Optimal. Leaf size=179 \[ \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{8 a^{5/2} \sqrt {1-a x}}-\frac {x \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{8 a^2 \sqrt {1-a x}}+\frac {x^3 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x}}+\frac {x^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{12 a \sqrt {1-a x}} \]
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Rubi [A] time = 0.25, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6134, 6128, 848, 50, 54, 215} \[ -\frac {x \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{8 a^2 \sqrt {1-a x}}+\frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{8 a^{5/2} \sqrt {1-a x}}+\frac {x^3 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x}}+\frac {x^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{12 a \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rule 848
Rule 6128
Rule 6134
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int e^{\tanh ^{-1}(a x)} x^{3/2} \sqrt {1-a x} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{3/2} \sqrt {1-a^2 x^2}}{\sqrt {1-a x}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int x^{3/2} \sqrt {1+a x} \, dx}{\sqrt {1-a x}}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x^3 \sqrt {1+a x}}{3 \sqrt {1-a x}}+\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{3/2}}{\sqrt {1+a x}} \, dx}{6 \sqrt {1-a x}}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{12 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^3 \sqrt {1+a x}}{3 \sqrt {1-a x}}-\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {x}}{\sqrt {1+a x}} \, dx}{8 a \sqrt {1-a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{8 a^2 \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{12 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^3 \sqrt {1+a x}}{3 \sqrt {1-a x}}+\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{16 a^2 \sqrt {1-a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{8 a^2 \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{12 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^3 \sqrt {1+a x}}{3 \sqrt {1-a x}}+\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{8 a^2 \sqrt {1-a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{8 a^2 \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{12 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^3 \sqrt {1+a x}}{3 \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{8 a^{5/2} \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 88, normalized size = 0.49 \[ \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\sqrt {a} \sqrt {x} \sqrt {a x+1} \left (8 a^2 x^2+2 a x-3\right )+3 \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right )}{24 a^{5/2} \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 292, normalized size = 1.63 \[ \left [\frac {3 \, {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (8 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{96 \, {\left (a^{4} x - a^{3}\right )}}, -\frac {3 \, {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (8 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{48 \, {\left (a^{4} x - a^{3}\right )}}\right ] \]
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 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}} x^{2}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 125, normalized size = 0.70 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (16 a^{\frac {5}{2}} x^{2} \sqrt {-\left (a x +1\right ) x}+4 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}-6 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}-3 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right )\right )}{48 a^{\frac {5}{2}} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}} \]
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 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}} x^{2}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^2\,\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{2} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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