Optimal. Leaf size=96 \[ 4 a^3 \sqrt {c-\frac {c}{a x}}-\frac {10 a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {8 a^3 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3} \]
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Rubi [A]
time = 0.24, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6302, 6268, 25,
528, 457, 78} \begin {gather*} -\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}+\frac {8 a^3 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {10 a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+4 a^3 \sqrt {c-\frac {c}{a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 78
Rule 457
Rule 528
Rule 6268
Rule 6302
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx\\ &=-\int \frac {\sqrt {c-\frac {c}{a x}} (1+a x)}{x^4 (1-a x)} \, dx\\ &=\frac {c \int \frac {1+a x}{\sqrt {c-\frac {c}{a x}} x^5} \, dx}{a}\\ &=\frac {c \int \frac {a+\frac {1}{x}}{\sqrt {c-\frac {c}{a x}} x^4} \, dx}{a}\\ &=-\frac {c \text {Subst}\left (\int \frac {x^2 (a+x)}{\sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {c \text {Subst}\left (\int \left (\frac {2 a^3}{\sqrt {c-\frac {c x}{a}}}-\frac {5 a^3 \sqrt {c-\frac {c x}{a}}}{c}+\frac {4 a^3 \left (c-\frac {c x}{a}\right )^{3/2}}{c^2}-\frac {a^3 \left (c-\frac {c x}{a}\right )^{5/2}}{c^3}\right ) \, dx,x,\frac {1}{x}\right )}{a}\\ &=4 a^3 \sqrt {c-\frac {c}{a x}}-\frac {10 a^3 \left (c-\frac {c}{a x}\right )^{3/2}}{3 c}+\frac {8 a^3 \left (c-\frac {c}{a x}\right )^{5/2}}{5 c^2}-\frac {2 a^3 \left (c-\frac {c}{a x}\right )^{7/2}}{7 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 44, normalized size = 0.46 \begin {gather*} \frac {2 \sqrt {c-\frac {c}{a x}} \left (15+39 a x+52 a^2 x^2+104 a^3 x^3\right )}{105 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.13, size = 211, normalized size = 2.20
method | result | size |
gosper | \(\frac {2 \left (104 a^{3} x^{3}+52 a^{2} x^{2}+39 a x +15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{105 x^{3}}\) | \(43\) |
trager | \(\frac {2 \left (104 a^{3} x^{3}+52 a^{2} x^{2}+39 a x +15\right ) \sqrt {-\frac {-a c x +c}{a x}}}{105 x^{3}}\) | \(45\) |
risch | \(\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (104 a^{4} x^{4}-52 a^{3} x^{3}-13 a^{2} x^{2}-24 a x -15\right )}{105 \left (a x -1\right ) x^{3}}\) | \(58\) |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (210 \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}} x^{5}+210 \sqrt {a \,x^{2}-x}\, a^{\frac {9}{2}} x^{5}-420 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {7}{2}} x^{3}-105 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{4} x^{5}+105 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{4} x^{5}-212 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}-108 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x -30 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{105 x^{4} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}}\) | \(211\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 44, normalized size = 0.46 \begin {gather*} \frac {2 \, {\left (104 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 39 \, a x + 15\right )} \sqrt {\frac {a c x - c}{a x}}}{105 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x^{4} \left (a x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 77, normalized size = 0.80 \begin {gather*} \frac {208\,a^3\,\sqrt {c-\frac {c}{a\,x}}}{105}+\frac {2\,\sqrt {c-\frac {c}{a\,x}}}{7\,x^3}+\frac {26\,a\,\sqrt {c-\frac {c}{a\,x}}}{35\,x^2}+\frac {104\,a^2\,\sqrt {c-\frac {c}{a\,x}}}{105\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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