Optimal. Leaf size=249 \[ \frac{b^5 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a b^4 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{2 a^2 b^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
[Out]
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Rubi [A] time = 0.158662, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b^5 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac{5 a b^4 x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{2 a^2 b^3 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}+\frac{5 a^3 b^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^5,x]
[Out]
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Rubi in Sympy [A] time = 25.9729, size = 216, normalized size = 0.87 \[ \frac{729 a^{3} b^{2} x^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{88 \left (a + b x^{3}\right )} + \frac{243 a^{2} b^{2} x^{2} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{44} + \frac{405 a b^{2} x^{2} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{88} + \frac{15 a \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{4 x^{4}} + \frac{45 b^{2} x^{2} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{11} - \frac{4 \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2)/x**5,x)
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Mathematica [A] time = 0.0497039, size = 83, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^3\right )^2} \left (-22 a^5-440 a^4 b x^3+440 a^3 b^2 x^6+176 a^2 b^3 x^9+55 a b^4 x^{12}+8 b^5 x^{15}\right )}{88 x^4 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)/x^5,x]
[Out]
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Maple [A] time = 0.009, size = 80, normalized size = 0.3 \[ -{\frac{-8\,{b}^{5}{x}^{15}-55\,a{b}^{4}{x}^{12}-176\,{a}^{2}{b}^{3}{x}^{9}-440\,{a}^{3}{b}^{2}{x}^{6}+440\,{a}^{4}b{x}^{3}+22\,{a}^{5}}{88\,{x}^{4} \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^6+2*a*b*x^3+a^2)^(5/2)/x^5,x)
[Out]
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Maxima [A] time = 0.764393, size = 80, normalized size = 0.32 \[ \frac{8 \, b^{5} x^{15} + 55 \, a b^{4} x^{12} + 176 \, a^{2} b^{3} x^{9} + 440 \, a^{3} b^{2} x^{6} - 440 \, a^{4} b x^{3} - 22 \, a^{5}}{88 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)/x^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270441, size = 80, normalized size = 0.32 \[ \frac{8 \, b^{5} x^{15} + 55 \, a b^{4} x^{12} + 176 \, a^{2} b^{3} x^{9} + 440 \, a^{3} b^{2} x^{6} - 440 \, a^{4} b x^{3} - 22 \, a^{5}}{88 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)/x^5,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**6+2*a*b*x**3+a**2)**(5/2)/x**5,x)
[Out]
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GIAC/XCAS [A] time = 0.299673, size = 144, normalized size = 0.58 \[ \frac{1}{11} \, b^{5} x^{11}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{8} \, a b^{4} x^{8}{\rm sign}\left (b x^{3} + a\right ) + 2 \, a^{2} b^{3} x^{5}{\rm sign}\left (b x^{3} + a\right ) + 5 \, a^{3} b^{2} x^{2}{\rm sign}\left (b x^{3} + a\right ) - \frac{20 \, a^{4} b x^{3}{\rm sign}\left (b x^{3} + a\right ) + a^{5}{\rm sign}\left (b x^{3} + a\right )}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)/x^5,x, algorithm="giac")
[Out]