Optimal. Leaf size=88 \[ \frac{b (A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}+\frac{\sqrt{a+b x^3} (A b-4 a B)}{12 a x^3}-\frac{A \left (a+b x^3\right )^{3/2}}{6 a x^6} \]
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Rubi [A] time = 0.211303, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{b (A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}+\frac{\sqrt{a+b x^3} (A b-4 a B)}{12 a x^3}-\frac{A \left (a+b x^3\right )^{3/2}}{6 a x^6} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x^3]*(A + B*x^3))/x^7,x]
[Out]
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Rubi in Sympy [A] time = 14.329, size = 76, normalized size = 0.86 \[ - \frac{A \left (a + b x^{3}\right )^{\frac{3}{2}}}{6 a x^{6}} + \frac{\sqrt{a + b x^{3}} \left (A b - 4 B a\right )}{12 a x^{3}} + \frac{b \left (A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{12 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**3+A)*(b*x**3+a)**(1/2)/x**7,x)
[Out]
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Mathematica [A] time = 0.332693, size = 82, normalized size = 0.93 \[ \frac{\sqrt{a+b x^3} \left (\frac{b (A b-4 a B) \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}-\frac{a \left (2 a \left (A+2 B x^3\right )+A b x^3\right )}{x^6}\right )}{12 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^7,x]
[Out]
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Maple [A] time = 0.012, size = 96, normalized size = 1.1 \[ A \left ( -{\frac{1}{6\,{x}^{6}}\sqrt{b{x}^{3}+a}}-{\frac{b}{12\,a{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{{b}^{2}}{12}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \right ) +B \left ( -{\frac{1}{3\,{x}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{b}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^3+A)*(b*x^3+a)^(1/2)/x^7,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((B*x^3 + A)*sqrt(b*x^3 + a)/x^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.27319, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, B a b - A b^{2}\right )} x^{6} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \,{\left ({\left (4 \, B a + A b\right )} x^{3} + 2 \, A a\right )} \sqrt{b x^{3} + a} \sqrt{a}}{24 \, a^{\frac{3}{2}} x^{6}}, \frac{{\left (4 \, B a b - A b^{2}\right )} x^{6} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) -{\left ({\left (4 \, B a + A b\right )} x^{3} + 2 \, A a\right )} \sqrt{b x^{3} + a} \sqrt{-a}}{12 \, \sqrt{-a} a x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^7,x, algorithm="fricas")
[Out]
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Sympy [A] time = 68.8284, size = 160, normalized size = 1.82 \[ - \frac{A a}{6 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{A \sqrt{b}}{4 x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{A b^{\frac{3}{2}}}{12 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{A b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{12 a^{\frac{3}{2}}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} - \frac{B b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**3+A)*(b*x**3+a)**(1/2)/x**7,x)
[Out]
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GIAC/XCAS [A] time = 0.221622, size = 162, normalized size = 1.84 \[ \frac{\frac{{\left (4 \, B a b^{2} - A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x^{3} + a} B a^{2} b^{2} +{\left (b x^{3} + a\right )}^{\frac{3}{2}} A b^{3} + \sqrt{b x^{3} + a} A a b^{3}}{a b^{2} x^{6}}}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^7,x, algorithm="giac")
[Out]