Optimal. Leaf size=68 \[ \frac{4 b^2 x^2}{3 a^3 \sqrt{a+b x^4}}+\frac{2 b}{3 a^2 x^2 \sqrt{a+b x^4}}-\frac{1}{6 a x^6 \sqrt{a+b x^4}} \]
[Out]
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Rubi [A] time = 0.0608464, antiderivative size = 68, 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 \[ \frac{4 b^2 x^2}{3 a^3 \sqrt{a+b x^4}}+\frac{2 b}{3 a^2 x^2 \sqrt{a+b x^4}}-\frac{1}{6 a x^6 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^7*(a + b*x^4)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 6.48862, size = 63, normalized size = 0.93 \[ - \frac{1}{6 a x^{6} \sqrt{a + b x^{4}}} + \frac{2 b}{3 a^{2} x^{2} \sqrt{a + b x^{4}}} + \frac{4 b^{2} x^{2}}{3 a^{3} \sqrt{a + b x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**7/(b*x**4+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0382332, size = 42, normalized size = 0.62 \[ \frac{-a^2+4 a b x^4+8 b^2 x^8}{6 a^3 x^6 \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^7*(a + b*x^4)^(3/2)),x]
[Out]
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Maple [A] time = 0.007, size = 37, normalized size = 0.5 \[ -{\frac{-8\,{b}^{2}{x}^{8}-4\,ab{x}^{4}+{a}^{2}}{6\,{a}^{3}{x}^{6}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^7/(b*x^4+a)^(3/2),x)
[Out]
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Maxima [A] time = 1.43974, size = 76, normalized size = 1.12 \[ \frac{b^{2} x^{2}}{2 \, \sqrt{b x^{4} + a} a^{3}} + \frac{\frac{6 \, \sqrt{b x^{4} + a} b}{x^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}{x^{6}}}{6 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/2)*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280812, size = 68, normalized size = 1. \[ \frac{{\left (8 \, b^{2} x^{8} + 4 \, a b x^{4} - a^{2}\right )} \sqrt{b x^{4} + a}}{6 \,{\left (a^{3} b x^{10} + a^{4} x^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/2)*x^7),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.35944, size = 233, normalized size = 3.43 \[ - \frac{a^{3} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{4}} + 1}}{6 a^{5} b^{4} x^{4} + 12 a^{4} b^{5} x^{8} + 6 a^{3} b^{6} x^{12}} + \frac{3 a^{2} b^{\frac{11}{2}} x^{4} \sqrt{\frac{a}{b x^{4}} + 1}}{6 a^{5} b^{4} x^{4} + 12 a^{4} b^{5} x^{8} + 6 a^{3} b^{6} x^{12}} + \frac{12 a b^{\frac{13}{2}} x^{8} \sqrt{\frac{a}{b x^{4}} + 1}}{6 a^{5} b^{4} x^{4} + 12 a^{4} b^{5} x^{8} + 6 a^{3} b^{6} x^{12}} + \frac{8 b^{\frac{15}{2}} x^{12} \sqrt{\frac{a}{b x^{4}} + 1}}{6 a^{5} b^{4} x^{4} + 12 a^{4} b^{5} x^{8} + 6 a^{3} b^{6} x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**7/(b*x**4+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.239089, size = 65, normalized size = 0.96 \[ -\frac{{\left (b + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} - 6 \, \sqrt{b + \frac{a}{x^{4}}} b}{6 \, a^{3}} - \frac{x^{2}}{256 \, \sqrt{b x^{4} + a} a^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/2)*x^7),x, algorithm="giac")
[Out]