Optimal. Leaf size=87 \[ -\frac {\sqrt {a+b x}}{3 x^3}-\frac {b \sqrt {a+b x}}{12 a x^2}+\frac {b^2 \sqrt {a+b x}}{8 a^2 x}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{5/2}} \]
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Rubi [A]
time = 0.02, 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 = {43, 44, 65, 214}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 214
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 \text {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 [A]
time = 0.10, size = 67, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {a+b x} \left (8 a^2+2 a b x-3 b^2 x^2\right )}{24 a^2 x^3}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(177\) vs. \(2(87)=174\).
time = 7.08, size = 135, normalized size = 1.55 \begin {gather*} -\frac {a b^{\frac {3}{2}} \left (1+\frac {a}{b x}\right )^{\frac {3}{2}}}{3 x^{\frac {3}{2}} \left (a+b x\right )^2}-\frac {b^3 \text {ArcSinh}\left [\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}}\right ]}{8 a^{\frac {5}{2}}}-\frac {5 b^{\frac {5}{2}} \left (1+\frac {a}{b x}\right )^{\frac {3}{2}}}{12 \sqrt {x} \left (a+b x\right )^2}+\frac {b^{\frac {7}{2}} \sqrt {x} \left (1+\frac {a}{b x}\right )^{\frac {3}{2}}}{24 a \left (a+b x\right )^2}+\frac {b^{\frac {9}{2}} x^{\frac {3}{2}} \left (1+\frac {a}{b x}\right )^{\frac {3}{2}}}{8 a^2 \left (a+b x\right )^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.09, size = 66, normalized size = 0.76
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-3 x^{2} b^{2}+2 a b x +8 a^{2}\right )}{24 x^{3} a^{2}}-\frac {b^{3} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 a^{\frac {5}{2}}}\) | \(56\) |
derivativedivides | \(2 b^{3} \left (-\frac {-\frac {\left (b x +a \right )^{\frac {5}{2}}}{16 a^{2}}+\frac {\left (b x +a \right )^{\frac {3}{2}}}{6 a}+\frac {\sqrt {b x +a}}{16}}{b^{3} x^{3}}-\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 a^{\frac {5}{2}}}\right )\) | \(66\) |
default | \(2 b^{3} \left (-\frac {-\frac {\left (b x +a \right )^{\frac {5}{2}}}{16 a^{2}}+\frac {\left (b x +a \right )^{\frac {3}{2}}}{6 a}+\frac {\sqrt {b x +a}}{16}}{b^{3} x^{3}}-\frac {\arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 a^{\frac {5}{2}}}\right )\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 121, normalized size = 1.39 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 145, normalized size = 1.67 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.28, size = 122, normalized size = 1.40 \begin {gather*} - \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 119, normalized size = 1.37 \begin {gather*} \frac {\frac {3 \sqrt {a+b x} \left (a+b x\right )^{2} b^{4}-8 \sqrt {a+b x} \left (a+b x\right ) a b^{4}-3 \sqrt {a+b x} a^{2} b^{4}}{24 a^{2} \left (a+b x-a\right )^{3}}+\frac {b^{4} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {-a}}\right )}{4 a^{2}\cdot 2 \sqrt {-a}}}{b} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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