Optimal. Leaf size=159 \[ \frac {3 a b^2 x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{5 \left (a+b x^2\right )^3}+\frac {a^2 b x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{\left (a+b x^2\right )^3}+\frac {b^3 x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{7 \left (a+b x^2\right )^3}+\frac {a^3 x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{\left (a+b x^2\right )^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1088, 194} \begin {gather*} \frac {b^3 x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{7 \left (a+b x^2\right )^3}+\frac {3 a b^2 x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{5 \left (a+b x^2\right )^3}+\frac {a^2 b x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{\left (a+b x^2\right )^3}+\frac {a^3 x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{\left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 194
Rule 1088
Rubi steps
\begin {align*} \int \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \int \left (2 a b+2 b^2 x^2\right )^3 \, dx}{\left (2 a b+2 b^2 x^2\right )^3}\\ &=\frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \int \left (8 a^3 b^3+24 a^2 b^4 x^2+24 a b^5 x^4+8 b^6 x^6\right ) \, dx}{\left (2 a b+2 b^2 x^2\right )^3}\\ &=\frac {a^3 x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{\left (a+b x^2\right )^3}+\frac {a^2 b x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{\left (a+b x^2\right )^3}+\frac {3 a b^2 x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{5 \left (a+b x^2\right )^3}+\frac {b^3 x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{7 \left (a+b x^2\right )^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 59, normalized size = 0.37 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (35 a^3 x+35 a^2 b x^3+21 a b^2 x^5+5 b^3 x^7\right )}{35 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 5.30, size = 59, normalized size = 0.37 \begin {gather*} \frac {\sqrt {\left (a+b x^2\right )^2} \left (35 a^3 x+35 a^2 b x^3+21 a b^2 x^5+5 b^3 x^7\right )}{35 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 31, normalized size = 0.19 \begin {gather*} \frac {1}{7} \, b^{3} x^{7} + \frac {3}{5} \, a b^{2} x^{5} + a^{2} b x^{3} + a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 63, normalized size = 0.40 \begin {gather*} \frac {1}{7} \, b^{3} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{5} \, a b^{2} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{2} b x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{3} x \mathrm {sgn}\left (b x^{2} + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 56, normalized size = 0.35 \begin {gather*} \frac {\left (5 b^{3} x^{6}+21 a \,b^{2} x^{4}+35 a^{2} b \,x^{2}+35 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} x}{35 \left (b \,x^{2}+a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.36, size = 31, normalized size = 0.19 \begin {gather*} \frac {1}{7} \, b^{3} x^{7} + \frac {3}{5} \, a b^{2} x^{5} + a^{2} b x^{3} + a^{3} x \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 {\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2} \,d x \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 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________