Optimal. Leaf size=112 \[ \frac {5 a^3 x \sqrt {a+b x^2}}{128 b}+\frac {5}{64} a^2 x^3 \sqrt {a+b x^2}+\frac {5}{48} a x^3 \left (a+b x^2\right )^{3/2}+\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2}-\frac {5 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {285, 327, 223,
212} \begin {gather*} -\frac {5 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}}+\frac {5 a^3 x \sqrt {a+b x^2}}{128 b}+\frac {5}{64} a^2 x^3 \sqrt {a+b x^2}+\frac {5}{48} a x^3 \left (a+b x^2\right )^{3/2}+\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 223
Rule 285
Rule 327
Rubi steps
\begin {align*} \int x^2 \left (a+b x^2\right )^{5/2} \, dx &=\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2}+\frac {1}{8} (5 a) \int x^2 \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac {5}{48} a x^3 \left (a+b x^2\right )^{3/2}+\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2}+\frac {1}{16} \left (5 a^2\right ) \int x^2 \sqrt {a+b x^2} \, dx\\ &=\frac {5}{64} a^2 x^3 \sqrt {a+b x^2}+\frac {5}{48} a x^3 \left (a+b x^2\right )^{3/2}+\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2}+\frac {1}{64} \left (5 a^3\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx\\ &=\frac {5 a^3 x \sqrt {a+b x^2}}{128 b}+\frac {5}{64} a^2 x^3 \sqrt {a+b x^2}+\frac {5}{48} a x^3 \left (a+b x^2\right )^{3/2}+\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2}-\frac {\left (5 a^4\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{128 b}\\ &=\frac {5 a^3 x \sqrt {a+b x^2}}{128 b}+\frac {5}{64} a^2 x^3 \sqrt {a+b x^2}+\frac {5}{48} a x^3 \left (a+b x^2\right )^{3/2}+\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2}-\frac {\left (5 a^4\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{128 b}\\ &=\frac {5 a^3 x \sqrt {a+b x^2}}{128 b}+\frac {5}{64} a^2 x^3 \sqrt {a+b x^2}+\frac {5}{48} a x^3 \left (a+b x^2\right )^{3/2}+\frac {1}{8} x^3 \left (a+b x^2\right )^{5/2}-\frac {5 a^4 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 85, normalized size = 0.76 \begin {gather*} \frac {x \sqrt {a+b x^2} \left (15 a^3+118 a^2 b x^2+136 a b^2 x^4+48 b^3 x^6\right )}{384 b}+\frac {5 a^4 \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{128 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.04, size = 90, normalized size = 0.80
method | result | size |
risch | \(\frac {x \left (48 b^{3} x^{6}+136 a \,b^{2} x^{4}+118 a^{2} b \,x^{2}+15 a^{3}\right ) \sqrt {b \,x^{2}+a}}{384 b}-\frac {5 a^{4} \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {3}{2}}}\) | \(73\) |
default | \(\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 85, normalized size = 0.76 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} x}{8 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a x}{48 \, b} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x}{192 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a^{3} x}{128 \, b} - \frac {5 \, a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.15, size = 167, normalized size = 1.49 \begin {gather*} \left [\frac {15 \, a^{4} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (48 \, b^{4} x^{7} + 136 \, a b^{3} x^{5} + 118 \, a^{2} b^{2} x^{3} + 15 \, a^{3} b x\right )} \sqrt {b x^{2} + a}}{768 \, b^{2}}, \frac {15 \, a^{4} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (48 \, b^{4} x^{7} + 136 \, a b^{3} x^{5} + 118 \, a^{2} b^{2} x^{3} + 15 \, a^{3} b x\right )} \sqrt {b x^{2} + a}}{384 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 7.37, size = 150, normalized size = 1.34 \begin {gather*} \frac {5 a^{\frac {7}{2}} x}{128 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {133 a^{\frac {5}{2}} x^{3}}{384 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {127 a^{\frac {3}{2}} b x^{5}}{192 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {23 \sqrt {a} b^{2} x^{7}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {5 a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{128 b^{\frac {3}{2}}} + \frac {b^{3} x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.81, size = 77, normalized size = 0.69 \begin {gather*} \frac {5 \, a^{4} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {3}{2}}} + \frac {1}{384} \, {\left (2 \, {\left (4 \, {\left (6 \, b^{2} x^{2} + 17 \, a b\right )} x^{2} + 59 \, a^{2}\right )} x^{2} + \frac {15 \, a^{3}}{b}\right )} \sqrt {b x^{2} + a} 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 x^2\,{\left (b\,x^2+a\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________