Optimal. Leaf size=261 \[ \frac {19 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{8 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {13 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{12 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{3 \sqrt {1-\frac {1}{a x}}}+\frac {45 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{8 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a^3 \sqrt {1-\frac {1}{a x}}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.22, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6317, 6315,
100, 156, 162, 65, 214, 212} \begin {gather*} \frac {45 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{8 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{a^3 \sqrt {1-\frac {1}{a x}}}+\frac {19 x \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{8 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {x^3 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-\frac {1}{a x}}}+\frac {13 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{12 a \sqrt {1-\frac {1}{a x}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 100
Rule 156
Rule 162
Rule 212
Rule 214
Rule 6315
Rule 6317
Rubi steps
\begin {align*} \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx &=\frac {\sqrt {c-\frac {c}{a x}} \int e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^2 \, dx}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^4 \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{3 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {-\frac {13}{2 a}-\frac {11 x}{2 a^2}}{x^3 \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {13 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{12 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{3 \sqrt {1-\frac {1}{a x}}}-\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {\frac {57}{4 a^2}+\frac {39 x}{4 a^3}}{x^2 \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{6 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {19 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{8 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {13 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{12 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{3 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} \text {Subst}\left (\int \frac {-\frac {135}{8 a^3}-\frac {57 x}{8 a^4}}{x \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{6 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {19 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{8 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {13 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{12 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{3 \sqrt {1-\frac {1}{a x}}}-\frac {\left (4 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}-\frac {\left (45 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{16 a^3 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {19 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{8 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {13 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{12 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{3 \sqrt {1-\frac {1}{a x}}}-\frac {\left (8 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{a^3 \sqrt {1-\frac {1}{a x}}}-\frac {\left (45 \sqrt {c-\frac {c}{a x}}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{8 a^2 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {19 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{8 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {13 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{12 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{3 \sqrt {1-\frac {1}{a x}}}+\frac {45 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{8 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a^3 \sqrt {1-\frac {1}{a x}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.39, size = 244, normalized size = 0.93 \begin {gather*} \frac {\frac {2 a^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2 \left (57+26 a x+8 a^2 x^2\right )}{-1+a x}-135 \sqrt {c} \log (1-a x)+96 \sqrt {2} \sqrt {c} \log \left ((-1+a x)^2\right )+135 \sqrt {c} \log \left (2 a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-a x+2 a^2 x^2\right )\right )-96 \sqrt {2} \sqrt {c} \log \left (2 \sqrt {2} a^2 \sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}} x^2+c \left (-1-2 a x+3 a^2 x^2\right )\right )}{48 a^3} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.14, size = 202, normalized size = 0.77
method | result | size |
default | \(\frac {\left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 \sqrt {x \left (a x +1\right )}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x^{2}+52 \sqrt {x \left (a x +1\right )}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, x +114 \sqrt {x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}-96 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {x \left (a x +1\right )}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}+135 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}\right )}{48 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) a^{\frac {7}{2}} \sqrt {x \left (a x +1\right )}\, \sqrt {\frac {1}{a}}}\) | \(202\) |
risch | \(\frac {\left (8 a^{2} x^{2}+26 a x +57\right ) x \sqrt {\frac {c \left (a x -1\right )}{a x}}}{24 a^{2} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {45 \ln \left (\frac {\frac {1}{2} a c +c \,a^{2} x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right )}{16 a^{2} \sqrt {a^{2} c}}-\frac {2 \sqrt {2}\, \ln \left (\frac {4 c +3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {2}\, \sqrt {c}\, \sqrt {a^{2} c \left (x -\frac {1}{a}\right )^{2}+3 \left (x -\frac {1}{a}\right ) a c +2 c}}{x -\frac {1}{a}}\right )}{a^{3} \sqrt {c}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c a x \left (a x +1\right )}}{\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(226\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.46, size = 552, normalized size = 2.11 \begin {gather*} \left [\frac {96 \, \sqrt {2} {\left (a x - 1\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^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 135 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (8 \, a^{4} x^{4} + 34 \, a^{3} x^{3} + 83 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{96 \, {\left (a^{4} x - a^{3}\right )}}, \frac {96 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 135 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (8 \, a^{4} x^{4} + 34 \, a^{3} x^{3} + 83 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{48 \, {\left (a^{4} x - a^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \sqrt {- c \left (-1 + \frac {1}{a x}\right )}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,\sqrt {c-\frac {c}{a\,x}}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________