Optimal. Leaf size=84 \[ \frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{3/2}}-\frac{b^2 \sqrt{a+b x}}{8 a x}-\frac{b \sqrt{a+b x}}{4 x^2}-\frac{(a+b x)^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.0222849, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {47, 51, 63, 208} \[ \frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{3/2}}-\frac{b^2 \sqrt{a+b x}}{8 a x}-\frac{b \sqrt{a+b x}}{4 x^2}-\frac{(a+b x)^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2}}{x^4} \, dx &=-\frac{(a+b x)^{3/2}}{3 x^3}+\frac{1}{2} b \int \frac{\sqrt{a+b x}}{x^3} \, dx\\ &=-\frac{b \sqrt{a+b x}}{4 x^2}-\frac{(a+b x)^{3/2}}{3 x^3}+\frac{1}{8} b^2 \int \frac{1}{x^2 \sqrt{a+b x}} \, dx\\ &=-\frac{b \sqrt{a+b x}}{4 x^2}-\frac{b^2 \sqrt{a+b x}}{8 a x}-\frac{(a+b x)^{3/2}}{3 x^3}-\frac{b^3 \int \frac{1}{x \sqrt{a+b x}} \, dx}{16 a}\\ &=-\frac{b \sqrt{a+b x}}{4 x^2}-\frac{b^2 \sqrt{a+b x}}{8 a x}-\frac{(a+b x)^{3/2}}{3 x^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{8 a}\\ &=-\frac{b \sqrt{a+b x}}{4 x^2}-\frac{b^2 \sqrt{a+b x}}{8 a x}-\frac{(a+b x)^{3/2}}{3 x^3}+\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0136337, size = 35, normalized size = 0.42 \[ \frac{2 b^3 (a+b x)^{5/2} \, _2F_1\left (\frac{5}{2},4;\frac{7}{2};\frac{b x}{a}+1\right )}{5 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 63, normalized size = 0.8 \begin{align*} 2\,{b}^{3} \left ({\frac{1}{{b}^{3}{x}^{3}} \left ( -1/16\,{\frac{ \left ( bx+a \right ) ^{5/2}}{a}}-1/6\, \left ( bx+a \right ) ^{3/2}+1/16\,a\sqrt{bx+a} \right ) }+1/16\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \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 = 1.52887, size = 351, normalized size = 4.18 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{3} x^{3} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \,{\left (3 \, a b^{2} x^{2} + 14 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x + a}}{48 \, a^{2} x^{3}}, -\frac{3 \, \sqrt{-a} b^{3} x^{3} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (3 \, a b^{2} x^{2} + 14 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x + a}}{24 \, a^{2} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.22523, size = 124, normalized size = 1.48 \begin{align*} - \frac{a^{2}}{3 \sqrt{b} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{11 a \sqrt{b}}{12 x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{17 b^{\frac{3}{2}}}{24 x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{b^{\frac{5}{2}}}{8 a \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19848, size = 113, normalized size = 1.35 \begin{align*} -\frac{\frac{3 \, b^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{3 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{4} + 8 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{4} - 3 \, \sqrt{b x + a} a^{2} b^{4}}{a b^{3} x^{3}}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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