Optimal. Leaf size=93 \[ \frac {2 c^2 (a x+1)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}-\frac {c^2 (a x+1)^6 \sqrt {c-a^2 c x^2}}{6 a \sqrt {1-a^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6143, 6140, 43} \[ \frac {2 c^2 (a x+1)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}-\frac {c^2 (a x+1)^6 \sqrt {c-a^2 c x^2}}{6 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int e^{3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int (1-a x) (1+a x)^4 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int \left (2 (1+a x)^4-(1+a x)^5\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 c^2 (1+a x)^5 \sqrt {c-a^2 c x^2}}{5 a \sqrt {1-a^2 x^2}}-\frac {c^2 (1+a x)^6 \sqrt {c-a^2 c x^2}}{6 a \sqrt {1-a^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 52, normalized size = 0.56 \[ -\frac {c^2 (a x+1)^5 (5 a x-7) \sqrt {c-a^2 c x^2}}{30 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 98, normalized size = 1.05 \[ \frac {{\left (5 \, a^{5} c^{2} x^{6} + 18 \, a^{4} c^{2} x^{5} + 15 \, a^{3} c^{2} x^{4} - 20 \, a^{2} c^{2} x^{3} - 45 \, a c^{2} x^{2} - 30 \, c^{2} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{30 \, {\left (a^{2} x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} {\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 81, normalized size = 0.87 \[ \frac {x \left (5 x^{5} a^{5}+18 x^{4} a^{4}+15 x^{3} a^{3}-20 a^{2} x^{2}-45 a x -30\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{30 \left (a x -1\right ) \left (a x +1\right ) \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.39, size = 237, normalized size = 2.55 \[ -\frac {1}{3} \, a^{2} c^{\frac {5}{2}} x^{3} + \frac {1}{12} \, {\left (\frac {2 \, a^{4} c^{3} x^{8}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {5 \, a^{2} c^{3} x^{6}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {3 \, c^{3} x^{4}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}}\right )} a^{3} + c^{\frac {5}{2}} x - \frac {1}{5} \, {\left (3 \, a^{2} c^{\frac {5}{2}} x^{5} - 5 \, c^{\frac {5}{2}} x^{3}\right )} a^{2} + \frac {3}{4} \, {\left (\frac {a^{4} c^{3} x^{6}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {3 \, a^{2} c^{3} x^{4}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {2 \, c^{3}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.08, size = 85, normalized size = 0.91 \[ \frac {\sqrt {c-a^2\,c\,x^2}\,\left (-\frac {a^5\,c^2\,x^6}{6}-\frac {3\,a^4\,c^2\,x^5}{5}-\frac {a^3\,c^2\,x^4}{2}+\frac {2\,a^2\,c^2\,x^3}{3}+\frac {3\,a\,c^2\,x^2}{2}+c^2\,x\right )}{\sqrt {1-a^2\,x^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________