Optimal. Leaf size=128 \[ -\frac {3 x}{8 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {x^3}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 \sqrt {a} b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1112, 288, 205} \[ -\frac {x^3}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {3 x}{8 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 \sqrt {a} b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 288
Rule 1112
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {x^4}{\left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {x^3}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3 \left (a b+b^2 x^2\right )\right ) \int \frac {x^2}{\left (a b+b^2 x^2\right )^2} \, dx}{4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {3 x}{8 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {x^3}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (3 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{8 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {3 x}{8 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {x^3}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3 \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 \sqrt {a} b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 84, normalized size = 0.66 \[ \frac {3 \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )-\sqrt {a} \sqrt {b} x \left (3 a+5 b x^2\right )}{8 \sqrt {a} b^{5/2} \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.01, size = 188, normalized size = 1.47 \[ \left [-\frac {10 \, a b^{2} x^{3} + 6 \, a^{2} b x + 3 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{16 \, {\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}, -\frac {5 \, a b^{2} x^{3} + 3 \, a^{2} b x - 3 \, {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{8 \, {\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 97, normalized size = 0.76 \[ -\frac {\left (-3 b^{2} x^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )-6 a b \,x^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+5 \sqrt {a b}\, b \,x^{3}-3 a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+3 \sqrt {a b}\, a x \right ) \left (b \,x^{2}+a \right )}{8 \sqrt {a b}\, \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.03, size = 59, normalized size = 0.46 \[ -\frac {5 \, b x^{3} + 3 \, a x}{8 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} + \frac {3 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^4}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________