Optimal. Leaf size=38 \[ \frac{2 a}{7 b^2 \left (a+b \sqrt{x}\right )^7}-\frac{1}{3 b^2 \left (a+b \sqrt{x}\right )^6} \]
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Rubi [A] time = 0.0210688, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac{2 a}{7 b^2 \left (a+b \sqrt{x}\right )^7}-\frac{1}{3 b^2 \left (a+b \sqrt{x}\right )^6} \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^8} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{(a+b x)^8} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^8}+\frac{1}{b (a+b x)^7}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a}{7 b^2 \left (a+b \sqrt{x}\right )^7}-\frac{1}{3 b^2 \left (a+b \sqrt{x}\right )^6}\\ \end{align*}
Mathematica [A] time = 0.017241, size = 28, normalized size = 0.74 \[ -\frac{a+7 b \sqrt{x}}{21 b^2 \left (a+b \sqrt{x}\right )^7} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.064, size = 399, normalized size = 10.5 \begin{align*} -{\frac{{a}^{8}}{7\, \left ({b}^{2}x-{a}^{2} \right ) ^{7}{b}^{2}}}+{b}^{8} \left ( -{\frac{2\,{a}^{6}}{3\,{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{6}}}-{\frac{{a}^{8}}{7\,{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{7}}}-{\frac{{a}^{2}}{{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{4}}}-{\frac{1}{3\,{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{3}}}-{\frac{6\,{a}^{4}}{5\,{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{5}}} \right ) -{\frac{1}{6\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-6}}+{\frac{1}{6\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-6}}+28\,{b}^{6}{a}^{2} \left ( -1/2\,{\frac{{a}^{4}}{{b}^{8} \left ({b}^{2}x-{a}^{2} \right ) ^{6}}}-1/7\,{\frac{{a}^{6}}{{b}^{8} \left ({b}^{2}x-{a}^{2} \right ) ^{7}}}-1/4\,{\frac{1}{ \left ({b}^{2}x-{a}^{2} \right ) ^{4}{b}^{8}}}-3/5\,{\frac{{a}^{2}}{{b}^{8} \left ({b}^{2}x-{a}^{2} \right ) ^{5}}} \right ) +{\frac{a}{7\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-7}}+{\frac{a}{7\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-7}}+28\,{a}^{6}{b}^{2} \left ( -1/6\,{\frac{1}{{b}^{4} \left ({b}^{2}x-{a}^{2} \right ) ^{6}}}-1/7\,{\frac{{a}^{2}}{{b}^{4} \left ({b}^{2}x-{a}^{2} \right ) ^{7}}} \right ) +70\,{a}^{4}{b}^{4} \left ( -1/3\,{\frac{{a}^{2}}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{6}}}-1/7\,{\frac{{a}^{4}}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{7}}}-1/5\,{\frac{1}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{5}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965735, size = 41, normalized size = 1.08 \begin{align*} -\frac{1}{3 \,{\left (b \sqrt{x} + a\right )}^{6} b^{2}} + \frac{2 \, a}{7 \,{\left (b \sqrt{x} + a\right )}^{7} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.30682, size = 350, normalized size = 9.21 \begin{align*} -\frac{7 \, b^{8} x^{4} + 140 \, a^{2} b^{6} x^{3} + 210 \, a^{4} b^{4} x^{2} + 28 \, a^{6} b^{2} x - a^{8} - 16 \,{\left (3 \, a b^{7} x^{3} + 14 \, a^{3} b^{5} x^{2} + 7 \, a^{5} b^{3} x\right )} \sqrt{x}}{21 \,{\left (b^{16} x^{7} - 7 \, a^{2} b^{14} x^{6} + 21 \, a^{4} b^{12} x^{5} - 35 \, a^{6} b^{10} x^{4} + 35 \, a^{8} b^{8} x^{3} - 21 \, a^{10} b^{6} x^{2} + 7 \, a^{12} b^{4} x - a^{14} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.7545, size = 199, normalized size = 5.24 \begin{align*} \begin{cases} - \frac{a}{21 a^{7} b^{2} + 147 a^{6} b^{3} \sqrt{x} + 441 a^{5} b^{4} x + 735 a^{4} b^{5} x^{\frac{3}{2}} + 735 a^{3} b^{6} x^{2} + 441 a^{2} b^{7} x^{\frac{5}{2}} + 147 a b^{8} x^{3} + 21 b^{9} x^{\frac{7}{2}}} - \frac{7 b \sqrt{x}}{21 a^{7} b^{2} + 147 a^{6} b^{3} \sqrt{x} + 441 a^{5} b^{4} x + 735 a^{4} b^{5} x^{\frac{3}{2}} + 735 a^{3} b^{6} x^{2} + 441 a^{2} b^{7} x^{\frac{5}{2}} + 147 a b^{8} x^{3} + 21 b^{9} x^{\frac{7}{2}}} & \text{for}\: b \neq 0 \\\frac{x}{a^{8}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10261, size = 30, normalized size = 0.79 \begin{align*} -\frac{7 \, b \sqrt{x} + a}{21 \,{\left (b \sqrt{x} + a\right )}^{7} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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