Optimal. Leaf size=90 \[ -\frac {\sqrt {a+b x}}{3 a x^3}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}-\frac {5 b^2 \sqrt {a+b x}}{8 a^3 x}+\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 90, normalized size of antiderivative = 1.00, 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 = {44, 65, 214}
\begin {gather*} \frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}}-\frac {5 b^2 \sqrt {a+b x}}{8 a^3 x}+\frac {5 b \sqrt {a+b x}}{12 a^2 x^2}-\frac {\sqrt {a+b x}}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 214
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 ) \text {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*}
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Mathematica [A]
time = 0.07, size = 67, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {a+b x} \left (8 a^2-10 a b x+15 b^2 x^2\right )}{24 a^3 x^3}+\frac {5 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 9.90, size = 137, normalized size = 1.52 \begin {gather*} \frac {5 b^3 \text {ArcSinh}\left [\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}}\right ]}{8 a^{\frac {7}{2}}}-\frac {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^{\frac {5}{2}} \left (1+\frac {a}{b x}\right )^{\frac {3}{2}}}{12 a \sqrt {x} \left (a+b x\right )^2}-\frac {5 b^{\frac {7}{2}} \sqrt {x} \left (1+\frac {a}{b x}\right )^{\frac {3}{2}}}{24 a^2 \left (a+b x\right )^2}-\frac {5 b^{\frac {9}{2}} x^{\frac {3}{2}} \left (1+\frac {a}{b x}\right )^{\frac {3}{2}}}{8 a^3 \left (a+b x\right )^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 90, normalized size = 1.00
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (15 x^{2} b^{2}-10 a b x +8 a^{2}\right )}{24 a^{3} x^{3}}+\frac {5 b^{3} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 a^{\frac {7}{2}}}\) | \(56\) |
derivativedivides | \(2 b^{3} \left (-\frac {\sqrt {b x +a}}{6 a \,b^{3} x^{3}}+\frac {\frac {5 \sqrt {b x +a}}{24 a \,b^{2} x^{2}}+\frac {5 \left (-\frac {3 \sqrt {b x +a}}{8 a b x}+\frac {3 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )}{6 a}}{a}\right )\) | \(90\) |
default | \(2 b^{3} \left (-\frac {\sqrt {b x +a}}{6 a \,b^{3} x^{3}}+\frac {\frac {5 \sqrt {b x +a}}{24 a \,b^{2} x^{2}}+\frac {5 \left (-\frac {3 \sqrt {b x +a}}{8 a b x}+\frac {3 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )}{6 a}}{a}\right )\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 121, normalized size = 1.34 \begin {gather*} -\frac {5 \, b^{3} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{16 \, a^{\frac {7}{2}}} - \frac {15 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{3} - 40 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{3} + 33 \, \sqrt {b x + a} a^{2} b^{3}}{24 \, {\left ({\left (b x + a\right )}^{3} a^{3} - 3 \, {\left (b x + a\right )}^{2} a^{4} + 3 \, {\left (b x + a\right )} a^{5} - a^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 145, normalized size = 1.61 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 7.93, size = 129, normalized size = 1.43 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 122, normalized size = 1.36 \begin {gather*} \frac {2 \left (\frac {-15 \sqrt {a+b x} \left (a+b x\right )^{2} b^{4}+40 \sqrt {a+b x} \left (a+b x\right ) a b^{4}-33 \sqrt {a+b x} a^{2} b^{4}}{48 a^{3} \left (a+b x-a\right )^{3}}-\frac {5 b^{4} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {-a}}\right )}{8 a^{3}\cdot 2 \sqrt {-a}}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 69, normalized size = 0.77 \begin {gather*} \frac {5\,{\left (a+b\,x\right )}^{3/2}}{3\,a^2\,x^3}-\frac {11\,\sqrt {a+b\,x}}{8\,a\,x^3}-\frac {5\,{\left (a+b\,x\right )}^{5/2}}{8\,a^3\,x^3}-\frac {b^3\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{8\,a^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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