3.554 \(\int \frac{\sqrt{e x} \left (A+B x^3\right )}{\left (a+b x^3\right )^{3/2}} \, dx\)
Optimal. Leaf size=85 \[ \frac{2 (e x)^{3/2} (A b-a B)}{3 a b e \sqrt{a+b x^3}}+\frac{2 B \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{3/2}} \]
[Out]
(2*(A*b - a*B)*(e*x)^(3/2))/(3*a*b*e*Sqrt[a + b*x^3]) + (2*B*Sqrt[e]*ArcTanh[(Sq
rt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])])/(3*b^(3/2))
_______________________________________________________________________________________
Rubi [A] time = 0.194777, antiderivative size = 85, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192
\[ \frac{2 (e x)^{3/2} (A b-a B)}{3 a b e \sqrt{a+b x^3}}+\frac{2 B \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 b^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[e*x]*(A + B*x^3))/(a + b*x^3)^(3/2),x]
[Out]
(2*(A*b - a*B)*(e*x)^(3/2))/(3*a*b*e*Sqrt[a + b*x^3]) + (2*B*Sqrt[e]*ArcTanh[(Sq
rt[b]*(e*x)^(3/2))/(e^(3/2)*Sqrt[a + b*x^3])])/(3*b^(3/2))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 19.5298, size = 75, normalized size = 0.88 \[ \frac{2 B \sqrt{e} \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{3 b^{\frac{3}{2}}} + \frac{2 \left (e x\right )^{\frac{3}{2}} \left (A b - B a\right )}{3 a b e \sqrt{a + b x^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**3+A)*(e*x)**(1/2)/(b*x**3+a)**(3/2),x)
[Out]
2*B*sqrt(e)*atanh(sqrt(b)*(e*x)**(3/2)/(e**(3/2)*sqrt(a + b*x**3)))/(3*b**(3/2))
+ 2*(e*x)**(3/2)*(A*b - B*a)/(3*a*b*e*sqrt(a + b*x**3))
_______________________________________________________________________________________
Mathematica [A] time = 0.155037, size = 80, normalized size = 0.94 \[ \frac{2 \sqrt{e x} \left (\frac{\sqrt{b} x^{3/2} (A b-a B)}{a \sqrt{a+b x^3}}+B \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a+b x^3}}\right )\right )}{3 b^{3/2} \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[e*x]*(A + B*x^3))/(a + b*x^3)^(3/2),x]
[Out]
(2*Sqrt[e*x]*((Sqrt[b]*(A*b - a*B)*x^(3/2))/(a*Sqrt[a + b*x^3]) + B*ArcTanh[(Sqr
t[b]*x^(3/2))/Sqrt[a + b*x^3]]))/(3*b^(3/2)*Sqrt[x])
_______________________________________________________________________________________
Maple [C] time = 0.041, size = 3654, normalized size = 43. \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^3+A)*(e*x)^(1/2)/(b*x^3+a)^(3/2),x)
[Out]
2/3*(e*x)^(1/2)/(b*x^3+a)^(1/2)/b^3*(-6*I*B*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-
b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3
^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)
^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*b
/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1
/2)+1)/(I*3^(1/2)-3))^(1/2))*((b*x^3+a)*e*x)^(1/2)*3^(1/2)*x^2*a*b^2+I*A*(1/b^2*
e*x*(-b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(
1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3)))^(1/2)*3^(1/2)*x^2*b^3-I*B*(1/b^2*e*x*
(-b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(1/2)
*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3)))^(1/2)*3^(1/2)*x^2*a*b^2+12*I*B*(-(I*3^(1/
2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+
2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a
*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*Ell
ipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),((I*3^(1/2
)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*((b*x^3+a)*e*x)^(1/2)*(-a
*b^2)^(1/3)*3^(1/2)*x*a*b-12*I*B*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2
)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(
-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*
3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*b/(I*3^(1/2
)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*
3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*((b*x^3+a)*e*x)^(1/2)*(-a*b^2)^(1
/3)*3^(1/2)*x*a*b+6*B*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(
1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^
2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/
(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(
-a*b^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^
(1/2))*((b*x^3+a)*e*x)^(1/2)*x^2*a*b^2+6*I*B*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(
-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*
3^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2
)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x
*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^
(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*((b*x^3+a)*e*x)^(1/2)
*(-a*b^2)^(2/3)*3^(1/2)*a-6*B*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(
1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*
x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(
1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1
)/(-b*x+(-a*b^2)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(
1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*((b*x^3+a)*e*x)^(1/2)*x^2*a*b^2-12*B
*(-a*b^2)^(1/3)*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*(
(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/
3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+
(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2
)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))
*((b*x^3+a)*e*x)^(1/2)*x*a*b+12*B*(-a*b^2)^(1/3)*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-
1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))
/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a
*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-
3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((
I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*((b*x^3+a)*e*x)^(
1/2)*x*a*b-6*I*B*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*
((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1
/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x
+(-a*b^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^
2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2)
)*((b*x^3+a)*e*x)^(1/2)*(-a*b^2)^(2/3)*3^(1/2)*a+6*B*(-a*b^2)^(2/3)*(-(I*3^(1/2)
-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*
b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b
^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*Ellip
ticF((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),((I*3^(1/2)+
3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*((b*x^3+a)*e*x)^(1/2)*a-6*B
*(-a*b^2)^(2/3)*(-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*(
(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/
3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+
(-a*b^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^
2)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(
1/2)+1)/(I*3^(1/2)-3))^(1/2))*((b*x^3+a)*e*x)^(1/2)*a+6*I*B*(-(I*3^(1/2)-3)*x*b/
(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*
b^2)^(1/3))/(I*3^(1/2)+1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-a*b^2)^(1/3
)-2*b*x-(-a*b^2)^(1/3))/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2)*EllipticPi((-
(I*3^(1/2)-3)*x*b/(I*3^(1/2)-1)/(-b*x+(-a*b^2)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^
(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*((b*x^
3+a)*e*x)^(1/2)*3^(1/2)*x^2*a*b^2-3*A*(1/b^2*e*x*(-b*x+(-a*b^2)^(1/3))*(I*3^(1/2
)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^
(1/3)))^(1/2)*x^2*b^3+3*B*(1/b^2*e*x*(-b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(
1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3)))^(1/2
)*x^2*a*b^2)/x/(I*3^(1/2)-3)/(1/b^2*e*x*(-b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2
)^(1/3)+2*b*x+(-a*b^2)^(1/3))*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3)))^(
1/2)/a
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ B \sqrt{e} \int \frac{x^{\frac{7}{2}}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} + \frac{2 \, A \sqrt{e} x^{\frac{3}{2}}}{3 \, \sqrt{b x^{3} + a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(e*x)/(b*x^3 + a)^(3/2),x, algorithm="maxima")
[Out]
B*sqrt(e)*integrate(x^(7/2)/(b*x^3 + a)^(3/2), x) + 2/3*A*sqrt(e)*x^(3/2)/(sqrt(
b*x^3 + a)*a)
_______________________________________________________________________________________
Fricas [A] time = 0.386239, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{b x^{3} + a}{\left (B a - A b\right )} \sqrt{e x} x -{\left (B a b x^{3} + B a^{2}\right )} \sqrt{\frac{e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e - 4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} \sqrt{\frac{e}{b}}\right )}{6 \,{\left (a b^{2} x^{3} + a^{2} b\right )}}, -\frac{2 \, \sqrt{b x^{3} + a}{\left (B a - A b\right )} \sqrt{e x} x -{\left (B a b x^{3} + B a^{2}\right )} \sqrt{-\frac{e}{b}} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{e x} x}{{\left (2 \, b x^{3} + a\right )} \sqrt{-\frac{e}{b}}}\right )}{3 \,{\left (a b^{2} x^{3} + a^{2} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(e*x)/(b*x^3 + a)^(3/2),x, algorithm="fricas")
[Out]
[-1/6*(4*sqrt(b*x^3 + a)*(B*a - A*b)*sqrt(e*x)*x - (B*a*b*x^3 + B*a^2)*sqrt(e/b)
*log(-8*b^2*e*x^6 - 8*a*b*e*x^3 - a^2*e - 4*(2*b^2*x^4 + a*b*x)*sqrt(b*x^3 + a)*
sqrt(e*x)*sqrt(e/b)))/(a*b^2*x^3 + a^2*b), -1/3*(2*sqrt(b*x^3 + a)*(B*a - A*b)*s
qrt(e*x)*x - (B*a*b*x^3 + B*a^2)*sqrt(-e/b)*arctan(2*sqrt(b*x^3 + a)*sqrt(e*x)*x
/((2*b*x^3 + a)*sqrt(-e/b))))/(a*b^2*x^3 + a^2*b)]
_______________________________________________________________________________________
Sympy [A] time = 57.2654, size = 95, normalized size = 1.12 \[ \frac{2 A \sqrt{e} x^{\frac{3}{2}}}{3 a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + B \left (\frac{2 \sqrt{e} \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 b^{\frac{3}{2}}} - \frac{2 \sqrt{e} x^{\frac{3}{2}}}{3 \sqrt{a} b \sqrt{1 + \frac{b x^{3}}{a}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**3+A)*(e*x)**(1/2)/(b*x**3+a)**(3/2),x)
[Out]
2*A*sqrt(e)*x**(3/2)/(3*a**(3/2)*sqrt(1 + b*x**3/a)) + B*(2*sqrt(e)*asinh(sqrt(b
)*x**(3/2)/sqrt(a))/(3*b**(3/2)) - 2*sqrt(e)*x**(3/2)/(3*sqrt(a)*b*sqrt(1 + b*x*
*3/a)))
_______________________________________________________________________________________
GIAC/XCAS [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)*sqrt(e*x)/(b*x^3 + a)^(3/2),x, algorithm="giac")
[Out]
Timed out