3.566 \(\int \frac{A+B x^3}{(e x)^{7/2} \left (a+b x^3\right )^{5/2}} \, dx\)

Optimal. Leaf size=320 \[ -\frac{16 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (14 A b-5 a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{135 \sqrt [4]{3} a^{10/3} e^4 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{16 \sqrt{e x} (14 A b-5 a B)}{135 a^3 e^4 \sqrt{a+b x^3}}-\frac{2 \sqrt{e x} (14 A b-5 a B)}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac{2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}} \]

[Out]

(-2*A)/(5*a*e*(e*x)^(5/2)*(a + b*x^3)^(3/2)) - (2*(14*A*b - 5*a*B)*Sqrt[e*x])/(4
5*a^2*e^4*(a + b*x^3)^(3/2)) - (16*(14*A*b - 5*a*B)*Sqrt[e*x])/(135*a^3*e^4*Sqrt
[a + b*x^3]) - (16*(14*A*b - 5*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]*Elli
pticF[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])/(135*3^(1/4)*a^(10/3)*e^4*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])

_______________________________________________________________________________________

Rubi [A]  time = 0.649666, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{16 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (14 A b-5 a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{135 \sqrt [4]{3} a^{10/3} e^4 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{16 \sqrt{e x} (14 A b-5 a B)}{135 a^3 e^4 \sqrt{a+b x^3}}-\frac{2 \sqrt{e x} (14 A b-5 a B)}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac{2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^3)/((e*x)^(7/2)*(a + b*x^3)^(5/2)),x]

[Out]

(-2*A)/(5*a*e*(e*x)^(5/2)*(a + b*x^3)^(3/2)) - (2*(14*A*b - 5*a*B)*Sqrt[e*x])/(4
5*a^2*e^4*(a + b*x^3)^(3/2)) - (16*(14*A*b - 5*a*B)*Sqrt[e*x])/(135*a^3*e^4*Sqrt
[a + b*x^3]) - (16*(14*A*b - 5*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]*Elli
pticF[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])/(135*3^(1/4)*a^(10/3)*e^4*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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 37.6652, size = 298, normalized size = 0.93 \[ - \frac{2 A}{5 a e \left (e x\right )^{\frac{5}{2}} \left (a + b x^{3}\right )^{\frac{3}{2}}} - \frac{2 \sqrt{e x} \left (14 A b - 5 B a\right )}{45 a^{2} e^{4} \left (a + b x^{3}\right )^{\frac{3}{2}}} - \frac{16 \sqrt{e x} \left (14 A b - 5 B a\right )}{135 a^{3} e^{4} \sqrt{a + b x^{3}}} - \frac{16 \cdot 3^{\frac{3}{4}} \sqrt{e x} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (14 A b - 5 B a\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{405 a^{\frac{10}{3}} e^{4} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \sqrt{a + b x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/(e*x)**(7/2)/(b*x**3+a)**(5/2),x)

[Out]

-2*A/(5*a*e*(e*x)**(5/2)*(a + b*x**3)**(3/2)) - 2*sqrt(e*x)*(14*A*b - 5*B*a)/(45
*a**2*e**4*(a + b*x**3)**(3/2)) - 16*sqrt(e*x)*(14*A*b - 5*B*a)/(135*a**3*e**4*s
qrt(a + b*x**3)) - 16*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)*(a**(1/3) + b**(1/3)*x)
*(14*A*b - 5*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)/(405*a**(10/3)*e**4*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))

_______________________________________________________________________________________

Mathematica [C]  time = 0.635807, size = 232, normalized size = 0.72 \[ -\frac{2 i \sqrt{e x} \left (3 i \sqrt [3]{-a} \left (a^2 \left (27 A-55 B x^3\right )+2 a b x^3 \left (77 A-20 B x^3\right )+112 A b^2 x^6\right )+16\ 3^{3/4} \sqrt [3]{b} x^4 \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-a}}{\sqrt [3]{b} x}-1\right )} \sqrt{\frac{\frac{(-a)^{2/3}}{b^{2/3}}+\frac{\sqrt [3]{-a} x}{\sqrt [3]{b}}+x^2}{x^2}} \left (a+b x^3\right ) (14 A b-5 a B) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{405 (-a)^{10/3} e^4 x^3 \left (a+b x^3\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(A + B*x^3)/((e*x)^(7/2)*(a + b*x^3)^(5/2)),x]

[Out]

(((-2*I)/405)*Sqrt[e*x]*((3*I)*(-a)^(1/3)*(112*A*b^2*x^6 + a^2*(27*A - 55*B*x^3)
+ 2*a*b*x^3*(77*A - 20*B*x^3)) + 16*3^(3/4)*b^(1/3)*(14*A*b - 5*a*B)*Sqrt[(-1)^
(5/6)*(-1 + (-a)^(1/3)/(b^(1/3)*x))]*x^4*Sqrt[((-a)^(2/3)/b^(2/3) + ((-a)^(1/3)*
x)/b^(1/3) + x^2)/x^2]*(a + b*x^3)*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-a)^(
1/3))/(b^(1/3)*x)]/3^(1/4)], (-1)^(1/3)]))/((-a)^(10/3)*e^4*x^3*(a + b*x^3)^(3/2
))

_______________________________________________________________________________________

Maple [C]  time = 0.063, size = 7299, normalized size = 22.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/(e*x)^(7/2)/(b*x^3+a)^(5/2),x)

[Out]

result too large to display

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

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="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{3} + A}{{\left (b^{2} e^{3} x^{9} + 2 \, a b e^{3} x^{6} + a^{2} e^{3} x^{3}\right )} \sqrt{b x^{3} + a} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^(5/2)*(e*x)^(7/2)),x, algorithm="fricas")

[Out]

integral((B*x^3 + A)/((b^2*e^3*x^9 + 2*a*b*e^3*x^6 + a^2*e^3*x^3)*sqrt(b*x^3 + a
)*sqrt(e*x)), x)

_______________________________________________________________________________________

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)**(7/2)/(b*x**3+a)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

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]

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