Optimal. Leaf size=171 \[ -\frac {3 a^{3/2} x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{5/2}}-\frac {3 a^2 x^3 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{5/2}}{(1-a x)^{5/2}}+\frac {4 a x^2 (a x+1)^{3/2} \left (c-\frac {c}{a x}\right )^{5/2}}{(1-a x)^{5/2}}-\frac {2 x (a x+1)^{3/2} \left (c-\frac {c}{a x}\right )^{5/2}}{3 (1-a x)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6134, 6129, 89, 78, 50, 54, 215} \[ -\frac {3 a^2 x^3 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{5/2}}{(1-a x)^{5/2}}-\frac {3 a^{3/2} x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{5/2}}+\frac {4 a x^2 (a x+1)^{3/2} \left (c-\frac {c}{a x}\right )^{5/2}}{(1-a x)^{5/2}}-\frac {2 x (a x+1)^{3/2} \left (c-\frac {c}{a x}\right )^{5/2}}{3 (1-a x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 54
Rule 78
Rule 89
Rule 215
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx &=\frac {\left (\left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac {e^{\tanh ^{-1}(a x)} (1-a x)^{5/2}}{x^{5/2}} \, dx}{(1-a x)^{5/2}}\\ &=\frac {\left (\left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac {(1-a x)^2 \sqrt {1+a x}}{x^{5/2}} \, dx}{(1-a x)^{5/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{5/2} x (1+a x)^{3/2}}{3 (1-a x)^{5/2}}+\frac {\left (2 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac {\sqrt {1+a x} \left (-3 a+\frac {3 a^2 x}{2}\right )}{x^{3/2}} \, dx}{3 (1-a x)^{5/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{5/2} x (1+a x)^{3/2}}{3 (1-a x)^{5/2}}+\frac {4 a \left (c-\frac {c}{a x}\right )^{5/2} x^2 (1+a x)^{3/2}}{(1-a x)^{5/2}}-\frac {\left (3 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac {\sqrt {1+a x}}{\sqrt {x}} \, dx}{(1-a x)^{5/2}}\\ &=-\frac {3 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x^3 \sqrt {1+a x}}{(1-a x)^{5/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{5/2} x (1+a x)^{3/2}}{3 (1-a x)^{5/2}}+\frac {4 a \left (c-\frac {c}{a x}\right )^{5/2} x^2 (1+a x)^{3/2}}{(1-a x)^{5/2}}-\frac {\left (3 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 (1-a x)^{5/2}}\\ &=-\frac {3 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x^3 \sqrt {1+a x}}{(1-a x)^{5/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{5/2} x (1+a x)^{3/2}}{3 (1-a x)^{5/2}}+\frac {4 a \left (c-\frac {c}{a x}\right )^{5/2} x^2 (1+a x)^{3/2}}{(1-a x)^{5/2}}-\frac {\left (3 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{(1-a x)^{5/2}}\\ &=-\frac {3 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x^3 \sqrt {1+a x}}{(1-a x)^{5/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{5/2} x (1+a x)^{3/2}}{3 (1-a x)^{5/2}}+\frac {4 a \left (c-\frac {c}{a x}\right )^{5/2} x^2 (1+a x)^{3/2}}{(1-a x)^{5/2}}-\frac {3 a^{3/2} \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 87, normalized size = 0.51 \[ \frac {c^2 \sqrt {c-\frac {c}{a x}} \left (\sqrt {a x+1} \left (3 a^2 x^2+10 a x-2\right )-9 a^{3/2} x^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right )}{3 a^2 x \sqrt {1-a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.68, size = 330, normalized size = 1.93 \[ \left [\frac {9 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (3 \, a^{2} c^{2} x^{2} + 10 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{12 \, {\left (a^{3} x^{2} - a^{2} x\right )}}, \frac {9 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (3 \, a^{2} c^{2} x^{2} + 10 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{6 \, {\left (a^{3} x^{2} - a^{2} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 136, normalized size = 0.80 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{2} \sqrt {-a^{2} x^{2}+1}\, \left (6 a^{\frac {5}{2}} x^{2} \sqrt {-\left (a x +1\right ) x}+9 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) x^{2} a^{2}+20 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}-4 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}\right )}{6 x \,a^{\frac {5}{2}} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________