Optimal. Leaf size=252 \[ -\frac {5 (1-a x)^{7/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{9/2} x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {115 (1-a x)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{16 \sqrt {2} a^{9/2} x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {35 \sqrt {a x+1} (1-a x)^{7/2}}{16 a^4 x^3 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {15 \sqrt {a x+1} (1-a x)^{5/2}}{16 a^3 x^2 \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {\sqrt {a x+1} (1-a x)^{3/2}}{4 a^2 x \left (c-\frac {c}{a x}\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 252, 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, 98, 149, 154, 157, 54, 215, 93, 206} \[ -\frac {35 \sqrt {a x+1} (1-a x)^{7/2}}{16 a^4 x^3 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {15 \sqrt {a x+1} (1-a x)^{5/2}}{16 a^3 x^2 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {5 (1-a x)^{7/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{9/2} x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {115 (1-a x)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{16 \sqrt {2} a^{9/2} x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {\sqrt {a x+1} (1-a x)^{3/2}}{4 a^2 x \left (c-\frac {c}{a x}\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 54
Rule 93
Rule 98
Rule 149
Rule 154
Rule 157
Rule 206
Rule 215
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx &=\frac {(1-a x)^{7/2} \int \frac {e^{-\tanh ^{-1}(a x)} x^{7/2}}{(1-a x)^{7/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{7/2} \int \frac {x^{7/2}}{(1-a x)^3 \sqrt {1+a x}} \, dx}{\left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{3/2} \sqrt {1+a x}}{4 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x}-\frac {(1-a x)^{7/2} \int \frac {x^{3/2} \left (\frac {5}{2}+5 a x\right )}{(1-a x)^2 \sqrt {1+a x}} \, dx}{4 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{3/2} \sqrt {1+a x}}{4 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x}-\frac {15 (1-a x)^{5/2} \sqrt {1+a x}}{16 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^2}-\frac {(1-a x)^{7/2} \int \frac {\sqrt {x} \left (-\frac {45 a}{4}-\frac {35 a^2 x}{2}\right )}{(1-a x) \sqrt {1+a x}} \, dx}{8 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{3/2} \sqrt {1+a x}}{4 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x}-\frac {15 (1-a x)^{5/2} \sqrt {1+a x}}{16 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^2}-\frac {35 (1-a x)^{7/2} \sqrt {1+a x}}{16 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^3}+\frac {(1-a x)^{7/2} \int \frac {\frac {35 a^2}{4}+20 a^3 x}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{8 a^6 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{3/2} \sqrt {1+a x}}{4 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x}-\frac {15 (1-a x)^{5/2} \sqrt {1+a x}}{16 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^2}-\frac {35 (1-a x)^{7/2} \sqrt {1+a x}}{16 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^3}-\frac {\left (5 (1-a x)^{7/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}+\frac {\left (115 (1-a x)^{7/2}\right ) \int \frac {1}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{32 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{3/2} \sqrt {1+a x}}{4 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x}-\frac {15 (1-a x)^{5/2} \sqrt {1+a x}}{16 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^2}-\frac {35 (1-a x)^{7/2} \sqrt {1+a x}}{16 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^3}-\frac {\left (5 (1-a x)^{7/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}+\frac {\left (115 (1-a x)^{7/2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+a x}}\right )}{16 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{3/2} \sqrt {1+a x}}{4 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x}-\frac {15 (1-a x)^{5/2} \sqrt {1+a x}}{16 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^2}-\frac {35 (1-a x)^{7/2} \sqrt {1+a x}}{16 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^3}-\frac {5 (1-a x)^{7/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{9/2} \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}+\frac {115 (1-a x)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{16 \sqrt {2} a^{9/2} \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.23, size = 139, normalized size = 0.55 \[ \frac {2 \sqrt {a} \sqrt {x} \sqrt {a x+1} \left (16 a^2 x^2-55 a x+35\right )+160 (a x-1)^2 \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )-115 \sqrt {2} (a x-1)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{32 a^{3/2} c^3 \sqrt {x} (1-a x)^{3/2} \sqrt {c-\frac {c}{a x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 600, normalized size = 2.38 \[ \left [-\frac {115 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 160 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 8 \, {\left (16 \, a^{3} x^{3} - 55 \, a^{2} x^{2} + 35 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{128 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}, -\frac {115 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 160 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 4 \, {\left (16 \, a^{3} x^{3} - 55 \, a^{2} x^{2} + 35 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{64 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 390, normalized size = 1.55 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (32 a^{\frac {7}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, x^{2}-110 a^{\frac {5}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, x -80 a^{3} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2}+115 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x^{2}+70 \sqrt {-\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}+160 a^{2} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x -230 a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x -80 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}+115 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {2}}{64 a^{\frac {3}{2}} c^{4} \left (a x -1\right )^{3} \sqrt {-\left (a x +1\right ) x}\, \sqrt {-\frac {1}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {1-a^2\,x^2}}{{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,\left (a\,x+1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {7}{2}} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________