Optimal. Leaf size=241 \[ -\frac{a^4 e^{7/2} (10 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{384 b^{5/2}}+\frac{a^3 e^2 (e x)^{3/2} \sqrt{a+b x^3} (10 A b-3 a B)}{384 b^2}+\frac{a^2 (e x)^{9/2} \sqrt{a+b x^3} (10 A b-3 a B)}{192 b e}+\frac{(e x)^{9/2} \left (a+b x^3\right )^{5/2} (10 A b-3 a B)}{120 b e}+\frac{a (e x)^{9/2} \left (a+b x^3\right )^{3/2} (10 A b-3 a B)}{144 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e} \]
[Out]
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Rubi [A] time = 0.471333, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{a^4 e^{7/2} (10 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{384 b^{5/2}}+\frac{a^3 e^2 (e x)^{3/2} \sqrt{a+b x^3} (10 A b-3 a B)}{384 b^2}+\frac{a^2 (e x)^{9/2} \sqrt{a+b x^3} (10 A b-3 a B)}{192 b e}+\frac{(e x)^{9/2} \left (a+b x^3\right )^{5/2} (10 A b-3 a B)}{120 b e}+\frac{a (e x)^{9/2} \left (a+b x^3\right )^{3/2} (10 A b-3 a B)}{144 b e}+\frac{B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e} \]
Antiderivative was successfully verified.
[In] Int[(e*x)^(7/2)*(a + b*x^3)^(5/2)*(A + B*x^3),x]
[Out]
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Rubi in Sympy [A] time = 41.4365, size = 218, normalized size = 0.9 \[ \frac{B \left (e x\right )^{\frac{9}{2}} \left (a + b x^{3}\right )^{\frac{7}{2}}}{15 b e} - \frac{a^{4} e^{\frac{7}{2}} \left (10 A b - 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{384 b^{\frac{5}{2}}} + \frac{a^{3} e^{2} \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}} \left (10 A b - 3 B a\right )}{384 b^{2}} + \frac{a^{2} \left (e x\right )^{\frac{9}{2}} \sqrt{a + b x^{3}} \left (10 A b - 3 B a\right )}{192 b e} + \frac{a \left (e x\right )^{\frac{9}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}} \left (10 A b - 3 B a\right )}{144 b e} + \frac{\left (e x\right )^{\frac{9}{2}} \left (a + b x^{3}\right )^{\frac{5}{2}} \left (10 A b - 3 B a\right )}{120 b e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(7/2)*(b*x**3+a)**(5/2)*(B*x**3+A),x)
[Out]
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Mathematica [A] time = 0.298754, size = 168, normalized size = 0.7 \[ \frac{e^2 (e x)^{3/2} \left (15 a^4 \sqrt{\frac{a}{x^3}+b} (3 a B-10 A b) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )-\sqrt{b} \left (a+b x^3\right ) \left (45 a^4 B-30 a^3 b \left (5 A+B x^3\right )-4 a^2 b^2 x^3 \left (295 A+186 B x^3\right )-16 a b^3 x^6 \left (85 A+63 B x^3\right )-96 b^4 x^9 \left (5 A+4 B x^3\right )\right )\right )}{5760 b^{5/2} \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
[In] Integrate[(e*x)^(7/2)*(a + b*x^3)^(5/2)*(A + B*x^3),x]
[Out]
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Maple [C] time = 0.072, size = 8117, normalized size = 33.7 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(7/2)*(b*x^3+a)^(5/2)*(B*x^3+A),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)^(5/2)*(e*x)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.688862, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} e^{3} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e + 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right ) - 4 \,{\left (384 \, B b^{4} e^{3} x^{13} + 48 \,{\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} e^{3} x^{10} + 8 \,{\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} e^{3} x^{7} + 10 \,{\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} e^{3} x^{4} - 15 \,{\left (3 \, B a^{4} - 10 \, A a^{3} b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{23040 \, b^{2}}, \frac{15 \,{\left (3 \, B a^{5} - 10 \, A a^{4} b\right )} e^{3} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right ) + 2 \,{\left (384 \, B b^{4} e^{3} x^{13} + 48 \,{\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} e^{3} x^{10} + 8 \,{\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} e^{3} x^{7} + 10 \,{\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} e^{3} x^{4} - 15 \,{\left (3 \, B a^{4} - 10 \, A a^{3} b\right )} e^{3} x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{11520 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^(7/2),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((e*x)**(7/2)*(b*x**3+a)**(5/2)*(B*x**3+A),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^(7/2),x, algorithm="giac")
[Out]