Optimal. Leaf size=96 \[ \frac{2 a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}-\frac{8 a^3 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}+\frac{10 a^3 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-4 a^3 \sqrt{c-\frac{c}{a x}} \]
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Rubi [A] time = 0.224457, antiderivative size = 96, normalized size of antiderivative = 1., 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 = {6133, 25, 514, 446, 77} \[ \frac{2 a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}-\frac{8 a^3 \left (c-\frac{c}{a x}\right )^{5/2}}{5 c^2}+\frac{10 a^3 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-4 a^3 \sqrt{c-\frac{c}{a x}} \]
Antiderivative was successfully verified.
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Rule 6133
Rule 25
Rule 514
Rule 446
Rule 77
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 \operatorname{Subst}\left (\int \frac{x^2 (a+x)}{\sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{c \operatorname{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*}
Mathematica [A] time = 0.0491571, size = 44, normalized size = 0.46 \[ -\frac{2 \left (104 a^3 x^3+52 a^2 x^2+39 a x+15\right ) \sqrt{c-\frac{c}{a x}}}{105 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 43, normalized size = 0.5 \begin{align*} -{\frac{208\,{x}^{3}{a}^{3}+104\,{a}^{2}{x}^{2}+78\,ax+30}{105\,{x}^{3}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (a x + 1\right )}^{2} \sqrt{c - \frac{c}{a x}}}{{\left (a^{2} x^{2} - 1\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79331, size = 104, normalized size = 1.08 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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