Optimal. Leaf size=128 \[ -\frac {2 x \left (1-a^2 x^2\right )^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}{(1-a x)^3}+\frac {\sqrt {a} x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{3/2}}+\frac {a x^2 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{3/2}}{(1-a x)^{3/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6134, 6128, 879, 848, 50, 54, 215} \[ -\frac {2 x \left (1-a^2 x^2\right )^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}{(1-a x)^3}+\frac {a x^2 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{3/2}}{(1-a x)^{3/2}}+\frac {\sqrt {a} x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rule 848
Rule 879
Rule 6128
Rule 6134
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx &=\frac {\left (\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {e^{\tanh ^{-1}(a x)} (1-a x)^{3/2}}{x^{3/2}} \, dx}{(1-a x)^{3/2}}\\ &=\frac {\left (\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {\sqrt {1-a x} \sqrt {1-a^2 x^2}}{x^{3/2}} \, dx}{(1-a x)^{3/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^3}+\frac {\left (a \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1-a x}} \, dx}{(1-a x)^{3/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^3}+\frac {\left (a \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {\sqrt {1+a x}}{\sqrt {x}} \, dx}{(1-a x)^{3/2}}\\ &=\frac {a \left (c-\frac {c}{a x}\right )^{3/2} x^2 \sqrt {1+a x}}{(1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^3}+\frac {\left (a \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 (1-a x)^{3/2}}\\ &=\frac {a \left (c-\frac {c}{a x}\right )^{3/2} x^2 \sqrt {1+a x}}{(1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^3}+\frac {\left (a \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{(1-a x)^{3/2}}\\ &=\frac {a \left (c-\frac {c}{a x}\right )^{3/2} x^2 \sqrt {1+a x}}{(1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^3}+\frac {\sqrt {a} \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 70, normalized size = 0.55 \[ \frac {c \sqrt {c-\frac {c}{a x}} \left (\sqrt {a x+1} (a x+2)-\sqrt {a} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right )}{a \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.60, size = 266, normalized size = 2.08 \[ \left [\frac {{\left (a c x - c\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (a c x + 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, \frac {{\left (a c x - c\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (a c x + 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \]
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 = 0.05, size = 108, normalized size = 0.84 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c \sqrt {-a^{2} x^{2}+1}\, \left (2 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}+\arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) x a +4 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}\right )}{2 a^{\frac {3}{2}} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}} \]
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 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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