Optimal. Leaf size=136 \[ \frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}-\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{9/2}}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2} \]
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Rubi [A] time = 0.0589333, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {266, 51, 63, 208} \[ \frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}-\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{9/2}}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \sqrt{x}} x^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^5 \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )}{4 a}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}+\frac{\left (35 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )}{24 a^2}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}-\frac{\left (35 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )}{32 a^3}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}+\frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}+\frac{\left (35 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sqrt{x}\right )}{64 a^4}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}+\frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}+\frac{\left (35 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sqrt{x}}\right )}{32 a^4}\\ &=-\frac{\sqrt{a+b \sqrt{x}}}{2 a x^2}+\frac{7 b \sqrt{a+b \sqrt{x}}}{12 a^2 x^{3/2}}-\frac{35 b^2 \sqrt{a+b \sqrt{x}}}{48 a^3 x}+\frac{35 b^3 \sqrt{a+b \sqrt{x}}}{32 a^4 \sqrt{x}}-\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{32 a^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0069564, size = 41, normalized size = 0.3 \[ -\frac{4 b^4 \sqrt{a+b \sqrt{x}} \, _2F_1\left (\frac{1}{2},5;\frac{3}{2};\frac{\sqrt{x} b}{a}+1\right )}{a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 124, normalized size = 0.9 \begin{align*} 4\,{b}^{4} \left ( -1/8\,{\frac{\sqrt{a+b\sqrt{x}}}{a{b}^{4}{x}^{2}}}-{\frac{7}{8\,a} \left ( -1/6\,{\frac{\sqrt{a+b\sqrt{x}}}{a{b}^{3}{x}^{3/2}}}-5/6\,{\frac{1}{a} \left ( -1/4\,{\frac{\sqrt{a+b\sqrt{x}}}{xa{b}^{2}}}-3/4\,{\frac{1}{a} \left ( -1/2\,{\frac{\sqrt{a+b\sqrt{x}}}{ab\sqrt{x}}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) } \right ) } \right ) } \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33949, size = 475, normalized size = 3.49 \begin{align*} \left [\frac{105 \, \sqrt{a} b^{4} x^{2} \log \left (\frac{b x - 2 \, \sqrt{b \sqrt{x} + a} \sqrt{a} \sqrt{x} + 2 \, a \sqrt{x}}{x}\right ) - 2 \,{\left (70 \, a^{2} b^{2} x + 48 \, a^{4} - 7 \,{\left (15 \, a b^{3} x + 8 \, a^{3} b\right )} \sqrt{x}\right )} \sqrt{b \sqrt{x} + a}}{192 \, a^{5} x^{2}}, \frac{105 \, \sqrt{-a} b^{4} x^{2} \arctan \left (\frac{\sqrt{b \sqrt{x} + a} \sqrt{-a}}{a}\right ) -{\left (70 \, a^{2} b^{2} x + 48 \, a^{4} - 7 \,{\left (15 \, a b^{3} x + 8 \, a^{3} b\right )} \sqrt{x}\right )} \sqrt{b \sqrt{x} + a}}{96 \, a^{5} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.2865, size = 173, normalized size = 1.27 \begin{align*} - \frac{1}{2 \sqrt{b} x^{\frac{9}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{\sqrt{b}}{12 a x^{\frac{7}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{7 b^{\frac{3}{2}}}{48 a^{2} x^{\frac{5}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{35 b^{\frac{5}{2}}}{96 a^{3} x^{\frac{3}{4}} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{35 b^{\frac{7}{2}}}{32 a^{4} \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} - \frac{35 b^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )}}{32 a^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11156, size = 127, normalized size = 0.93 \begin{align*} \frac{1}{96} \, b^{4}{\left (\frac{105 \, \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{105 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} - 385 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a + 511 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{2} - 279 \, \sqrt{b \sqrt{x} + a} a^{3}}{a^{4} b^{4} x^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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