Optimal. Leaf size=151 \[ -\frac{a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]
[Out]
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Rubi [A] time = 0.132317, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^6 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^8,x]
[Out]
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Rubi in Sympy [A] time = 14.5282, size = 121, normalized size = 0.8 \[ \frac{a b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{140 x^{5} \left (a + b x\right )} - \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{28 x^{5}} - \frac{b \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{14 x^{6}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/x**8,x)
[Out]
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Mathematica [A] time = 0.0219406, size = 55, normalized size = 0.36 \[ -\frac{\sqrt{(a+b x)^2} \left (20 a^3+70 a^2 b x+84 a b^2 x^2+35 b^3 x^3\right )}{140 x^7 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^8,x]
[Out]
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Maple [A] time = 0.008, size = 52, normalized size = 0.3 \[ -{\frac{35\,{b}^{3}{x}^{3}+84\,a{b}^{2}{x}^{2}+70\,{a}^{2}bx+20\,{a}^{3}}{140\,{x}^{7} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^(3/2)/x^8,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^2*x^2 + 2*a*b*x + a^2)^(3/2)/x^8,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225095, size = 47, normalized size = 0.31 \[ -\frac{35 \, b^{3} x^{3} + 84 \, a b^{2} x^{2} + 70 \, a^{2} b x + 20 \, a^{3}}{140 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/x^8,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{8}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/x**8,x)
[Out]
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GIAC/XCAS [A] time = 0.208405, size = 100, normalized size = 0.66 \[ \frac{b^{7}{\rm sign}\left (b x + a\right )}{140 \, a^{4}} - \frac{35 \, b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 84 \, a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 70 \, a^{2} b x{\rm sign}\left (b x + a\right ) + 20 \, a^{3}{\rm sign}\left (b x + a\right )}{140 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/x^8,x, algorithm="giac")
[Out]