Optimal. Leaf size=115 \[ -\frac{\left (a+b x^3\right )^{3/2} (4 a B+A b)}{12 a x^3}+\frac{b \sqrt{a+b x^3} (4 a B+A b)}{4 a}-\frac{b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{A \left (a+b x^3\right )^{5/2}}{6 a x^6} \]
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Rubi [A] time = 0.251023, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\left (a+b x^3\right )^{3/2} (4 a B+A b)}{12 a x^3}+\frac{b \sqrt{a+b x^3} (4 a B+A b)}{4 a}-\frac{b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 \sqrt{a}}-\frac{A \left (a+b x^3\right )^{5/2}}{6 a x^6} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^3)^(3/2)*(A + B*x^3))/x^7,x]
[Out]
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Rubi in Sympy [A] time = 17.1355, size = 100, normalized size = 0.87 \[ - \frac{A \left (a + b x^{3}\right )^{\frac{5}{2}}}{6 a x^{6}} + \frac{b \sqrt{a + b x^{3}} \left (A b + 4 B a\right )}{4 a} - \frac{\left (a + b x^{3}\right )^{\frac{3}{2}} \left (A b + 4 B a\right )}{12 a x^{3}} - \frac{b \left (A b + 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{4 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a)**(3/2)*(B*x**3+A)/x**7,x)
[Out]
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Mathematica [A] time = 0.417434, size = 84, normalized size = 0.73 \[ \frac{1}{12} \sqrt{a+b x^3} \left (-\frac{4 a B+5 A b}{x^3}-\frac{3 b (4 a B+A b) \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{a \sqrt{\frac{b x^3}{a}+1}}-\frac{2 a A}{x^6}+8 b B\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/x^7,x]
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Maple [A] time = 0.012, size = 107, normalized size = 0.9 \[ A \left ( -{\frac{a}{6\,{x}^{6}}\sqrt{b{x}^{3}+a}}-{\frac{5\,b}{12\,{x}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{{b}^{2}}{4}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \right ) +B \left ( -{\frac{a}{3\,{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{2\,b}{3}\sqrt{b{x}^{3}+a}}-\sqrt{a}b{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a)^(3/2)*(B*x^3+A)/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)*(b*x^3 + a)^(3/2)/x^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.264952, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (4 \, B a b + A b^{2}\right )} x^{6} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \,{\left (8 \, B b x^{6} -{\left (4 \, B a + 5 \, A b\right )} x^{3} - 2 \, A a\right )} \sqrt{b x^{3} + a} \sqrt{a}}{24 \, \sqrt{a} x^{6}}, \frac{3 \,{\left (4 \, B a b + A b^{2}\right )} x^{6} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left (8 \, B b x^{6} -{\left (4 \, B a + 5 \, A b\right )} x^{3} - 2 \, A a\right )} \sqrt{b x^{3} + a} \sqrt{-a}}{12 \, \sqrt{-a} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)/x^7,x, algorithm="fricas")
[Out]
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Sympy [A] time = 88.5395, size = 243, normalized size = 2.11 \[ - \frac{A a^{2}}{6 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{A a \sqrt{b}}{4 x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} - \frac{A b^{\frac{3}{2}}}{12 x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{A b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{4 \sqrt{a}} - B \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )} - \frac{B a \sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} + \frac{2 B a \sqrt{b}}{3 x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{2 B b^{\frac{3}{2}} x^{\frac{3}{2}}}{3 \sqrt{\frac{a}{b x^{3}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a)**(3/2)*(B*x**3+A)/x**7,x)
[Out]
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GIAC/XCAS [A] time = 0.219918, size = 177, normalized size = 1.54 \[ \frac{8 \, \sqrt{b x^{3} + a} B b^{2} + \frac{3 \,{\left (4 \, B a b^{2} + A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x^{3} + a} B a^{2} b^{2} + 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A b^{3} - 3 \, \sqrt{b x^{3} + a} A a b^{3}}{b^{2} x^{6}}}{12 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)/x^7,x, algorithm="giac")
[Out]