Optimal. Leaf size=161 \[ -\frac{a^2 \left (c+d x^3\right )^{3/2}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}+\frac{a (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{7/2} \sqrt{b c-a d}}-\frac{a \sqrt{c+d x^3} (4 b c-5 a d)}{3 b^3 (b c-a d)}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b^2 d} \]
[Out]
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Rubi [A] time = 0.532934, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2 \left (c+d x^3\right )^{3/2}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}+\frac{a (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{7/2} \sqrt{b c-a d}}-\frac{a \sqrt{c+d x^3} (4 b c-5 a d)}{3 b^3 (b c-a d)}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b^2 d} \]
Antiderivative was successfully verified.
[In] Int[(x^8*Sqrt[c + d*x^3])/(a + b*x^3)^2,x]
[Out]
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Rubi in Sympy [A] time = 46.6628, size = 139, normalized size = 0.86 \[ \frac{a^{2} \left (c + d x^{3}\right )^{\frac{3}{2}}}{3 b^{2} \left (a + b x^{3}\right ) \left (a d - b c\right )} - \frac{a \sqrt{c + d x^{3}} \left (5 a d - 4 b c\right )}{3 b^{3} \left (a d - b c\right )} + \frac{a \left (5 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 b^{\frac{7}{2}} \sqrt{a d - b c}} + \frac{2 \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8*(d*x**3+c)**(1/2)/(b*x**3+a)**2,x)
[Out]
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Mathematica [A] time = 0.293466, size = 111, normalized size = 0.69 \[ \frac{\sqrt{c+d x^3} \left (-\frac{3 a^2}{a+b x^3}-12 a+\frac{2 b \left (c+d x^3\right )}{d}\right )}{9 b^3}+\frac{a (4 b c-5 a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{7/2} \sqrt{b c-a d}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^8*Sqrt[c + d*x^3])/(a + b*x^3)^2,x]
[Out]
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Maple [C] time = 0.057, size = 917, normalized size = 5.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8*(d*x^3+c)^(1/2)/(b*x^3+a)^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(sqrt(d*x^3 + c)*x^8/(b*x^3 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226139, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, b^{2} d x^{6} + 2 \,{\left (b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 15 \, a^{2} d\right )} \sqrt{d x^{3} + c} \sqrt{b^{2} c - a b d} - 3 \,{\left (4 \, a^{2} b c d - 5 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 5 \, a^{2} b d^{2}\right )} x^{3}\right )} \log \left (\frac{{\left (b d x^{3} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{3} + c}{\left (b^{2} c - a b d\right )}}{b x^{3} + a}\right )}{18 \,{\left (b^{4} d x^{3} + a b^{3} d\right )} \sqrt{b^{2} c - a b d}}, \frac{{\left (2 \, b^{2} d x^{6} + 2 \,{\left (b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 15 \, a^{2} d\right )} \sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d} + 3 \,{\left (4 \, a^{2} b c d - 5 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 5 \, a^{2} b d^{2}\right )} x^{3}\right )} \arctan \left (-\frac{b c - a d}{\sqrt{d x^{3} + c} \sqrt{-b^{2} c + a b d}}\right )}{9 \,{\left (b^{4} d x^{3} + a b^{3} d\right )} \sqrt{-b^{2} c + a b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)*x^8/(b*x^3 + a)^2,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**8*(d*x**3+c)**(1/2)/(b*x**3+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.218818, size = 184, normalized size = 1.14 \[ -\frac{\sqrt{d x^{3} + c} a^{2} d}{3 \,{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} b^{3}} - \frac{{\left (4 \, a b c - 5 \, a^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{3 \, \sqrt{-b^{2} c + a b d} b^{3}} + \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} b^{4} d^{2} - 6 \, \sqrt{d x^{3} + c} a b^{3} d^{3}\right )}}{9 \, b^{6} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)*x^8/(b*x^3 + a)^2,x, algorithm="giac")
[Out]