Optimal. Leaf size=225 \[ -\frac {a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4 (54-227 a x) \sqrt {1+a x}}{105 (1-a x)^{9/2}}-\frac {10 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 \sqrt {1+a x}}{21 (1-a x)^{5/2}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 \sqrt {1+a x}}{5 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}-\frac {7 a^{7/2} \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6269, 6264, 99,
155, 148, 56, 221} \begin {gather*} -\frac {7 a^{7/2} x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}}-\frac {a^3 x^4 (54-227 a x) \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{9/2}}{105 (1-a x)^{9/2}}-\frac {10 a^2 x^3 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{9/2}}{21 (1-a x)^{5/2}}+\frac {2 a x^2 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{9/2}}{5 (1-a x)^{3/2}}-\frac {2 x \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{9/2}}{7 \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 56
Rule 99
Rule 148
Rule 155
Rule 221
Rule 6264
Rule 6269
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx &=\frac {\left (\left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {e^{\tanh ^{-1}(a x)} (1-a x)^{9/2}}{x^{9/2}} \, dx}{(1-a x)^{9/2}}\\ &=\frac {\left (\left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x)^4 \sqrt {1+a x}}{x^{9/2}} \, dx}{(1-a x)^{9/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}+\frac {\left (2 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x)^3 \left (-\frac {7 a}{2}-\frac {9 a^2 x}{2}\right )}{x^{7/2} \sqrt {1+a x}} \, dx}{7 (1-a x)^{9/2}}\\ &=\frac {2 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 \sqrt {1+a x}}{5 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}+\frac {\left (4 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x)^2 \left (\frac {25 a^2}{4}+\frac {59 a^3 x}{4}\right )}{x^{5/2} \sqrt {1+a x}} \, dx}{35 (1-a x)^{9/2}}\\ &=-\frac {10 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 \sqrt {1+a x}}{21 (1-a x)^{5/2}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 \sqrt {1+a x}}{5 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}+\frac {\left (8 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x) \left (\frac {27 a^3}{8}-\frac {227 a^4 x}{8}\right )}{x^{3/2} \sqrt {1+a x}} \, dx}{105 (1-a x)^{9/2}}\\ &=-\frac {a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4 (54-227 a x) \sqrt {1+a x}}{105 (1-a x)^{9/2}}-\frac {10 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 \sqrt {1+a x}}{21 (1-a x)^{5/2}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 \sqrt {1+a x}}{5 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}-\frac {\left (7 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 (1-a x)^{9/2}}\\ &=-\frac {a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4 (54-227 a x) \sqrt {1+a x}}{105 (1-a x)^{9/2}}-\frac {10 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 \sqrt {1+a x}}{21 (1-a x)^{5/2}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 \sqrt {1+a x}}{5 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}-\frac {\left (7 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{(1-a x)^{9/2}}\\ &=-\frac {a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4 (54-227 a x) \sqrt {1+a x}}{105 (1-a x)^{9/2}}-\frac {10 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 \sqrt {1+a x}}{21 (1-a x)^{5/2}}+\frac {2 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 \sqrt {1+a x}}{5 (1-a x)^{3/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x \sqrt {1+a x}}{7 \sqrt {1-a x}}-\frac {7 a^{7/2} \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.05, size = 101, normalized size = 0.45 \begin {gather*} \frac {c^4 \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+a x} \left (-30+162 a x-601 a^2 x^2-688 a^3 x^3+105 a^4 x^4\right )+245 a^2 x^2 \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-a x\right )\right )}{105 a^4 x^3 \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.71, size = 172, normalized size = 0.76
method | result | size |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \sqrt {-a^{2} x^{2}+1}\, \left (210 a^{\frac {9}{2}} \sqrt {-\left (a x +1\right ) x}\, x^{4}+735 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a^{4} x^{4}+584 a^{\frac {7}{2}} x^{3} \sqrt {-\left (a x +1\right ) x}-712 a^{\frac {5}{2}} x^{2} \sqrt {-\left (a x +1\right ) x}+324 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}-60 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}\right )}{210 x^{3} a^{\frac {9}{2}} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}}\) | \(172\) |
risch | \(\frac {\left (105 a^{5} x^{5}+397 a^{4} x^{4}-64 a^{3} x^{3}-194 a^{2} x^{2}+132 a x -30\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{105 x^{3} \sqrt {-a c x \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, a^{4}}-\frac {7 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-c x a}}\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{2 \sqrt {a^{2} c}\, \sqrt {-a^{2} x^{2}+1}}\) | \(208\) |
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 = 0.42, size = 386, normalized size = 1.72 \begin {gather*} \left [\frac {735 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\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 (105 \, a^{4} c^{4} x^{4} + 292 \, a^{3} c^{4} x^{3} - 356 \, a^{2} c^{4} x^{2} + 162 \, a c^{4} x - 30 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{420 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}, \frac {735 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\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 (105 \, a^{4} c^{4} x^{4} + 292 \, a^{3} c^{4} x^{3} - 356 \, a^{2} c^{4} x^{2} + 162 \, a c^{4} x - 30 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{210 \, {\left (a^{5} x^{4} - a^{4} x^{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 {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {9}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________