Optimal. Leaf size=152 \[ \frac{a (a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{4 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{3/2} (a B+4 A b)}{6 a e^4}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (a B+4 A b)}{4 e^4}-\frac{2 A \left (a+b x^3\right )^{5/2}}{3 a e (e x)^{3/2}} \]
[Out]
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Rubi [A] time = 0.306196, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a (a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{4 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \left (a+b x^3\right )^{3/2} (a B+4 A b)}{6 a e^4}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (a B+4 A b)}{4 e^4}-\frac{2 A \left (a+b x^3\right )^{5/2}}{3 a e (e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^3)^(3/2)*(A + B*x^3))/(e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 28.4392, size = 138, normalized size = 0.91 \[ - \frac{2 A \left (a + b x^{3}\right )^{\frac{5}{2}}}{3 a e \left (e x\right )^{\frac{3}{2}}} + \frac{a \left (4 A b + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{4 \sqrt{b} e^{\frac{5}{2}}} + \frac{\left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}} \left (4 A b + B a\right )}{4 e^{4}} + \frac{\left (e x\right )^{\frac{3}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}} \left (4 A b + B a\right )}{6 a e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a)**(3/2)*(B*x**3+A)/(e*x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.239739, size = 111, normalized size = 0.73 \[ \frac{x \left (3 a x^3 \sqrt{\frac{a}{x^3}+b} (a B+4 A b) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )+\sqrt{b} \left (a+b x^3\right ) \left (-8 a A+5 a B x^3+4 A b x^3+2 b B x^6\right )\right )}{12 \sqrt{b} (e x)^{5/2} \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^3)^(3/2)*(A + B*x^3))/(e*x)^(5/2),x]
[Out]
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Maple [C] time = 0.048, size = 7108, normalized size = 46.8 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a)^(3/2)*(B*x^3+A)/(e*x)^(5/2),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((B*x^3 + A)*(b*x^3 + a)^(3/2)/(e*x)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.658036, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (B a^{2} + 4 \, A a b\right )} e x^{2} \log \left (-4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} -{\left (8 \, b^{2} x^{6} + 8 \, a b x^{3} + a^{2}\right )} \sqrt{b e}\right ) + 4 \,{\left (2 \, B b x^{6} +{\left (5 \, B a + 4 \, A b\right )} x^{3} - 8 \, A a\right )} \sqrt{b x^{3} + a} \sqrt{b e} \sqrt{e x}}{48 \, \sqrt{b e} e^{3} x^{2}}, \frac{3 \,{\left (B a^{2} + 4 \, A a b\right )} e x^{2} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{-b e} \sqrt{e x} x}{2 \, b e x^{3} + a e}\right ) + 2 \,{\left (2 \, B b x^{6} +{\left (5 \, B a + 4 \, A b\right )} x^{3} - 8 \, A a\right )} \sqrt{b x^{3} + a} \sqrt{-b e} \sqrt{e x}}{24 \, \sqrt{-b e} e^{3} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)/(e*x)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 130.744, size = 289, normalized size = 1.9 \[ - \frac{2 A a^{\frac{3}{2}}}{3 e^{\frac{5}{2}} x^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{A \sqrt{a} b x^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{3 e^{\frac{5}{2}}} - \frac{2 A \sqrt{a} b x^{\frac{3}{2}}}{3 e^{\frac{5}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{A a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{e^{\frac{5}{2}}} + \frac{B a^{\frac{3}{2}} x^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{3 e^{\frac{5}{2}}} + \frac{B a^{\frac{3}{2}} x^{\frac{3}{2}}}{12 e^{\frac{5}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{B \sqrt{a} b x^{\frac{9}{2}}}{4 e^{\frac{5}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{B a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{4 \sqrt{b} e^{\frac{5}{2}}} + \frac{B b^{2} x^{\frac{15}{2}}}{6 \sqrt{a} e^{\frac{5}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**3+a)**(3/2)*(B*x**3+A)/(e*x)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{\left (e x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)/(e*x)^(5/2),x, algorithm="giac")
[Out]