Optimal. Leaf size=59 \[ \frac {a^2 \left (a+b x^2\right )^{3/2}}{3 b^3}+\frac {\left (a+b x^2\right )^{7/2}}{7 b^3}-\frac {2 a \left (a+b x^2\right )^{5/2}}{5 b^3} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {a^2 \left (a+b x^2\right )^{3/2}}{3 b^3}+\frac {\left (a+b x^2\right )^{7/2}}{7 b^3}-\frac {2 a \left (a+b x^2\right )^{5/2}}{5 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^5 \sqrt {a+b x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 \sqrt {a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2 \sqrt {a+b x}}{b^2}-\frac {2 a (a+b x)^{3/2}}{b^2}+\frac {(a+b x)^{5/2}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 \left (a+b x^2\right )^{3/2}}{3 b^3}-\frac {2 a \left (a+b x^2\right )^{5/2}}{5 b^3}+\frac {\left (a+b x^2\right )^{7/2}}{7 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 0.66 \[ \frac {\left (a+b x^2\right )^{3/2} \left (8 a^2-12 a b x^2+15 b^2 x^4\right )}{105 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 46, normalized size = 0.78 \[ \frac {{\left (15 \, b^{3} x^{6} + 3 \, a b^{2} x^{4} - 4 \, a^{2} b x^{2} + 8 \, a^{3}\right )} \sqrt {b x^{2} + a}}{105 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 43, normalized size = 0.73 \[ \frac {15 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} - 42 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2}}{105 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.61 \[ \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (15 b^{2} x^{4}-12 a b \,x^{2}+8 a^{2}\right )}{105 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 53, normalized size = 0.90 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} x^{4}}{7 \, b} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x^{2}}{35 \, b^{2}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2}}{105 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.66, size = 44, normalized size = 0.75 \[ \sqrt {b\,x^2+a}\,\left (\frac {x^6}{7}+\frac {8\,a^3}{105\,b^3}+\frac {a\,x^4}{35\,b}-\frac {4\,a^2\,x^2}{105\,b^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.70, size = 87, normalized size = 1.47 \[ \begin {cases} \frac {8 a^{3} \sqrt {a + b x^{2}}}{105 b^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + b x^{2}}}{105 b^{2}} + \frac {a x^{4} \sqrt {a + b x^{2}}}{35 b} + \frac {x^{6} \sqrt {a + b x^{2}}}{7} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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