Optimal. Leaf size=73 \[ -\frac{2 d \sqrt{a+b x^3}}{3 b^2 c^2}-\frac{2 \left (a c^2-d^2\right ) \log \left (c \sqrt{a+b x^3}+d\right )}{3 b^2 c^3}+\frac{x^3}{3 b c} \]
[Out]
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Rubi [A] time = 0.367028, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{2 d \sqrt{a+b x^3}}{3 b^2 c^2}-\frac{2 \left (a c^2-d^2\right ) \log \left (c \sqrt{a+b x^3}+d\right )}{3 b^2 c^3}+\frac{x^3}{3 b c} \]
Antiderivative was successfully verified.
[In] Int[x^5/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{2 \int ^{\sqrt{a + b x^{3}}} x\, dx}{3 b^{2} c} - \frac{2 \int ^{\sqrt{a + b x^{3}}} d\, dx}{3 b^{2} c^{2}} + \frac{2 \left (- a c^{2} + d^{2}\right ) \log{\left (c \sqrt{a + b x^{3}} + d \right )}}{3 b^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.102777, size = 95, normalized size = 1.3 \[ \frac{\left (d^2-a c^2\right ) \log \left (a c^2+b c^2 x^3-d^2\right )+\left (2 d^2-2 a c^2\right ) \tanh ^{-1}\left (\frac{c \sqrt{a+b x^3}}{d}\right )+c \left (b c x^3-2 d \sqrt{a+b x^3}\right )}{3 b^2 c^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]
[Out]
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Maple [C] time = 0.021, size = 943, normalized size = 12.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x)
[Out]
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Maxima [A] time = 0.701437, size = 84, normalized size = 1.15 \[ \frac{\frac{{\left (b x^{3} + a\right )} c - 2 \, \sqrt{b x^{3} + a} d}{c^{2}} - \frac{2 \,{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{3} + a} c + d\right )}{c^{3}}}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.279972, size = 159, normalized size = 2.18 \[ \frac{b c^{2} x^{3} - 2 \, \sqrt{b x^{3} + a} c d -{\left (a c^{2} - d^{2}\right )} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) -{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{3} + a} c + d\right ) +{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{3} + a} c - d\right )}{3 \, b^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.271632, size = 97, normalized size = 1.33 \[ -\frac{\frac{2 \,{\left (a c^{2} - d^{2}\right )}{\rm ln}\left ({\left | \sqrt{b x^{3} + a} c + d \right |}\right )}{b c^{3}} - \frac{{\left (b x^{3} + a\right )} b c - 2 \, \sqrt{b x^{3} + a} b d}{b^{2} c^{2}}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d),x, algorithm="giac")
[Out]