Optimal. Leaf size=172 \[ \frac {32 a^3 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{315 x \sqrt {1-a x}}-\frac {16 a^2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{105 x^2 \sqrt {1-a x}}-\frac {2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{9 x^4 \sqrt {1-a x}}+\frac {4 a (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{21 x^3 \sqrt {1-a x}} \]
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Rubi [A] time = 0.23, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6134, 6128, 848, 45, 37} \[ -\frac {16 a^2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{105 x^2 \sqrt {1-a x}}+\frac {32 a^3 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{315 x \sqrt {1-a x}}+\frac {4 a (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{21 x^3 \sqrt {1-a x}}-\frac {2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{9 x^4 \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 848
Rule 6128
Rule 6134
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {e^{\tanh ^{-1}(a x)} \sqrt {1-a x}}{x^{11/2}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1-a^2 x^2}}{x^{11/2} \sqrt {1-a x}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1+a x}}{x^{11/2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{9 x^4 \sqrt {1-a x}}-\frac {\left (2 a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1+a x}}{x^{9/2}} \, dx}{3 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{9 x^4 \sqrt {1-a x}}+\frac {4 a \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{21 x^3 \sqrt {1-a x}}+\frac {\left (8 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1+a x}}{x^{7/2}} \, dx}{21 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{9 x^4 \sqrt {1-a x}}+\frac {4 a \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{21 x^3 \sqrt {1-a x}}-\frac {16 a^2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{105 x^2 \sqrt {1-a x}}-\frac {\left (16 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1+a x}}{x^{5/2}} \, dx}{105 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{9 x^4 \sqrt {1-a x}}+\frac {4 a \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{21 x^3 \sqrt {1-a x}}-\frac {16 a^2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{105 x^2 \sqrt {1-a x}}+\frac {32 a^3 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{315 x \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 63, normalized size = 0.37 \[ \frac {2 (a x+1)^{3/2} \left (16 a^3 x^3-24 a^2 x^2+30 a x-35\right ) \sqrt {c-\frac {c}{a x}}}{315 x^4 \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 74, normalized size = 0.43 \[ -\frac {2 \, {\left (16 \, a^{4} x^{4} - 8 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 5 \, a x - 35\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{315 \, {\left (a x^{5} - x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}}}{\sqrt {-a^{2} x^{2} + 1} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 62, normalized size = 0.36 \[ \frac {2 \left (a x +1\right )^{2} \left (16 x^{3} a^{3}-24 a^{2} x^{2}+30 a x -35\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{315 x^{4} \sqrt {-a^{2} x^{2}+1}} \]
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 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}}}{\sqrt {-a^{2} x^{2} + 1} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 68, normalized size = 0.40 \[ -\frac {\sqrt {c-\frac {c}{a\,x}}\,\left (-\frac {32\,a^5\,x^5}{315}-\frac {16\,a^4\,x^4}{315}+\frac {4\,a^3\,x^3}{315}-\frac {2\,a^2\,x^2}{315}+\frac {16\,a\,x}{63}+\frac {2}{9}\right )}{x^4\,\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x^{5} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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