Optimal. Leaf size=139 \[ \frac{20 b^3}{3 a^6 x^{3/2}}-\frac{3 b^2}{a^5 x^2}+\frac{14 b^6}{a^8 \left (a+b \sqrt{x}\right )}+\frac{b^6}{a^7 \left (a+b \sqrt{x}\right )^2}+\frac{42 b^5}{a^8 \sqrt{x}}-\frac{15 b^4}{a^7 x}-\frac{56 b^6 \log \left (a+b \sqrt{x}\right )}{a^9}+\frac{28 b^6 \log (x)}{a^9}+\frac{6 b}{5 a^4 x^{5/2}}-\frac{1}{3 a^3 x^3} \]
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Rubi [A] time = 0.0993551, antiderivative size = 139, normalized size of antiderivative = 1., 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} \[ \frac{20 b^3}{3 a^6 x^{3/2}}-\frac{3 b^2}{a^5 x^2}+\frac{14 b^6}{a^8 \left (a+b \sqrt{x}\right )}+\frac{b^6}{a^7 \left (a+b \sqrt{x}\right )^2}+\frac{42 b^5}{a^8 \sqrt{x}}-\frac{15 b^4}{a^7 x}-\frac{56 b^6 \log \left (a+b \sqrt{x}\right )}{a^9}+\frac{28 b^6 \log (x)}{a^9}+\frac{6 b}{5 a^4 x^{5/2}}-\frac{1}{3 a^3 x^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^3 x^4} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^7 (a+b x)^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^7}-\frac{3 b}{a^4 x^6}+\frac{6 b^2}{a^5 x^5}-\frac{10 b^3}{a^6 x^4}+\frac{15 b^4}{a^7 x^3}-\frac{21 b^5}{a^8 x^2}+\frac{28 b^6}{a^9 x}-\frac{b^7}{a^7 (a+b x)^3}-\frac{7 b^7}{a^8 (a+b x)^2}-\frac{28 b^7}{a^9 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{b^6}{a^7 \left (a+b \sqrt{x}\right )^2}+\frac{14 b^6}{a^8 \left (a+b \sqrt{x}\right )}-\frac{1}{3 a^3 x^3}+\frac{6 b}{5 a^4 x^{5/2}}-\frac{3 b^2}{a^5 x^2}+\frac{20 b^3}{3 a^6 x^{3/2}}-\frac{15 b^4}{a^7 x}+\frac{42 b^5}{a^8 \sqrt{x}}-\frac{56 b^6 \log \left (a+b \sqrt{x}\right )}{a^9}+\frac{28 b^6 \log (x)}{a^9}\\ \end{align*}
Mathematica [A] time = 0.153078, size = 128, normalized size = 0.92 \[ \frac{\frac{a \left (28 a^4 b^3 x^{3/2}-70 a^3 b^4 x^2+280 a^2 b^5 x^{5/2}-14 a^5 b^2 x+8 a^6 b \sqrt{x}-5 a^7+1260 a b^6 x^3+840 b^7 x^{7/2}\right )}{x^3 \left (a+b \sqrt{x}\right )^2}-840 b^6 \log \left (a+b \sqrt{x}\right )+420 b^6 \log (x)}{15 a^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 122, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{a}^{3}{x}^{3}}}+{\frac{6\,b}{5\,{a}^{4}}{x}^{-{\frac{5}{2}}}}-3\,{\frac{{b}^{2}}{{a}^{5}{x}^{2}}}+{\frac{20\,{b}^{3}}{3\,{a}^{6}}{x}^{-{\frac{3}{2}}}}-15\,{\frac{{b}^{4}}{{a}^{7}x}}+28\,{\frac{{b}^{6}\ln \left ( x \right ) }{{a}^{9}}}-56\,{\frac{{b}^{6}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{9}}}+42\,{\frac{{b}^{5}}{{a}^{8}\sqrt{x}}}+{\frac{{b}^{6}}{{a}^{7}} \left ( a+b\sqrt{x} \right ) ^{-2}}+14\,{\frac{{b}^{6}}{{a}^{8} \left ( a+b\sqrt{x} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983174, size = 178, normalized size = 1.28 \begin{align*} \frac{840 \, b^{7} x^{\frac{7}{2}} + 1260 \, a b^{6} x^{3} + 280 \, a^{2} b^{5} x^{\frac{5}{2}} - 70 \, a^{3} b^{4} x^{2} + 28 \, a^{4} b^{3} x^{\frac{3}{2}} - 14 \, a^{5} b^{2} x + 8 \, a^{6} b \sqrt{x} - 5 \, a^{7}}{15 \,{\left (a^{8} b^{2} x^{4} + 2 \, a^{9} b x^{\frac{7}{2}} + a^{10} x^{3}\right )}} - \frac{56 \, b^{6} \log \left (b \sqrt{x} + a\right )}{a^{9}} + \frac{28 \, b^{6} \log \left (x\right )}{a^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35731, size = 462, normalized size = 3.32 \begin{align*} -\frac{420 \, a^{2} b^{8} x^{4} - 630 \, a^{4} b^{6} x^{3} + 140 \, a^{6} b^{4} x^{2} + 35 \, a^{8} b^{2} x + 5 \, a^{10} + 840 \,{\left (b^{10} x^{5} - 2 \, a^{2} b^{8} x^{4} + a^{4} b^{6} x^{3}\right )} \log \left (b \sqrt{x} + a\right ) - 840 \,{\left (b^{10} x^{5} - 2 \, a^{2} b^{8} x^{4} + a^{4} b^{6} x^{3}\right )} \log \left (\sqrt{x}\right ) - 2 \,{\left (420 \, a b^{9} x^{4} - 700 \, a^{3} b^{7} x^{3} + 224 \, a^{5} b^{5} x^{2} + 32 \, a^{7} b^{3} x + 9 \, a^{9} b\right )} \sqrt{x}}{15 \,{\left (a^{9} b^{4} x^{5} - 2 \, a^{11} b^{2} x^{4} + a^{13} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 98.2033, size = 707, normalized size = 5.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11283, size = 166, normalized size = 1.19 \begin{align*} -\frac{56 \, b^{6} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{9}} + \frac{28 \, b^{6} \log \left ({\left | x \right |}\right )}{a^{9}} + \frac{840 \, a b^{7} x^{\frac{7}{2}} + 1260 \, a^{2} b^{6} x^{3} + 280 \, a^{3} b^{5} x^{\frac{5}{2}} - 70 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3} x^{\frac{3}{2}} - 14 \, a^{6} b^{2} x + 8 \, a^{7} b \sqrt{x} - 5 \, a^{8}}{15 \,{\left (b \sqrt{x} + a\right )}^{2} a^{9} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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