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

Optimal. Leaf size=624 \[ -\frac{4 \left (1-\sqrt{3}\right ) \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (10 A b-a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{27 \sqrt [4]{3} a^{8/3} b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{8 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (10 A b-a B) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{9\ 3^{3/4} a^{8/3} b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{8 \left (1+\sqrt{3}\right ) \sqrt{e x} \sqrt{a+b x^3} (10 A b-a B)}{27 a^3 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}-\frac{8 (e x)^{5/2} (10 A b-a B)}{27 a^3 e^4 \sqrt{a+b x^3}}-\frac{2 (e x)^{5/2} (10 A b-a B)}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac{2 A}{a e \sqrt{e x} \left (a+b x^3\right )^{3/2}} \]

[Out]

(-2*A)/(a*e*Sqrt[e*x]*(a + b*x^3)^(3/2)) - (2*(10*A*b - a*B)*(e*x)^(5/2))/(9*a^2
*e^4*(a + b*x^3)^(3/2)) - (8*(10*A*b - a*B)*(e*x)^(5/2))/(27*a^3*e^4*Sqrt[a + b*
x^3]) + (8*(1 + Sqrt[3])*(10*A*b - a*B)*Sqrt[e*x]*Sqrt[a + b*x^3])/(27*a^3*b^(2/
3)*e^2*(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)) - (8*(10*A*b - a*B)*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]*EllipticE[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])/(9*3^(3/4)*a^(8/3)*b^(2/3)*
e^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]) - (4*(1 - Sqrt[3])*(10*A*b - a*B)*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])/(27*3^(1/4)*a^(8/3)*b^(2/3)*e^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 = 1.48848, antiderivative size = 624, 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 \[ -\frac{4 \left (1-\sqrt{3}\right ) \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (10 A b-a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{27 \sqrt [4]{3} a^{8/3} b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{8 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (10 A b-a B) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{9\ 3^{3/4} a^{8/3} b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{8 \left (1+\sqrt{3}\right ) \sqrt{e x} \sqrt{a+b x^3} (10 A b-a B)}{27 a^3 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}-\frac{8 (e x)^{5/2} (10 A b-a B)}{27 a^3 e^4 \sqrt{a+b x^3}}-\frac{2 (e x)^{5/2} (10 A b-a B)}{9 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac{2 A}{a e \sqrt{e x} \left (a+b x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(a*e*Sqrt[e*x]*(a + b*x^3)^(3/2)) - (2*(10*A*b - a*B)*(e*x)^(5/2))/(9*a^2
*e^4*(a + b*x^3)^(3/2)) - (8*(10*A*b - a*B)*(e*x)^(5/2))/(27*a^3*e^4*Sqrt[a + b*
x^3]) + (8*(1 + Sqrt[3])*(10*A*b - a*B)*Sqrt[e*x]*Sqrt[a + b*x^3])/(27*a^3*b^(2/
3)*e^2*(a^(1/3) + (1 + Sqrt[3])*b^(1/3)*x)) - (8*(10*A*b - a*B)*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]*EllipticE[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])/(9*3^(3/4)*a^(8/3)*b^(2/3)*
e^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]) - (4*(1 - Sqrt[3])*(10*A*b - a*B)*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])/(27*3^(1/4)*a^(8/3)*b^(2/3)*e^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 = 86.4614, size = 571, normalized size = 0.92 \[ - \frac{2 A}{a e \sqrt{e x} \left (a + b x^{3}\right )^{\frac{3}{2}}} - \frac{2 \left (e x\right )^{\frac{5}{2}} \left (10 A b - B a\right )}{9 a^{2} e^{4} \left (a + b x^{3}\right )^{\frac{3}{2}}} - \frac{8 \left (e x\right )^{\frac{5}{2}} \left (10 A b - B a\right )}{27 a^{3} e^{4} \sqrt{a + b x^{3}}} + \frac{\sqrt{e x} \left (\frac{16}{27} + \frac{16 \sqrt{3}}{27}\right ) \sqrt{a + b x^{3}} \left (10 A b - B a\right )}{2 a^{3} b^{\frac{2}{3}} e^{2} \left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )} - \frac{8 \sqrt [4]{3} \sqrt{e x} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (10 A b - B a\right ) E\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{27 a^{\frac{8}{3}} b^{\frac{2}{3}} e^{2} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \sqrt{a + b x^{3}}} - \frac{4 \cdot 3^{\frac{3}{4}} \sqrt{e x} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (- \sqrt{3} + 1\right ) \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (10 A b - B a\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{81 a^{\frac{8}{3}} b^{\frac{2}{3}} e^{2} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \sqrt{a + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/(a*e*sqrt(e*x)*(a + b*x**3)**(3/2)) - 2*(e*x)**(5/2)*(10*A*b - B*a)/(9*a**2
*e**4*(a + b*x**3)**(3/2)) - 8*(e*x)**(5/2)*(10*A*b - B*a)/(27*a**3*e**4*sqrt(a
+ b*x**3)) + sqrt(e*x)*(16/27 + 16*sqrt(3)/27)*sqrt(a + b*x**3)*(10*A*b - B*a)/(
2*a**3*b**(2/3)*e**2*(a**(1/3) + b**(1/3)*x*(1 + sqrt(3)))) - 8*3**(1/4)*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)*(10*A*b - B*a)*elliptic_e(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)/(27*a**(8/3)*b**(2/3)*e**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)) - 4*3**(3/4)*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)*(-sqrt(3) + 1)*(a**(1/3) + b**(1/3)*x)*(10*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)/(81*a**(8/3)*b**(2/3)*e**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))

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Mathematica [A]  time = 4.27213, size = 401, normalized size = 0.64 \[ \frac{2 x \left (\frac{4 (10 A b-a B) \left (-(-1)^{2/3} a^{2/3} \sqrt{\frac{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{b} x \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt [3]{b} x}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2 \left (\left (1+\sqrt [3]{-1}\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{b} x}{\sqrt [3]{b} x+\sqrt [3]{a}}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )-\left (1+(-1)^{2/3}\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{b} x}{\sqrt [3]{b} x+\sqrt [3]{a}}}\right )|\frac{\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )-\left ((-1)^{2/3}-1\right ) \sqrt [3]{a} \sqrt [3]{b} x \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )\right )}{\left ((-1)^{2/3}-1\right ) \sqrt [3]{a} b}+\frac{a^2 \left (7 B x^3-27 A\right )+a \left (4 b B x^6-70 A b x^3\right )-40 A b^2 x^6}{a+b x^3}\right )}{27 a^3 (e x)^{3/2} \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*((-40*A*b^2*x^6 + a^2*(-27*A + 7*B*x^3) + a*(-70*A*b*x^3 + 4*b*B*x^6))/(a +
b*x^3) + (4*(10*A*b - a*B)*(-((-1 + (-1)^(2/3))*a^(1/3)*b^(1/3)*x*((-1)^(1/3)*a
^(1/3) - b^(1/3)*x)*((-1)^(2/3)*a^(1/3) + b^(1/3)*x)) - (-1)^(2/3)*a^(2/3)*(a^(1
/3) + b^(1/3)*x)^2*Sqrt[((1 + (-1)^(1/3))*b^(1/3)*x*(a^(1/3) - (-1)^(1/3)*b^(1/3
)*x))/(a^(1/3) + b^(1/3)*x)^2]*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/(a^(1/3) +
b^(1/3)*x)]*((1 + (-1)^(1/3))*EllipticE[ArcSin[Sqrt[((1 + (-1)^(1/3))*b^(1/3)*x)
/(a^(1/3) + b^(1/3)*x)]], (-1)^(1/3)/(-1 + (-1)^(1/3))] - (1 + (-1)^(2/3))*Ellip
ticF[ArcSin[Sqrt[((1 + (-1)^(1/3))*b^(1/3)*x)/(a^(1/3) + b^(1/3)*x)]], (-1)^(1/3
)/(-1 + (-1)^(1/3))])))/((-1 + (-1)^(2/3))*a^(1/3)*b)))/(27*a^3*(e*x)^(3/2)*Sqrt
[a + b*x^3])

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Maple [C]  time = 0.08, size = 10961, normalized size = 17.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{3} + A}{{\left (b^{2} e x^{7} + 2 \, a b e x^{4} + a^{2} e x\right )} \sqrt{b x^{3} + a} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^3 + A)/((b^2*e*x^7 + 2*a*b*e*x^4 + a^2*e*x)*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((B*x**3+A)/(e*x)**(3/2)/(b*x**3+a)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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