Optimal. Leaf size=117 \[ -\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right )}{6 a^2 c^{3/2}}-\frac{\sqrt{c+d x^6}}{6 a c x^6} \]
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Rubi [A] time = 0.357349, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^6}}{\sqrt{b c-a d}}\right )}{3 a^2 \sqrt{b c-a d}}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^6}}{\sqrt{c}}\right )}{6 a^2 c^{3/2}}-\frac{\sqrt{c+d x^6}}{6 a c x^6} \]
Antiderivative was successfully verified.
[In] Int[1/(x^7*(a + b*x^6)*Sqrt[c + d*x^6]),x]
[Out]
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Rubi in Sympy [A] time = 44.2555, size = 100, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{6}}}{6 a c x^{6}} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{6}}}{\sqrt{a d - b c}} \right )}}{3 a^{2} \sqrt{a d - b c}} + \frac{\left (\frac{a d}{2} + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{6}}}{\sqrt{c}} \right )}}{3 a^{2} c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**7/(b*x**6+a)/(d*x**6+c)**(1/2),x)
[Out]
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Mathematica [C] time = 0.790981, size = 410, normalized size = 3.5 \[ \frac{\frac{5 b d x^6 \left (a \left (3 c+2 d x^6\right )+b x^6 \left (c+3 d x^6\right )\right ) F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^6},-\frac{a}{b x^6}\right )-3 \left (a+b x^6\right ) \left (c+d x^6\right ) \left (2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^6},-\frac{a}{b x^6}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^6},-\frac{a}{b x^6}\right )\right )}{a c \left (-5 b d x^6 F_1\left (\frac{3}{2};\frac{1}{2},1;\frac{5}{2};-\frac{c}{d x^6},-\frac{a}{b x^6}\right )+2 a d F_1\left (\frac{5}{2};\frac{1}{2},2;\frac{7}{2};-\frac{c}{d x^6},-\frac{a}{b x^6}\right )+b c F_1\left (\frac{5}{2};\frac{3}{2},1;\frac{7}{2};-\frac{c}{d x^6},-\frac{a}{b x^6}\right )\right )}+\frac{6 b d x^{12} F_1\left (1;\frac{1}{2},1;2;-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{x^6 \left (2 b c F_1\left (2;\frac{1}{2},2;3;-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (2;\frac{3}{2},1;3;-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-4 a c F_1\left (1;\frac{1}{2},1;2;-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}}{18 x^6 \left (a+b x^6\right ) \sqrt{c+d x^6}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^7*(a + b*x^6)*Sqrt[c + d*x^6]),x]
[Out]
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Maple [F] time = 0.125, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{7} \left ( b{x}^{6}+a \right ) }{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^7/(b*x^6+a)/(d*x^6+c)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{6} + a\right )} \sqrt{d x^{6} + c} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.248566, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, b c^{\frac{3}{2}} x^{6} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{6} + 2 \, b c - a d - 2 \, \sqrt{d x^{6} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{6} + a}\right ) +{\left (2 \, b c + a d\right )} x^{6} \log \left (\frac{{\left (d x^{6} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{6} + c} c}{x^{6}}\right ) - 2 \, \sqrt{d x^{6} + c} a \sqrt{c}}{12 \, a^{2} c^{\frac{3}{2}} x^{6}}, -\frac{4 \, b c^{\frac{3}{2}} x^{6} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{6} + c} b}\right ) -{\left (2 \, b c + a d\right )} x^{6} \log \left (\frac{{\left (d x^{6} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{6} + c} c}{x^{6}}\right ) + 2 \, \sqrt{d x^{6} + c} a \sqrt{c}}{12 \, a^{2} c^{\frac{3}{2}} x^{6}}, \frac{b \sqrt{-c} c x^{6} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b d x^{6} + 2 \, b c - a d - 2 \, \sqrt{d x^{6} + c}{\left (b c - a d\right )} \sqrt{\frac{b}{b c - a d}}}{b x^{6} + a}\right ) -{\left (2 \, b c + a d\right )} x^{6} \arctan \left (\frac{c}{\sqrt{d x^{6} + c} \sqrt{-c}}\right ) - \sqrt{d x^{6} + c} a \sqrt{-c}}{6 \, a^{2} \sqrt{-c} c x^{6}}, -\frac{2 \, b \sqrt{-c} c x^{6} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}{\sqrt{d x^{6} + c} b}\right ) +{\left (2 \, b c + a d\right )} x^{6} \arctan \left (\frac{c}{\sqrt{d x^{6} + c} \sqrt{-c}}\right ) + \sqrt{d x^{6} + c} a \sqrt{-c}}{6 \, a^{2} \sqrt{-c} c x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^7),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(1/x**7/(b*x**6+a)/(d*x**6+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.217019, size = 159, normalized size = 1.36 \[ \frac{1}{6} \, d^{2}{\left (\frac{2 \, b^{2} \arctan \left (\frac{\sqrt{d x^{6} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} d^{2}} - \frac{{\left (2 \, b c + a d\right )} \arctan \left (\frac{\sqrt{d x^{6} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c d^{2}} - \frac{\sqrt{d x^{6} + c}}{a c d^{2} x^{6}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^7),x, algorithm="giac")
[Out]