Optimal. Leaf size=87 \[ -\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{5/2}}+\frac {b^2 \sqrt {a+b x}}{8 a^2 x}-\frac {\sqrt {a+b x}}{3 x^3}-\frac {b \sqrt {a+b x}}{12 a x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, 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} \[ \frac {b^2 \sqrt {a+b x}}{8 a^2 x}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{5/2}}-\frac {b \sqrt {a+b x}}{12 a x^2}-\frac {\sqrt {a+b x}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x}}{x^4} \, dx &=-\frac {\sqrt {a+b x}}{3 x^3}+\frac {1}{6} b \int \frac {1}{x^3 \sqrt {a+b x}} \, dx\\ &=-\frac {\sqrt {a+b x}}{3 x^3}-\frac {b \sqrt {a+b x}}{12 a x^2}-\frac {b^2 \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{8 a}\\ &=-\frac {\sqrt {a+b x}}{3 x^3}-\frac {b \sqrt {a+b x}}{12 a x^2}+\frac {b^2 \sqrt {a+b x}}{8 a^2 x}+\frac {b^3 \int \frac {1}{x \sqrt {a+b x}} \, dx}{16 a^2}\\ &=-\frac {\sqrt {a+b x}}{3 x^3}-\frac {b \sqrt {a+b x}}{12 a x^2}+\frac {b^2 \sqrt {a+b x}}{8 a^2 x}+\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^2}\\ &=-\frac {\sqrt {a+b x}}{3 x^3}-\frac {b \sqrt {a+b x}}{12 a x^2}+\frac {b^2 \sqrt {a+b x}}{8 a^2 x}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 35, normalized size = 0.40 \[ \frac {2 b^3 (a+b x)^{3/2} \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};\frac {b x}{a}+1\right )}{3 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 145, normalized size = 1.67 \[ \left [\frac {3 \, \sqrt {a} b^{3} x^{3} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (3 \, a b^{2} x^{2} - 2 \, a^{2} b x - 8 \, a^{3}\right )} \sqrt {b x + a}}{48 \, a^{3} x^{3}}, \frac {3 \, \sqrt {-a} b^{3} x^{3} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b^{2} x^{2} - 2 \, a^{2} b x - 8 \, a^{3}\right )} \sqrt {b x + a}}{24 \, a^{3} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 84, normalized size = 0.97 \[ \frac {\frac {3 \, b^{4} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {3 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{4} - 8 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{4} - 3 \, \sqrt {b x + a} a^{2} b^{4}}{a^{2} b^{3} x^{3}}}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 65, normalized size = 0.75 \[ 2 \left (-\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 a^{\frac {5}{2}}}+\frac {-\frac {\left (b x +a \right )^{\frac {3}{2}}}{6 a}+\frac {\left (b x +a \right )^{\frac {5}{2}}}{16 a^{2}}-\frac {\sqrt {b x +a}}{16}}{b^{3} x^{3}}\right ) b^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 121, normalized size = 1.39 \[ \frac {b^{3} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{16 \, a^{\frac {5}{2}}} + \frac {3 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{3} - 8 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{3} - 3 \, \sqrt {b x + a} a^{2} b^{3}}{24 \, {\left ({\left (b x + a\right )}^{3} a^{2} - 3 \, {\left (b x + a\right )}^{2} a^{3} + 3 \, {\left (b x + a\right )} a^{4} - a^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 66, normalized size = 0.76 \[ \frac {{\left (a+b\,x\right )}^{5/2}}{8\,a^2\,x^3}-\frac {{\left (a+b\,x\right )}^{3/2}}{3\,a\,x^3}-\frac {\sqrt {a+b\,x}}{8\,x^3}+\frac {b^3\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{8\,a^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.68, size = 122, normalized size = 1.40 \[ - \frac {a}{3 \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 \sqrt {b}}{12 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {b^{\frac {3}{2}}}{24 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {b^{\frac {5}{2}}}{8 a^{2} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8 a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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