Optimal. Leaf size=201 \[ \frac {\sqrt [4]{1-a^2 x^2}}{a^4 c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}+\frac {2 \sqrt {1-a x} \sqrt [4]{1-a^2 x^2}}{a^4 c \sqrt [4]{c-a^2 c x^2}}-\frac {2 (1-a x)^{3/2} \sqrt [4]{1-a^2 x^2}}{3 a^4 c \sqrt [4]{c-a^2 c x^2}}+\frac {\sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{\sqrt {2} a^4 c \sqrt [4]{c-a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.32, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {6288, 6285,
89, 45, 797, 79, 65, 213} \begin {gather*} -\frac {2 \sqrt [4]{1-a^2 x^2} (1-a x)^{3/2}}{3 a^4 c \sqrt [4]{c-a^2 c x^2}}+\frac {2 \sqrt [4]{1-a^2 x^2} \sqrt {1-a x}}{a^4 c \sqrt [4]{c-a^2 c x^2}}+\frac {\sqrt [4]{1-a^2 x^2}}{a^4 c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}+\frac {\sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{\sqrt {2} a^4 c \sqrt [4]{c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 65
Rule 79
Rule 89
Rule 213
Rule 797
Rule 6285
Rule 6288
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^{5/4}} \, dx &=\frac {\sqrt [4]{1-a^2 x^2} \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)} x^3}{\left (1-a^2 x^2\right )^{5/4}} \, dx}{c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {\sqrt [4]{1-a^2 x^2} \int \frac {x^3}{(1-a x)^{3/2} (1+a x)} \, dx}{c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {\sqrt [4]{1-a^2 x^2} \int \left (-\frac {x}{a^2 \sqrt {1-a x}}-\frac {x}{a^2 \sqrt {1-a x} \left (-1+a^2 x^2\right )}\right ) \, dx}{c \sqrt [4]{c-a^2 c x^2}}\\ &=-\frac {\sqrt [4]{1-a^2 x^2} \int \frac {x}{\sqrt {1-a x}} \, dx}{a^2 c \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2} \int \frac {x}{\sqrt {1-a x} \left (-1+a^2 x^2\right )} \, dx}{a^2 c \sqrt [4]{c-a^2 c x^2}}\\ &=-\frac {\sqrt [4]{1-a^2 x^2} \int \frac {x}{(-1-a x) (1-a x)^{3/2}} \, dx}{a^2 c \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2} \int \left (\frac {1}{a \sqrt {1-a x}}-\frac {\sqrt {1-a x}}{a}\right ) \, dx}{a^2 c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {\sqrt [4]{1-a^2 x^2}}{a^4 c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}+\frac {2 \sqrt {1-a x} \sqrt [4]{1-a^2 x^2}}{a^4 c \sqrt [4]{c-a^2 c x^2}}-\frac {2 (1-a x)^{3/2} \sqrt [4]{1-a^2 x^2}}{3 a^4 c \sqrt [4]{c-a^2 c x^2}}+\frac {\sqrt [4]{1-a^2 x^2} \int \frac {1}{(-1-a x) \sqrt {1-a x}} \, dx}{2 a^3 c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {\sqrt [4]{1-a^2 x^2}}{a^4 c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}+\frac {2 \sqrt {1-a x} \sqrt [4]{1-a^2 x^2}}{a^4 c \sqrt [4]{c-a^2 c x^2}}-\frac {2 (1-a x)^{3/2} \sqrt [4]{1-a^2 x^2}}{3 a^4 c \sqrt [4]{c-a^2 c x^2}}-\frac {\sqrt [4]{1-a^2 x^2} \text {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {1-a x}\right )}{a^4 c \sqrt [4]{c-a^2 c x^2}}\\ &=\frac {\sqrt [4]{1-a^2 x^2}}{a^4 c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}}+\frac {2 \sqrt {1-a x} \sqrt [4]{1-a^2 x^2}}{a^4 c \sqrt [4]{c-a^2 c x^2}}-\frac {2 (1-a x)^{3/2} \sqrt [4]{1-a^2 x^2}}{3 a^4 c \sqrt [4]{c-a^2 c x^2}}+\frac {\sqrt [4]{1-a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{\sqrt {2} a^4 c \sqrt [4]{c-a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 84, normalized size = 0.42 \begin {gather*} -\frac {\sqrt [4]{1-a^2 x^2} \left (2 \left (-5+a x+a^2 x^2\right )+3 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2} (1-a x)\right )\right )}{3 a^4 c \sqrt {1-a x} \sqrt [4]{c-a^2 c x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, x^{3}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{4}}}\, dx\]
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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int \frac {x^3\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}}{{\left (c-a^2\,c\,x^2\right )}^{5/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________