Optimal. Leaf size=167 \[ \frac {x \left (a+b x^2\right )}{8 a^2 \left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}+\frac {9 x \left (a+b x^2\right )}{32 a^3 \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}}+\frac {19 \sqrt {a-b x^2} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a+b x^2}}\right )}{32 \sqrt {2} a^3 \sqrt {b} \sqrt {a^2-b^2 x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1152, 414, 527, 12, 377, 208} \[ \frac {9 x \left (a+b x^2\right )}{32 a^3 \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}}+\frac {x \left (a+b x^2\right )}{8 a^2 \left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}+\frac {19 \sqrt {a-b x^2} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a+b x^2}}\right )}{32 \sqrt {2} a^3 \sqrt {b} \sqrt {a^2-b^2 x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 208
Rule 377
Rule 414
Rule 527
Rule 1152
Rubi steps
\begin {align*} \int \frac {1}{\left (a-b x^2\right )^{5/2} \sqrt {a^2-b^2 x^4}} \, dx &=\frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {1}{\left (a-b x^2\right )^3 \sqrt {a+b x^2}} \, dx}{\sqrt {a^2-b^2 x^4}}\\ &=\frac {x \left (a+b x^2\right )}{8 a^2 \left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}+\frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {7 a b+2 b^2 x^2}{\left (a-b x^2\right )^2 \sqrt {a+b x^2}} \, dx}{8 a^2 b \sqrt {a^2-b^2 x^4}}\\ &=\frac {x \left (a+b x^2\right )}{8 a^2 \left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}+\frac {9 x \left (a+b x^2\right )}{32 a^3 \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}}+\frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {19 a^2 b^2}{\left (a-b x^2\right ) \sqrt {a+b x^2}} \, dx}{32 a^4 b^2 \sqrt {a^2-b^2 x^4}}\\ &=\frac {x \left (a+b x^2\right )}{8 a^2 \left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}+\frac {9 x \left (a+b x^2\right )}{32 a^3 \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}}+\frac {\left (19 \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {1}{\left (a-b x^2\right ) \sqrt {a+b x^2}} \, dx}{32 a^2 \sqrt {a^2-b^2 x^4}}\\ &=\frac {x \left (a+b x^2\right )}{8 a^2 \left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}+\frac {9 x \left (a+b x^2\right )}{32 a^3 \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}}+\frac {\left (19 \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{a-2 a b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{32 a^2 \sqrt {a^2-b^2 x^4}}\\ &=\frac {x \left (a+b x^2\right )}{8 a^2 \left (a-b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}}+\frac {9 x \left (a+b x^2\right )}{32 a^3 \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}}+\frac {19 \sqrt {a-b x^2} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a+b x^2}}\right )}{32 \sqrt {2} a^3 \sqrt {b} \sqrt {a^2-b^2 x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 122, normalized size = 0.73 \[ \frac {\sqrt {a^2-b^2 x^4} \left (2 \sqrt {b} x \left (13 a-9 b x^2\right ) \sqrt {a+b x^2}+19 \sqrt {2} \left (a-b x^2\right )^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a+b x^2}}\right )\right )}{64 a^3 \sqrt {b} \left (a-b x^2\right )^{5/2} \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.96, size = 376, normalized size = 2.25 \[ \left [\frac {19 \, \sqrt {2} {\left (b^{3} x^{6} - 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - a^{3}\right )} \sqrt {b} \log \left (-\frac {3 \, b^{2} x^{4} - 2 \, a b x^{2} - 2 \, \sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} \sqrt {b} x - a^{2}}{b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}\right ) + 4 \, \sqrt {-b^{2} x^{4} + a^{2}} {\left (9 \, b^{2} x^{3} - 13 \, a b x\right )} \sqrt {-b x^{2} + a}}{128 \, {\left (a^{3} b^{4} x^{6} - 3 \, a^{4} b^{3} x^{4} + 3 \, a^{5} b^{2} x^{2} - a^{6} b\right )}}, \frac {19 \, \sqrt {2} {\left (b^{3} x^{6} - 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - a^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} \sqrt {-b}}{2 \, {\left (b^{2} x^{3} - a b x\right )}}\right ) + 2 \, \sqrt {-b^{2} x^{4} + a^{2}} {\left (9 \, b^{2} x^{3} - 13 \, a b x\right )} \sqrt {-b x^{2} + a}}{64 \, {\left (a^{3} b^{4} x^{6} - 3 \, a^{4} b^{3} x^{4} + 3 \, a^{5} b^{2} x^{2} - a^{6} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} {\left (-b x^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 739, normalized size = 4.43 \[ \frac {\sqrt {-b \,x^{2}+a}\, \sqrt {-b^{2} x^{4}+a^{2}}\, \left (19 \sqrt {2}\, \sqrt {a}\, b^{\frac {5}{2}} x^{4} \ln \left (\frac {2 \left (a -\sqrt {a b}\, x +\sqrt {2}\, \sqrt {b \,x^{2}+a}\, \sqrt {a}\right ) b}{b x +\sqrt {a b}}\right )-19 \sqrt {2}\, \sqrt {a}\, b^{\frac {5}{2}} x^{4} \ln \left (\frac {2 \left (a +\sqrt {a b}\, x +\sqrt {2}\, \sqrt {b \,x^{2}+a}\, \sqrt {a}\right ) b}{b x -\sqrt {a b}}\right )-16 \sqrt {a b}\, b^{2} x^{4} \ln \left (\frac {b x +\sqrt {-\frac {\left (b x +\sqrt {-a b}\right ) \left (-b x +\sqrt {-a b}\right )}{b}}\, \sqrt {b}}{\sqrt {b}}\right )+16 \sqrt {a b}\, b^{2} x^{4} \ln \left (\frac {b x +\sqrt {b \,x^{2}+a}\, \sqrt {b}}{\sqrt {b}}\right )-38 \sqrt {2}\, a^{\frac {3}{2}} b^{\frac {3}{2}} x^{2} \ln \left (\frac {2 \left (a -\sqrt {a b}\, x +\sqrt {2}\, \sqrt {b \,x^{2}+a}\, \sqrt {a}\right ) b}{b x +\sqrt {a b}}\right )+38 \sqrt {2}\, a^{\frac {3}{2}} b^{\frac {3}{2}} x^{2} \ln \left (\frac {2 \left (a +\sqrt {a b}\, x +\sqrt {2}\, \sqrt {b \,x^{2}+a}\, \sqrt {a}\right ) b}{b x -\sqrt {a b}}\right )+32 \sqrt {a b}\, a b \,x^{2} \ln \left (\frac {b x +\sqrt {-\frac {\left (b x +\sqrt {-a b}\right ) \left (-b x +\sqrt {-a b}\right )}{b}}\, \sqrt {b}}{\sqrt {b}}\right )-32 \sqrt {a b}\, a b \,x^{2} \ln \left (\frac {b x +\sqrt {b \,x^{2}+a}\, \sqrt {b}}{\sqrt {b}}\right )+36 \sqrt {a b}\, \sqrt {b \,x^{2}+a}\, b^{\frac {3}{2}} x^{3}+19 \sqrt {2}\, a^{\frac {5}{2}} \sqrt {b}\, \ln \left (\frac {2 \left (a -\sqrt {a b}\, x +\sqrt {2}\, \sqrt {b \,x^{2}+a}\, \sqrt {a}\right ) b}{b x +\sqrt {a b}}\right )-19 \sqrt {2}\, a^{\frac {5}{2}} \sqrt {b}\, \ln \left (\frac {2 \left (a +\sqrt {a b}\, x +\sqrt {2}\, \sqrt {b \,x^{2}+a}\, \sqrt {a}\right ) b}{b x -\sqrt {a b}}\right )-16 \sqrt {a b}\, a^{2} \ln \left (\frac {b x +\sqrt {-\frac {\left (b x +\sqrt {-a b}\right ) \left (-b x +\sqrt {-a b}\right )}{b}}\, \sqrt {b}}{\sqrt {b}}\right )+16 \sqrt {a b}\, a^{2} \ln \left (\frac {b x +\sqrt {b \,x^{2}+a}\, \sqrt {b}}{\sqrt {b}}\right )-52 \sqrt {a b}\, \sqrt {b \,x^{2}+a}\, a \sqrt {b}\, x \right ) b^{\frac {9}{2}}}{16 \left (b \,x^{2}-a \right ) \sqrt {b \,x^{2}+a}\, \left (\sqrt {-a b}+\sqrt {a b}\right )^{3} \left (-\sqrt {-a b}+\sqrt {a b}\right )^{3} \sqrt {a b}\, \left (b x -\sqrt {a b}\right )^{2} \left (b x +\sqrt {a b}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} {\left (-b x^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a^2-b^2\,x^4}\,{\left (a-b\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )} \left (a - b x^{2}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________