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 = 0.16, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {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
Rubi steps
\begin {align*} \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 = 0.06, 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 = 0.81, size = 211, normalized size = 2.20
method | result | size |
gosper | \(-\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (104 a^{3} x^{3}+52 a^{2} x^{2}+39 a x +15\right )}{105 x^{3}}\) | \(43\) |
trager | \(-\frac {2 \left (104 a^{3} x^{3}+52 a^{2} x^{2}+39 a x +15\right ) \sqrt {-\frac {-c x a +c}{a x}}}{105 x^{3}}\) | \(45\) |
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 | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (210 \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}} x^{5}+210 a^{\frac {9}{2}} \sqrt {a \,x^{2}-x}\, x^{5}-420 a^{\frac {7}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{3}-105 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{4} x^{5}+105 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) a^{4} x^{5}-212 a^{\frac {5}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x^{2}-108 a^{\frac {3}{2}} \left (a \,x^{2}-x \right )^{\frac {3}{2}} x -30 \left (a \,x^{2}-x \right )^{\frac {3}{2}} \sqrt {a}\right )}{105 x^{4} \sqrt {\left (a x -1\right ) x}\, \sqrt {a}}\) | \(211\) |
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 = 0.35, 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 - \frac {c}{a x}}}{a x^{5} - x^{4}}\, dx - \int \frac {a x \sqrt {c - \frac {c}{a x}}}{a x^{5} - x^{4}}\, 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 = 0.91, 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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