Optimal. Leaf size=213 \[ \frac {2672 a^3 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{105 \sqrt {1-a x}}-\frac {1336 a^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{105 x \sqrt {1-a x}}+\frac {334 a^2 \sqrt {c-\frac {c}{a x}}}{35 x \sqrt {1-a x} \sqrt {a x+1}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{7 x^3 \sqrt {1-a x} \sqrt {a x+1}}+\frac {44 a \sqrt {c-\frac {c}{a x}}}{35 x^2 \sqrt {1-a x} \sqrt {a x+1}} \]
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Rubi [A] time = 0.24, antiderivative size = 213, 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 = {6134, 6129, 89, 78, 45, 37} \[ \frac {2672 a^3 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{105 \sqrt {1-a x}}-\frac {1336 a^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{105 x \sqrt {1-a x}}+\frac {334 a^2 \sqrt {c-\frac {c}{a x}}}{35 x \sqrt {1-a x} \sqrt {a x+1}}+\frac {44 a \sqrt {c-\frac {c}{a x}}}{35 x^2 \sqrt {1-a x} \sqrt {a x+1}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{7 x^3 \sqrt {1-a x} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rule 89
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^4} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {e^{-3 \tanh ^{-1}(a x)} \sqrt {1-a x}}{x^{9/2}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {(1-a x)^2}{x^{9/2} (1+a x)^{3/2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{7 x^3 \sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {-11 a+\frac {7 a^2 x}{2}}{x^{7/2} (1+a x)^{3/2}} \, dx}{7 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{7 x^3 \sqrt {1-a x} \sqrt {1+a x}}+\frac {44 a \sqrt {c-\frac {c}{a x}}}{35 x^2 \sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (167 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{x^{5/2} (1+a x)^{3/2}} \, dx}{35 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{7 x^3 \sqrt {1-a x} \sqrt {1+a x}}+\frac {44 a \sqrt {c-\frac {c}{a x}}}{35 x^2 \sqrt {1-a x} \sqrt {1+a x}}+\frac {334 a^2 \sqrt {c-\frac {c}{a x}}}{35 x \sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (668 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{x^{5/2} \sqrt {1+a x}} \, dx}{35 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{7 x^3 \sqrt {1-a x} \sqrt {1+a x}}+\frac {44 a \sqrt {c-\frac {c}{a x}}}{35 x^2 \sqrt {1-a x} \sqrt {1+a x}}+\frac {334 a^2 \sqrt {c-\frac {c}{a x}}}{35 x \sqrt {1-a x} \sqrt {1+a x}}-\frac {1336 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{105 x \sqrt {1-a x}}-\frac {\left (1336 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{x^{3/2} \sqrt {1+a x}} \, dx}{105 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{7 x^3 \sqrt {1-a x} \sqrt {1+a x}}+\frac {44 a \sqrt {c-\frac {c}{a x}}}{35 x^2 \sqrt {1-a x} \sqrt {1+a x}}+\frac {334 a^2 \sqrt {c-\frac {c}{a x}}}{35 x \sqrt {1-a x} \sqrt {1+a x}}+\frac {2672 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{105 \sqrt {1-a x}}-\frac {1336 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{105 x \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 0.31 \[ \frac {2 \left (1336 a^4 x^4+668 a^3 x^3-167 a^2 x^2+66 a x-15\right ) \sqrt {c-\frac {c}{a x}}}{105 x^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 76, normalized size = 0.36 \[ -\frac {2 \, {\left (1336 \, a^{4} x^{4} + 668 \, a^{3} x^{3} - 167 \, a^{2} x^{2} + 66 \, a x - 15\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{105 \, {\left (a^{2} x^{5} - x^{3}\right )}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 77, normalized size = 0.36 \[ \frac {2 \left (1336 x^{4} a^{4}+668 x^{3} a^{3}-167 a^{2} x^{2}+66 a x -15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{105 \left (a x +1\right )^{2} x^{3} \left (a x -1\right )^{2}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )}^{3} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 120, normalized size = 0.56 \[ -\frac {\sqrt {c-\frac {c}{a\,x}}\,\left (\frac {44\,x\,\sqrt {1-a^2\,x^2}}{35\,a}-\frac {334\,x^2\,\sqrt {1-a^2\,x^2}}{105}-\frac {2\,\sqrt {1-a^2\,x^2}}{7\,a^2}+\frac {1336\,a\,x^3\,\sqrt {1-a^2\,x^2}}{105}+\frac {2672\,a^2\,x^4\,\sqrt {1-a^2\,x^2}}{105}\right )}{x^5-\frac {x^3}{a^2}} \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{4} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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