Optimal. Leaf size=90 \[ -\frac{5 b^2 \sqrt{a+b x}}{8 a^3 x}+\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}-\frac{\sqrt{a+b x}}{3 a x^3} \]
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Rubi [A] time = 0.0253039, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {51, 63, 208} \[ -\frac{5 b^2 \sqrt{a+b x}}{8 a^3 x}+\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}-\frac{\sqrt{a+b x}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt{a+b x}} \, dx &=-\frac{\sqrt{a+b x}}{3 a x^3}-\frac{(5 b) \int \frac{1}{x^3 \sqrt{a+b x}} \, dx}{6 a}\\ &=-\frac{\sqrt{a+b x}}{3 a x^3}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}+\frac{\left (5 b^2\right ) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{8 a^2}\\ &=-\frac{\sqrt{a+b x}}{3 a x^3}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}-\frac{5 b^2 \sqrt{a+b x}}{8 a^3 x}-\frac{\left (5 b^3\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{16 a^3}\\ &=-\frac{\sqrt{a+b x}}{3 a x^3}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}-\frac{5 b^2 \sqrt{a+b x}}{8 a^3 x}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{8 a^3}\\ &=-\frac{\sqrt{a+b x}}{3 a x^3}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}-\frac{5 b^2 \sqrt{a+b x}}{8 a^3 x}+\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0084621, size = 33, normalized size = 0.37 \[ \frac{2 b^3 \sqrt{a+b x} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{b x}{a}+1\right )}{a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 90, normalized size = 1. \begin{align*} 2\,{b}^{3} \left ( -1/6\,{\frac{\sqrt{bx+a}}{a{b}^{3}{x}^{3}}}-5/6\,{\frac{1}{a} \left ( -1/4\,{\frac{\sqrt{bx+a}}{a{b}^{2}{x}^{2}}}-3/4\,{\frac{1}{a} \left ( -1/2\,{\frac{\sqrt{bx+a}}{abx}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \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 = 1.49112, size = 356, normalized size = 3.96 \begin{align*} \left [\frac{15 \, \sqrt{a} b^{3} x^{3} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \,{\left (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x + a}}{48 \, a^{4} x^{3}}, -\frac{15 \, \sqrt{-a} b^{3} x^{3} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x + a}}{24 \, a^{4} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.4177, size = 129, normalized size = 1.43 \begin{align*} - \frac{1}{3 \sqrt{b} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{\sqrt{b}}{12 a x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 b^{\frac{3}{2}}}{24 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 b^{\frac{5}{2}}}{8 a^{3} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{5 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8 a^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20976, size = 113, normalized size = 1.26 \begin{align*} -\frac{\frac{15 \, b^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{15 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{4} - 40 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{4} + 33 \, \sqrt{b x + a} a^{2} b^{4}}{a^{3} b^{3} x^{3}}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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