Optimal. Leaf size=292 \[ -\frac {107 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{64 a^3 \sqrt {1-a x}}-\frac {21 \sqrt {c-\frac {c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt {1-a x}}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}-\frac {363 \sqrt {c-\frac {c}{a x}} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{64 a^{7/2} \sqrt {1-a x}}+\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{a^{7/2} \sqrt {1-a x}} \]
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Rubi [A]
time = 0.21, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6269, 6264,
103, 159, 163, 56, 221, 95, 212} \begin {gather*} -\frac {363 \sqrt {x} \sqrt {c-\frac {c}{a x}} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{64 a^{7/2} \sqrt {1-a x}}+\frac {4 \sqrt {2} \sqrt {x} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{a^{7/2} \sqrt {1-a x}}-\frac {21 x (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{32 a^3 \sqrt {1-a x}}-\frac {107 x \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{64 a^3 \sqrt {1-a x}}-\frac {11 x^2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{24 a^2 \sqrt {1-a x}}-\frac {x^3 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{4 a \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 95
Rule 103
Rule 159
Rule 163
Rule 212
Rule 221
Rule 6264
Rule 6269
Rubi steps
\begin {align*} \int 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 e^{3 \tanh ^{-1}(a x)} x^{5/2} \sqrt {1-a x} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{5/2} (1+a x)^{3/2}}{1-a x} \, dx}{\sqrt {1-a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}+\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{3/2} \sqrt {1+a x} \left (\frac {5}{2}+\frac {11 a x}{2}\right )}{1-a x} \, dx}{4 a \sqrt {1-a x}}\\ &=-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}-\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {x} \sqrt {1+a x} \left (-\frac {33 a}{4}-\frac {63 a^2 x}{4}\right )}{1-a x} \, dx}{12 a^3 \sqrt {1-a x}}\\ &=-\frac {21 \sqrt {c-\frac {c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt {1-a x}}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}+\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1+a x} \left (\frac {63 a^2}{8}+\frac {321 a^3 x}{8}\right )}{\sqrt {x} (1-a x)} \, dx}{24 a^5 \sqrt {1-a x}}\\ &=-\frac {107 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{64 a^3 \sqrt {1-a x}}-\frac {21 \sqrt {c-\frac {c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt {1-a x}}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}-\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {-\frac {447 a^3}{16}-\frac {1089 a^4 x}{16}}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{24 a^6 \sqrt {1-a x}}\\ &=-\frac {107 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{64 a^3 \sqrt {1-a x}}-\frac {21 \sqrt {c-\frac {c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt {1-a x}}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}-\frac {\left (363 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{128 a^3 \sqrt {1-a x}}+\frac {\left (4 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{a^3 \sqrt {1-a x}}\\ &=-\frac {107 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{64 a^3 \sqrt {1-a x}}-\frac {21 \sqrt {c-\frac {c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt {1-a x}}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}-\frac {\left (363 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{64 a^3 \sqrt {1-a x}}+\frac {\left (8 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{1-2 a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+a x}}\right )}{a^3 \sqrt {1-a x}}\\ &=-\frac {107 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{64 a^3 \sqrt {1-a x}}-\frac {21 \sqrt {c-\frac {c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt {1-a x}}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}-\frac {363 \sqrt {c-\frac {c}{a x}} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{64 a^{7/2} \sqrt {1-a x}}+\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{a^{7/2} \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 130, normalized size = 0.45 \begin {gather*} -\frac {\sqrt {c-\frac {c}{a x}} \left (\sqrt {a} x \sqrt {1+a x} \left (447+214 a x+136 a^2 x^2+48 a^3 x^3\right )+1089 \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )-768 \sqrt {2} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )\right )}{192 a^{7/2} \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.96, size = 247, normalized size = 0.85
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (96 a^{\frac {9}{2}} \sqrt {-\left (a x +1\right ) x}\, \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{3}+272 a^{\frac {7}{2}} \sqrt {-\left (a x +1\right ) x}\, \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2}+428 a^{\frac {5}{2}} \sqrt {-\left (a x +1\right ) x}\, \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x +894 \sqrt {-\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}-1089 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}+1536 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {2}}{768 a^{\frac {9}{2}} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}\, \sqrt {-\frac {1}{a}}}\) | \(247\) |
risch | \(-\frac {\left (48 a^{3} x^{3}+136 a^{2} x^{2}+214 a x +447\right ) x \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{192 a^{3} \sqrt {-a c x \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}}-\frac {\left (\frac {363 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-c x a}}\right )}{128 a^{3} \sqrt {a^{2} c}}-\frac {4 \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right )}{a^{4} \sqrt {-2 c}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a^{2} x^{2}+1}}\) | \(270\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 500, normalized size = 1.71 \begin {gather*} \left [\frac {768 \, \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^{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 ) + 1089 \, {\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 (48 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} + 447 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{768 \, {\left (a^{5} x - a^{4}\right )}}, -\frac {768 \, \sqrt {2} {\left (a x - 1\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 ) - 1089 \, {\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 (48 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} + 447 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{384 \, {\left (a^{5} x - a^{4}\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {x^{3} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,\sqrt {c-\frac {c}{a\,x}}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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