Optimal. Leaf size=136 \[ \frac {3 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}}-\frac {3 a^4 x \sqrt {a+b x^2}}{256 b^2}+\frac {a^3 x^3 \sqrt {a+b x^2}}{128 b}+\frac {1}{32} a^2 x^5 \sqrt {a+b x^2}+\frac {1}{16} a x^5 \left (a+b x^2\right )^{3/2}+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2} \]
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Rubi [A] time = 0.05, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {279, 321, 217, 206} \[ -\frac {3 a^4 x \sqrt {a+b x^2}}{256 b^2}+\frac {3 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}}+\frac {a^3 x^3 \sqrt {a+b x^2}}{128 b}+\frac {1}{32} a^2 x^5 \sqrt {a+b x^2}+\frac {1}{16} a x^5 \left (a+b x^2\right )^{3/2}+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 279
Rule 321
Rubi steps
\begin {align*} \int x^4 \left (a+b x^2\right )^{5/2} \, dx &=\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}+\frac {1}{2} a \int x^4 \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac {1}{16} a x^5 \left (a+b x^2\right )^{3/2}+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}+\frac {1}{16} \left (3 a^2\right ) \int x^4 \sqrt {a+b x^2} \, dx\\ &=\frac {1}{32} a^2 x^5 \sqrt {a+b x^2}+\frac {1}{16} a x^5 \left (a+b x^2\right )^{3/2}+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}+\frac {1}{32} a^3 \int \frac {x^4}{\sqrt {a+b x^2}} \, dx\\ &=\frac {a^3 x^3 \sqrt {a+b x^2}}{128 b}+\frac {1}{32} a^2 x^5 \sqrt {a+b x^2}+\frac {1}{16} a x^5 \left (a+b x^2\right )^{3/2}+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}-\frac {\left (3 a^4\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{128 b}\\ &=-\frac {3 a^4 x \sqrt {a+b x^2}}{256 b^2}+\frac {a^3 x^3 \sqrt {a+b x^2}}{128 b}+\frac {1}{32} a^2 x^5 \sqrt {a+b x^2}+\frac {1}{16} a x^5 \left (a+b x^2\right )^{3/2}+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}+\frac {\left (3 a^5\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{256 b^2}\\ &=-\frac {3 a^4 x \sqrt {a+b x^2}}{256 b^2}+\frac {a^3 x^3 \sqrt {a+b x^2}}{128 b}+\frac {1}{32} a^2 x^5 \sqrt {a+b x^2}+\frac {1}{16} a x^5 \left (a+b x^2\right )^{3/2}+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}+\frac {\left (3 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{256 b^2}\\ &=-\frac {3 a^4 x \sqrt {a+b x^2}}{256 b^2}+\frac {a^3 x^3 \sqrt {a+b x^2}}{128 b}+\frac {1}{32} a^2 x^5 \sqrt {a+b x^2}+\frac {1}{16} a x^5 \left (a+b x^2\right )^{3/2}+\frac {1}{10} x^5 \left (a+b x^2\right )^{5/2}+\frac {3 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 105, normalized size = 0.77 \[ \frac {\sqrt {a+b x^2} \left (\frac {15 a^{9/2} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {\frac {b x^2}{a}+1}}+\sqrt {b} x \left (-15 a^4+10 a^3 b x^2+248 a^2 b^2 x^4+336 a b^3 x^6+128 b^4 x^8\right )\right )}{1280 b^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 190, normalized size = 1.40 \[ \left [\frac {15 \, a^{5} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (128 \, b^{5} x^{9} + 336 \, a b^{4} x^{7} + 248 \, a^{2} b^{3} x^{5} + 10 \, a^{3} b^{2} x^{3} - 15 \, a^{4} b x\right )} \sqrt {b x^{2} + a}}{2560 \, b^{3}}, -\frac {15 \, a^{5} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (128 \, b^{5} x^{9} + 336 \, a b^{4} x^{7} + 248 \, a^{2} b^{3} x^{5} + 10 \, a^{3} b^{2} x^{3} - 15 \, a^{4} b x\right )} \sqrt {b x^{2} + a}}{1280 \, b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 91, normalized size = 0.67 \[ -\frac {3 \, a^{5} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {5}{2}}} + \frac {1}{1280} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, b^{2} x^{2} + 21 \, a b\right )} x^{2} + 31 \, a^{2}\right )} x^{2} + \frac {5 \, a^{3}}{b}\right )} x^{2} - \frac {15 \, a^{4}}{b^{2}}\right )} \sqrt {b x^{2} + a} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 113, normalized size = 0.83 \[ \frac {3 a^{5} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {5}{2}}}+\frac {3 \sqrt {b \,x^{2}+a}\, a^{4} x}{256 b^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{3} x}{128 b^{2}}+\frac {\left (b \,x^{2}+a \right )^{\frac {7}{2}} x^{3}}{10 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{2} x}{160 b^{2}}-\frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a x}{80 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 105, normalized size = 0.77 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {7}{2}} x^{3}}{10 \, b} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x}{80 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} x}{160 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} x}{128 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} a^{4} x}{256 \, b^{2}} + \frac {3 \, a^{5} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\left (b\,x^2+a\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.11, size = 175, normalized size = 1.29 \[ - \frac {3 a^{\frac {9}{2}} x}{256 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {a^{\frac {7}{2}} x^{3}}{256 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {129 a^{\frac {5}{2}} x^{5}}{640 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {73 a^{\frac {3}{2}} b x^{7}}{160 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {29 \sqrt {a} b^{2} x^{9}}{80 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{5} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{256 b^{\frac {5}{2}}} + \frac {b^{3} x^{11}}{10 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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