Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^2\right )^{5/2}}{5 b^3}+\frac{\left (a+b x^2\right )^{9/2}}{9 b^3}-\frac{2 a \left (a+b x^2\right )^{7/2}}{7 b^3} \]
[Out]
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Rubi [A] time = 0.0935262, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a^2 \left (a+b x^2\right )^{5/2}}{5 b^3}+\frac{\left (a+b x^2\right )^{9/2}}{9 b^3}-\frac{2 a \left (a+b x^2\right )^{7/2}}{7 b^3} \]
Antiderivative was successfully verified.
[In] Int[x^5*(a + b*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 11.7695, size = 51, normalized size = 0.86 \[ \frac{a^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{3}} - \frac{2 a \left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b^{3}} + \frac{\left (a + b x^{2}\right )^{\frac{9}{2}}}{9 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(b*x**2+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0355322, size = 39, normalized size = 0.66 \[ \frac{\left (a+b x^2\right )^{5/2} \left (8 a^2-20 a b x^2+35 b^2 x^4\right )}{315 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^5*(a + b*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.006, size = 36, normalized size = 0.6 \[{\frac{35\,{b}^{2}{x}^{4}-20\,ab{x}^{2}+8\,{a}^{2}}{315\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(b*x^2+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)*x^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238168, size = 77, normalized size = 1.31 \[ \frac{{\left (35 \, b^{4} x^{8} + 50 \, a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + 8 \, a^{4}\right )} \sqrt{b x^{2} + a}}{315 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)*x^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.34492, size = 109, normalized size = 1.85 \[ \begin{cases} \frac{8 a^{4} \sqrt{a + b x^{2}}}{315 b^{3}} - \frac{4 a^{3} x^{2} \sqrt{a + b x^{2}}}{315 b^{2}} + \frac{a^{2} x^{4} \sqrt{a + b x^{2}}}{105 b} + \frac{10 a x^{6} \sqrt{a + b x^{2}}}{63} + \frac{b x^{8} \sqrt{a + b x^{2}}}{9} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{6}}{6} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(b*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.206692, size = 143, normalized size = 2.42 \[ \frac{\frac{3 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} a}{b^{2}} + \frac{35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}}{b^{2}}}{315 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(3/2)*x^5,x, algorithm="giac")
[Out]