, a^{2} b^{6} x^{3} - 105 \, a^{4} b^{4} x^{2} - 35 \, a^{6} b^{2} x - 10 \, a^{8} + 420 \, {\left (b^{8} x^{4} - a^{2} b^{6} x^{3}\right )} \log \left (b \sqrt {x} + a\right ) - 420 \, {\left (b^{8} x^{4} - a^{2} b^{6} x^{3}\right )} \log \left (\sqrt {x}\right ) - 4 \, {\left (105 \, a b^{7} x^{3} - 70 \, a^{3} b^{5} x^{2} - 14 \, a^{5} b^{3} x - 6 \, a^{7} b\right )} \sqrt {x}}{30 \, {\left (a^{8} b^{2} x^{4} - a^{10} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 112, normalized size = 0.91 \[ -\frac {14 \, b^{6} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{a^{8}} + \frac {7 \, b^{6} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac {420 \, a b^{6} x^{3} + 210 \, a^{2} b^{5} x^{\frac {5}{2}} - 70 \, a^{3} b^{4} x^{2} + 35 \, a^{4} b^{3} x^{\frac {3}{2}} - 21 \, a^{5} b^{2} x + 14 \, a^{6} b \sqrt {x} - 10 \, a^{7}}{30 \, {\left (b \sqrt {x} + a\right )} a^{8} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 106, normalized size = 0.86 \[ \frac {2 b^{6}}{\left (b \sqrt {x}+a \right ) a^{7}}+\frac {7 b^{6} \ln \relax (x )}{a^{8}}-\frac {14 b^{6} \ln \left (b \sqrt {x}+a \right )}{a^{8}}+\frac {12 b^{5}}{a^{7} \sqrt {x}}-\frac {5 b^{4}}{a^{6} x}+\frac {8 b^{3}}{3 a^{5} x^{\frac {3}{2}}}-\frac {3 b^{2}}{2 a^{4} x^{2}}+\frac {4 b}{5 a^{3} x^{\frac {5}{2}}}-\frac {1}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 110, normalized size = 0.89 \[ \frac {420 \, b^{6} x^{3} + 210 \, a b^{5} x^{\frac {5}{2}} - 70 \, a^{2} b^{4} x^{2} + 35 \, a^{3} b^{3} x^{\frac {3}{2}} - 21 \, a^{4} b^{2} x + 14 \, a^{5} b \sqrt {x} - 10 \, a^{6}}{30 \, {\left (a^{7} b x^{\frac {7}{2}} + a^{8} x^{3}\right )}} - \frac {14 \, b^{6} \log \left (b \sqrt {x} + a\right )}{a^{8}} + \frac {7 \, b^{6} \log \relax (x)}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 103, normalized size = 0.84 \[ \frac {\frac {7\,b\,\sqrt {x}}{15\,a^2}-\frac {1}{3\,a}-\frac {7\,b^2\,x}{10\,a^3}-\frac {7\,b^4\,x^2}{3\,a^5}+\frac {7\,b^3\,x^{3/2}}{6\,a^4}+\frac {14\,b^6\,x^3}{a^7}+\frac {7\,b^5\,x^{5/2}}{a^6}}{a\,x^3+b\,x^{7/2}}-\frac {28\,b^6\,\mathrm {atanh}\left (\frac {2\,b\,\sqrt {x}}{a}+1\right )}{a^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.65, size = 400, normalized size = 3.25 \[ \begin {cases} \frac {\tilde {\infty }}{x^{4}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{4 b^{2} x^{4}} & \text {for}\: a = 0 \\- \frac {1}{3 a^{2} x^{3}} & \text {for}\: b = 0 \\- \frac {10 a^{7} \sqrt {x}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} + \frac {14 a^{6} b x}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} - \frac {21 a^{5} b^{2} x^{\frac {3}{2}}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} + \frac {35 a^{4} b^{3} x^{2}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} - \frac {70 a^{3} b^{4} x^{\frac {5}{2}}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} + \frac {210 a^{2} b^{5} x^{3}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} + \frac {210 a b^{6} x^{\frac {7}{2}} \log {\relax (x )}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} - \frac {420 a b^{6} x^{\frac {7}{2}} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} + \frac {420 a b^{6} x^{\frac {7}{2}}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} + \frac {210 b^{7} x^{4} \log {\relax (x )}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} - \frac {420 b^{7} x^{4} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{30 a^{9} x^{\frac {7}{2}} + 30 a^{8} b x^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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