Optimal. Leaf size=154 \[ -\frac{b d \left (3 a c^2-d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2} \left (a c^2-d^2\right )^2}-\frac{a c-d \sqrt{a+b x^3}}{3 a x^3 \left (a c^2-d^2\right )}+\frac{2 b c^3 \log \left (c \sqrt{a+b x^3}+d\right )}{3 \left (a c^2-d^2\right )^2}-\frac{b c^3 \log (x)}{\left (a c^2-d^2\right )^2} \]
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Rubi [A] time = 0.617852, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{b d \left (3 a c^2-d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2} \left (a c^2-d^2\right )^2}-\frac{a c-d \sqrt{a+b x^3}}{3 a x^3 \left (a c^2-d^2\right )}+\frac{2 b c^3 \log \left (c \sqrt{a+b x^3}+d\right )}{3 \left (a c^2-d^2\right )^2}-\frac{b c^3 \log (x)}{\left (a c^2-d^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])),x]
[Out]
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Rubi in Sympy [A] time = 42.5427, size = 136, normalized size = 0.88 \[ - \frac{b c^{3} \log{\left (- b x^{3} \right )}}{3 \left (- a c^{2} + d^{2}\right )^{2}} + \frac{2 b c^{3} \log{\left (c \sqrt{a + b x^{3}} + d \right )}}{3 \left (- a c^{2} + d^{2}\right )^{2}} + \frac{a c - d \sqrt{a + b x^{3}}}{3 a x^{3} \left (- a c^{2} + d^{2}\right )} + \frac{b d \left (- 3 a c^{2} + d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{3 a^{\frac{3}{2}} \left (- a c^{2} + d^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)
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Mathematica [C] time = 6.76996, size = 860, normalized size = 5.58 \[ \frac{5 b^2 d x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right ) c^4}{3 \left (a c^2-d^2\right ) \sqrt{b x^3+a} \left (b c^2 x^3+a c^2-d^2\right ) \left (-5 b c^2 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right ) x^3+2 \left (a c^2-d^2\right ) F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right )+a c^2 F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right )\right )}-\frac{b \log (x) c^3}{\left (a c^2-d^2\right )^2}+\frac{b \log \left (b c^2 x^3+a c^2-d^2\right ) c^3}{3 \left (a c^2-d^2\right )^2}+\frac{2 b^2 d x^3 F_1\left (1;\frac{1}{2},1;2;-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right ) c^2}{3 \sqrt{b x^3+a} \left (b c^2 x^3+a c^2-d^2\right ) \left (b \left (\left (d^2-a c^2\right ) F_1\left (2;\frac{3}{2},1;3;-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )-2 a c^2 F_1\left (2;\frac{1}{2},2;3;-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )\right ) x^3+4 a \left (a c^2-d^2\right ) F_1\left (1;\frac{1}{2},1;2;-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )\right )}-\frac{5 b^2 d^3 x^3 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right ) c^2}{9 a \left (a c^2-d^2\right ) \sqrt{b x^3+a} \left (b c^2 x^3+a c^2-d^2\right ) \left (-5 b c^2 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right ) x^3+2 \left (a c^2-d^2\right ) F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right )+a c^2 F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{a}{b x^3},\frac{d^2-a c^2}{b c^2 x^3}\right )\right )}-\frac{c}{3 \left (a c^2-d^2\right ) x^3}+\frac{d \sqrt{b x^3+a}}{3 a \left (a c^2-d^2\right ) x^3} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^4*(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])),x]
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Maple [C] time = 0.053, size = 863, normalized size = 5.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b c x^{3} + a c + \sqrt{b x^{3} + a} d\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^4),x, algorithm="maxima")
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Fricas [A] time = 0.394049, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, a^{\frac{3}{2}} b c^{3} x^{3} \log \left (\sqrt{b x^{3} + a} c + d\right ) - 2 \, a^{\frac{3}{2}} b c^{3} x^{3} \log \left (\sqrt{b x^{3} + a} c - d\right ) -{\left (3 \, a b c^{2} d - b d^{3}\right )} x^{3} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \,{\left (a c^{2} d - d^{3}\right )} \sqrt{b x^{3} + a} \sqrt{a} + 2 \,{\left (a b c^{3} x^{3} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) - 3 \, a b c^{3} x^{3} \log \left (x\right ) - a^{2} c^{3} + a c d^{2}\right )} \sqrt{a}}{6 \,{\left (a^{3} c^{4} - 2 \, a^{2} c^{2} d^{2} + a d^{4}\right )} \sqrt{a} x^{3}}, \frac{\sqrt{-a} a b c^{3} x^{3} \log \left (\sqrt{b x^{3} + a} c + d\right ) - \sqrt{-a} a b c^{3} x^{3} \log \left (\sqrt{b x^{3} + a} c - d\right ) +{\left (3 \, a b c^{2} d - b d^{3}\right )} x^{3} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left (a c^{2} d - d^{3}\right )} \sqrt{b x^{3} + a} \sqrt{-a} +{\left (a b c^{3} x^{3} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) - 3 \, a b c^{3} x^{3} \log \left (x\right ) - a^{2} c^{3} + a c d^{2}\right )} \sqrt{-a}}{3 \,{\left (a^{3} c^{4} - 2 \, a^{2} c^{2} d^{2} + a d^{4}\right )} \sqrt{-a} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^4),x, algorithm="fricas")
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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(1/x**4/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)
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GIAC/XCAS [A] time = 0.289417, size = 275, normalized size = 1.79 \[ \frac{1}{3} \,{\left (\frac{2 \, c^{4}{\rm ln}\left ({\left | \sqrt{b x^{3} + a} c + d \right |}\right )}{a^{2} c^{5} - 2 \, a c^{3} d^{2} + c d^{4}} - \frac{c^{3}{\rm ln}\left (b x^{3}\right )}{a^{2} c^{4} - 2 \, a c^{2} d^{2} + d^{4}} + \frac{{\left (3 \, a c^{2} d - d^{3}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{{\left (a^{3} c^{4} - 2 \, a^{2} c^{2} d^{2} + a d^{4}\right )} \sqrt{-a}} - \frac{a^{2} c^{3} - a c d^{2} -{\left (a c^{2} d - d^{3}\right )} \sqrt{b x^{3} + a}}{{\left (a c^{2} - d^{2}\right )}^{2} a b x^{3}}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^4),x, algorithm="giac")
[Out]