Optimal. Leaf size=191 \[ \frac {4 \sqrt {2} a^{5/2} \sqrt {x} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {1-a x}}-\frac {4 a^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}-\frac {2 (a x+1)^{5/2} \sqrt {c-\frac {c}{a x}}}{5 x^2 \sqrt {1-a x}}-\frac {2 a (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x}} \]
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Rubi [A] time = 0.28, antiderivative size = 191, 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, 96, 94, 93, 206} \[ -\frac {4 a^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}+\frac {4 \sqrt {2} a^{5/2} \sqrt {x} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {1-a x}}-\frac {2 (a x+1)^{5/2} \sqrt {c-\frac {c}{a x}}}{5 x^2 \sqrt {1-a x}}-\frac {2 a (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 206
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, 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^{7/2}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {(1+a x)^{3/2}}{x^{7/2} (1-a x)} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{5/2}}{5 x^2 \sqrt {1-a x}}+\frac {\left (a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {(1+a x)^{3/2}}{x^{5/2} (1-a x)} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 a \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{5/2}}{5 x^2 \sqrt {1-a x}}+\frac {\left (2 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1+a x}}{x^{3/2} (1-a x)} \, dx}{\sqrt {1-a x}}\\ &=-\frac {4 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {2 a \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{5/2}}{5 x^2 \sqrt {1-a x}}+\frac {\left (4 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {4 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {2 a \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{5/2}}{5 x^2 \sqrt {1-a x}}+\frac {\left (8 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+a x}}\right )}{\sqrt {1-a x}}\\ &=-\frac {4 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {2 a \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{3 x \sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{5/2}}{5 x^2 \sqrt {1-a x}}+\frac {4 \sqrt {2} a^{5/2} \sqrt {c-\frac {c}{a x}} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{\sqrt {1-a x}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 101, normalized size = 0.53 \[ \frac {2 \sqrt {c-\frac {c}{a x}} \left (30 \sqrt {2} a^{5/2} x^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )-\sqrt {a x+1} \left (38 a^2 x^2+11 a x+3\right )\right )}{15 x^2 \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 337, normalized size = 1.76 \[ \left [\frac {15 \, \sqrt {2} {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, {\left (38 \, a^{2} x^{2} + 11 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{15 \, {\left (a x^{3} - x^{2}\right )}}, -\frac {2 \, {\left (15 \, \sqrt {2} {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - {\left (38 \, a^{2} x^{2} + 11 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}\right )}}{15 \, {\left (a x^{3} - x^{2}\right )}}\right ] \]
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 = 0.06, size = 181, normalized size = 0.95 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, \left (38 a^{2} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2} \sqrt {-\left (a x +1\right ) x}+30 a^{2} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x^{3}+11 a \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x \sqrt {-\left (a x +1\right ) x}+3 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\right ) \sqrt {2}}{15 x^{2} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}\, \sqrt {-\frac {1}{a}}} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3} \sqrt {c - \frac {c}{a x}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c-\frac {c}{a\,x}}\,{\left (a\,x+1\right )}^3}{x^3\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}{x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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