Optimal. Leaf size=249 \[ -\frac {9 (1-a x)^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{5/2} x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {51 (1-a x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{4 \sqrt {2} a^{5/2} x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {21 \sqrt {a x+1} (1-a x)^{3/2}}{8 a^2 x \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {9 (a x+1)^{3/2} \sqrt {1-a x}}{8 a^2 x \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {(a x+1)^{3/2}}{2 a \sqrt {1-a x} \left (c-\frac {c}{a x}\right )^{3/2}} \]
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Rubi [A] time = 0.21, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6134, 6129, 97, 149, 154, 157, 54, 215, 93, 206} \[ -\frac {9 (1-a x)^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{5/2} x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {51 (1-a x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{4 \sqrt {2} a^{5/2} x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {21 \sqrt {a x+1} (1-a x)^{3/2}}{8 a^2 x \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {9 (a x+1)^{3/2} \sqrt {1-a x}}{8 a^2 x \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {(a x+1)^{3/2}}{2 a \sqrt {1-a x} \left (c-\frac {c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 97
Rule 149
Rule 154
Rule 157
Rule 206
Rule 215
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx &=\frac {(1-a x)^{3/2} \int \frac {e^{3 \tanh ^{-1}(a x)} x^{3/2}}{(1-a x)^{3/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac {(1-a x)^{3/2} \int \frac {x^{3/2} (1+a x)^{3/2}}{(1-a x)^3} \, dx}{\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac {(1+a x)^{3/2}}{2 a \left (c-\frac {c}{a x}\right )^{3/2} \sqrt {1-a x}}-\frac {(1-a x)^{3/2} \int \frac {\sqrt {x} \sqrt {1+a x} \left (\frac {3}{2}+3 a x\right )}{(1-a x)^2} \, dx}{2 a \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=\frac {(1+a x)^{3/2}}{2 a \left (c-\frac {c}{a x}\right )^{3/2} \sqrt {1-a x}}-\frac {9 \sqrt {1-a x} (1+a x)^{3/2}}{8 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}-\frac {(1-a x)^{3/2} \int \frac {\sqrt {1+a x} \left (-\frac {9 a}{4}-\frac {21 a^2 x}{2}\right )}{\sqrt {x} (1-a x)} \, dx}{4 a^3 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac {21 (1-a x)^{3/2} \sqrt {1+a x}}{8 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}+\frac {(1+a x)^{3/2}}{2 a \left (c-\frac {c}{a x}\right )^{3/2} \sqrt {1-a x}}-\frac {9 \sqrt {1-a x} (1+a x)^{3/2}}{8 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}+\frac {(1-a x)^{3/2} \int \frac {\frac {15 a^2}{2}+18 a^3 x}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{4 a^4 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac {21 (1-a x)^{3/2} \sqrt {1+a x}}{8 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}+\frac {(1+a x)^{3/2}}{2 a \left (c-\frac {c}{a x}\right )^{3/2} \sqrt {1-a x}}-\frac {9 \sqrt {1-a x} (1+a x)^{3/2}}{8 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}-\frac {\left (9 (1-a x)^{3/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}+\frac {\left (51 (1-a x)^{3/2}\right ) \int \frac {1}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{8 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac {21 (1-a x)^{3/2} \sqrt {1+a x}}{8 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}+\frac {(1+a x)^{3/2}}{2 a \left (c-\frac {c}{a x}\right )^{3/2} \sqrt {1-a x}}-\frac {9 \sqrt {1-a x} (1+a x)^{3/2}}{8 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}-\frac {\left (9 (1-a x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{a^2 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}+\frac {\left (51 (1-a x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+a x}}\right )}{4 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac {21 (1-a x)^{3/2} \sqrt {1+a x}}{8 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}+\frac {(1+a x)^{3/2}}{2 a \left (c-\frac {c}{a x}\right )^{3/2} \sqrt {1-a x}}-\frac {9 \sqrt {1-a x} (1+a x)^{3/2}}{8 a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}-\frac {9 (1-a x)^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{5/2} \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}+\frac {51 (1-a x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{4 \sqrt {2} a^{5/2} \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 139, normalized size = 0.56 \[ -\frac {-2 \sqrt {a} \sqrt {x} \sqrt {a x+1} \left (4 a^2 x^2-23 a x+15\right )-72 (a x-1)^2 \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )+51 \sqrt {2} (a x-1)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{8 a^{3/2} c \sqrt {x} (1-a x)^{3/2} \sqrt {c-\frac {c}{a x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 600, normalized size = 2.41 \[ \left [-\frac {51 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 72 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 8 \, {\left (4 \, a^{3} x^{3} - 23 \, a^{2} x^{2} + 15 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{32 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}}, -\frac {51 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 72 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 4 \, {\left (4 \, a^{3} x^{3} - 23 \, a^{2} x^{2} + 15 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{16 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}}\right ] \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 390, normalized size = 1.57 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (8 a^{\frac {7}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, x^{2}-36 a^{3} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2}-46 a^{\frac {5}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, x +51 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x^{2}+72 a^{2} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x +30 \sqrt {-\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}-102 a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x -36 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}+51 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {2}}{16 a^{\frac {3}{2}} c^{2} \left (a x -1\right )^{3} \sqrt {-\left (a x +1\right ) x}\, \sqrt {-\frac {1}{a}}} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a\,x+1\right )}^3}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a x + 1\right )^{3}}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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