Optimal. Leaf size=184 \[ \frac {\sqrt {1-a^2 x^2}}{8 a^4 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 a^4 c^2 (1-a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^4 c^2 (1+a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^4 c^2 \sqrt {c-a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6288, 6285, 90,
213} \begin {gather*} -\frac {\sqrt {1-a^2 x^2}}{2 a^4 c^2 (1-a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^4 c^2 (a x+1) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^4 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^4 c^2 \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 90
Rule 213
Rule 6285
Rule 6288
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {x^3}{(1-a x)^3 (1+a x)^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (-\frac {1}{4 a^3 (-1+a x)^3}-\frac {1}{2 a^3 (-1+a x)^2}-\frac {1}{8 a^3 (1+a x)^2}-\frac {3}{8 a^3 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2}}{8 a^4 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 a^4 c^2 (1-a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^4 c^2 (1+a x) \sqrt {c-a^2 c x^2}}-\frac {\left (3 \sqrt {1-a^2 x^2}\right ) \int \frac {1}{-1+a^2 x^2} \, dx}{8 a^3 c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2}}{8 a^4 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 a^4 c^2 (1-a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^4 c^2 (1+a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^4 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 85, normalized size = 0.46 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (-2-a x+5 a^2 x^2+3 (-1+a x)^2 (1+a x) \tanh ^{-1}(a x)\right )}{8 a^4 c^2 (-1+a x)^2 (1+a x) \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 166, normalized size = 0.90
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (3 \ln \left (a x +1\right ) a^{3} x^{3}-3 \ln \left (a x -1\right ) a^{3} x^{3}-3 \ln \left (a x +1\right ) a^{2} x^{2}+3 \ln \left (a x -1\right ) a^{2} x^{2}+10 a^{2} x^{2}-3 \ln \left (a x +1\right ) a x +3 \ln \left (a x -1\right ) a x -2 a x +3 \ln \left (a x +1\right )-3 \ln \left (a x -1\right )-4\right )}{16 \left (a^{2} x^{2}-1\right ) c^{3} a^{4} \left (a x -1\right )^{2} \left (a x +1\right )}\) | \(166\) |
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 = 0.44, size = 461, normalized size = 2.51 \begin {gather*} \left [\frac {3 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left (a^{3} x^{3} + a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right ) - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - 3 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{32 \, {\left (a^{9} c^{3} x^{5} - a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{6} c^{3} x^{2} + a^{5} c^{3} x - a^{4} c^{3}\right )}}, \frac {3 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} a \sqrt {-c} x}{a^{4} c x^{4} - c}\right ) - 2 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - 3 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{16 \, {\left (a^{9} c^{3} x^{5} - a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{6} c^{3} x^{2} + a^{5} c^{3} x - a^{4} c^{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 {x^{3} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,\left (a\,x+1\right )}{{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________