3.536 \(\int (e x)^{5/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=404 \[ -\frac{27\ 3^{3/4} a^{11/3} e^2 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{11264 b^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{81 a^3 e^2 \sqrt{e x} \sqrt{a+b x^3} (4 A b-a B)}{5632 b^2}+\frac{27 a^2 (e x)^{7/2} \sqrt{a+b x^3} (4 A b-a B)}{1408 b e}+\frac{15 a (e x)^{7/2} \left (a+b x^3\right )^{3/2} (4 A b-a B)}{704 b e}+\frac{(e x)^{7/2} \left (a+b x^3\right )^{5/2} (4 A b-a B)}{44 b e}+\frac{B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e} \]

[Out]

(81*a^3*(4*A*b - a*B)*e^2*Sqrt[e*x]*Sqrt[a + b*x^3])/(5632*b^2) + (27*a^2*(4*A*b
- a*B)*(e*x)^(7/2)*Sqrt[a + b*x^3])/(1408*b*e) + (15*a*(4*A*b - a*B)*(e*x)^(7/2
)*(a + b*x^3)^(3/2))/(704*b*e) + ((4*A*b - a*B)*(e*x)^(7/2)*(a + b*x^3)^(5/2))/(
44*b*e) + (B*(e*x)^(7/2)*(a + b*x^3)^(7/2))/(14*b*e) - (27*3^(3/4)*a^(11/3)*(4*A
*b - a*B)*e^2*Sqrt[e*x]*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3)
+ (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4
])/(11264*b^2*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^
(1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi [A]  time = 0.882428, antiderivative size = 404, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{27\ 3^{3/4} a^{11/3} e^2 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{11264 b^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{81 a^3 e^2 \sqrt{e x} \sqrt{a+b x^3} (4 A b-a B)}{5632 b^2}+\frac{27 a^2 (e x)^{7/2} \sqrt{a+b x^3} (4 A b-a B)}{1408 b e}+\frac{15 a (e x)^{7/2} \left (a+b x^3\right )^{3/2} (4 A b-a B)}{704 b e}+\frac{(e x)^{7/2} \left (a+b x^3\right )^{5/2} (4 A b-a B)}{44 b e}+\frac{B (e x)^{7/2} \left (a+b x^3\right )^{7/2}}{14 b e} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(5/2)*(a + b*x^3)^(5/2)*(A + B*x^3),x]

[Out]

(81*a^3*(4*A*b - a*B)*e^2*Sqrt[e*x]*Sqrt[a + b*x^3])/(5632*b^2) + (27*a^2*(4*A*b
- a*B)*(e*x)^(7/2)*Sqrt[a + b*x^3])/(1408*b*e) + (15*a*(4*A*b - a*B)*(e*x)^(7/2
)*(a + b*x^3)^(3/2))/(704*b*e) + ((4*A*b - a*B)*(e*x)^(7/2)*(a + b*x^3)^(5/2))/(
44*b*e) + (B*(e*x)^(7/2)*(a + b*x^3)^(7/2))/(14*b*e) - (27*3^(3/4)*a^(11/3)*(4*A
*b - a*B)*e^2*Sqrt[e*x]*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)^2]*EllipticF[ArcCos[(a^(1/3)
+ (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)], (2 + Sqrt[3])/4
])/(11264*b^2*Sqrt[(b^(1/3)*x*(a^(1/3) + b^(1/3)*x))/(a^(1/3) + (1 + Sqrt[3])*b^
(1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi in Sympy [A]  time = 52.6039, size = 360, normalized size = 0.89 \[ \frac{B \left (e x\right )^{\frac{7}{2}} \left (a + b x^{3}\right )^{\frac{7}{2}}}{14 b e} - \frac{27 \cdot 3^{\frac{3}{4}} a^{\frac{11}{3}} e^{2} \sqrt{e x} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (4 A b - B a\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{11264 b^{2} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \sqrt{a + b x^{3}}} + \frac{81 a^{3} e^{2} \sqrt{e x} \sqrt{a + b x^{3}} \left (4 A b - B a\right )}{5632 b^{2}} + \frac{27 a^{2} \left (e x\right )^{\frac{7}{2}} \sqrt{a + b x^{3}} \left (4 A b - B a\right )}{1408 b e} + \frac{15 a \left (e x\right )^{\frac{7}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}} \left (4 A b - B a\right )}{704 b e} + \frac{\left (e x\right )^{\frac{7}{2}} \left (a + b x^{3}\right )^{\frac{5}{2}} \left (4 A b - B a\right )}{44 b e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(5/2)*(b*x**3+a)**(5/2)*(B*x**3+A),x)

[Out]

B*(e*x)**(7/2)*(a + b*x**3)**(7/2)/(14*b*e) - 27*3**(3/4)*a**(11/3)*e**2*sqrt(e*
x)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a**(1/3) + b**(1/3)*x*
(1 + sqrt(3)))**2)*(a**(1/3) + b**(1/3)*x)*(4*A*b - B*a)*elliptic_f(acos((a**(1/
3) + b**(1/3)*x*(-sqrt(3) + 1))/(a**(1/3) + b**(1/3)*x*(1 + sqrt(3)))), sqrt(3)/
4 + 1/2)/(11264*b**2*sqrt(b**(1/3)*x*(a**(1/3) + b**(1/3)*x)/(a**(1/3) + b**(1/3
)*x*(1 + sqrt(3)))**2)*sqrt(a + b*x**3)) + 81*a**3*e**2*sqrt(e*x)*sqrt(a + b*x**
3)*(4*A*b - B*a)/(5632*b**2) + 27*a**2*(e*x)**(7/2)*sqrt(a + b*x**3)*(4*A*b - B*
a)/(1408*b*e) + 15*a*(e*x)**(7/2)*(a + b*x**3)**(3/2)*(4*A*b - B*a)/(704*b*e) +
(e*x)**(7/2)*(a + b*x**3)**(5/2)*(4*A*b - B*a)/(44*b*e)

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Mathematica [C]  time = 0.921583, size = 276, normalized size = 0.68 \[ \frac{e^2 \sqrt{e x} \left (-\sqrt [3]{-a} \left (a+b x^3\right ) \left (567 a^4 B-324 a^3 b \left (7 A+B x^3\right )-8 a^2 b^2 x^3 \left (1246 A+727 B x^3\right )-32 a b^3 x^6 \left (329 A+236 B x^3\right )-256 b^4 x^9 \left (14 A+11 B x^3\right )\right )+189 i 3^{3/4} a^4 \sqrt [3]{b} x \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{-a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} x}} \sqrt{\frac{\frac{(-a)^{2/3}}{b^{2/3}}+\frac{\sqrt [3]{-a} x}{\sqrt [3]{b}}+x^2}{x^2}} (4 A b-a B) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{39424 \sqrt [3]{-a} b^2 \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*x)^(5/2)*(a + b*x^3)^(5/2)*(A + B*x^3),x]

[Out]

(e^2*Sqrt[e*x]*(-((-a)^(1/3)*(a + b*x^3)*(567*a^4*B - 324*a^3*b*(7*A + B*x^3) -
256*b^4*x^9*(14*A + 11*B*x^3) - 32*a*b^3*x^6*(329*A + 236*B*x^3) - 8*a^2*b^2*x^3
*(1246*A + 727*B*x^3))) + (189*I)*3^(3/4)*a^4*b^(1/3)*(4*A*b - a*B)*x*Sqrt[((-1)
^(5/6)*((-a)^(1/3) - b^(1/3)*x))/(b^(1/3)*x)]*Sqrt[((-a)^(2/3)/b^(2/3) + ((-a)^(
1/3)*x)/b^(1/3) + x^2)/x^2]*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-a)^(1/3))/(
b^(1/3)*x)]/3^(1/4)], (-1)^(1/3)]))/(39424*(-a)^(1/3)*b^2*Sqrt[a + b*x^3])

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Maple [C]  time = 0.067, size = 5063, normalized size = 12.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(5/2)*(b*x^3+a)^(5/2)*(B*x^3+A),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B b^{2} e^{2} x^{11} +{\left (2 \, B a b + A b^{2}\right )} e^{2} x^{8} +{\left (B a^{2} + 2 \, A a b\right )} e^{2} x^{5} + A a^{2} e^{2} x^{2}\right )} \sqrt{b x^{3} + a} \sqrt{e x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^(5/2),x, algorithm="fricas")

[Out]

integral((B*b^2*e^2*x^11 + (2*B*a*b + A*b^2)*e^2*x^8 + (B*a^2 + 2*A*a*b)*e^2*x^5
+ A*a^2*e^2*x^2)*sqrt(b*x^3 + a)*sqrt(e*x), x)

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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)**(5/2)*(b*x**3+a)**(5/2)*(B*x**3+A),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^(5/2), x)