3.525 \(\int \frac{\sqrt{a+b x^3} \left (A+B x^3\right )}{x^{11/2}} \, dx\)
Optimal. Leaf size=79 \[ -\frac{2 A \left (a+b x^3\right )^{3/2}}{9 a x^{9/2}}-\frac{2 B \sqrt{a+b x^3}}{3 x^{3/2}}+\frac{2}{3} \sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a+b x^3}}\right ) \]
[Out]
(-2*B*Sqrt[a + b*x^3])/(3*x^(3/2)) - (2*A*(a + b*x^3)^(3/2))/(9*a*x^(9/2)) + (2*
Sqrt[b]*B*ArcTanh[(Sqrt[b]*x^(3/2))/Sqrt[a + b*x^3]])/3
_______________________________________________________________________________________
Rubi [A] time = 0.14234, antiderivative size = 79, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25
\[ -\frac{2 A \left (a+b x^3\right )^{3/2}}{9 a x^{9/2}}-\frac{2 B \sqrt{a+b x^3}}{3 x^{3/2}}+\frac{2}{3} \sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a+b x^3}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x^3]*(A + B*x^3))/x^(11/2),x]
[Out]
(-2*B*Sqrt[a + b*x^3])/(3*x^(3/2)) - (2*A*(a + b*x^3)^(3/2))/(9*a*x^(9/2)) + (2*
Sqrt[b]*B*ArcTanh[(Sqrt[b]*x^(3/2))/Sqrt[a + b*x^3]])/3
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.4074, size = 73, normalized size = 0.92 \[ - \frac{2 A \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 a x^{\frac{9}{2}}} + \frac{2 B \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a + b x^{3}}} \right )}}{3} - \frac{2 B \sqrt{a + b x^{3}}}{3 x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**3+A)*(b*x**3+a)**(1/2)/x**(11/2),x)
[Out]
-2*A*(a + b*x**3)**(3/2)/(9*a*x**(9/2)) + 2*B*sqrt(b)*atanh(sqrt(b)*x**(3/2)/sqr
t(a + b*x**3))/3 - 2*B*sqrt(a + b*x**3)/(3*x**(3/2))
_______________________________________________________________________________________
Mathematica [A] time = 0.170781, size = 74, normalized size = 0.94 \[ \frac{2}{3} \sqrt{b} B \tanh ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a+b x^3}}\right )-\frac{2 \sqrt{a+b x^3} \left (a \left (A+3 B x^3\right )+A b x^3\right )}{9 a x^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^(11/2),x]
[Out]
(-2*Sqrt[a + b*x^3]*(A*b*x^3 + a*(A + 3*B*x^3)))/(9*a*x^(9/2)) + (2*Sqrt[b]*B*Ar
cTanh[(Sqrt[b]*x^(3/2))/Sqrt[a + b*x^3]])/3
_______________________________________________________________________________________
Maple [C] time = 0.091, size = 3759, normalized size = 47.6 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^3+A)*(b*x^3+a)^(1/2)/x^(11/2),x)
[Out]
-2/9*(b*x^3+a)^(1/2)/x^(9/2)/b*(36*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))*(-a*b^2)^(1/3)*3^(1/2)*x^6*
a*b-3*A*(1/b^2*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)*((b*x^3+a)*x)^(1/2
)*x^3*b^2+I*A*(1/b^2*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)*((b*x^3+a)*x
)^(1/2)*3^(1/2)*a*b-3*A*(1/b^2*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)*((
b*x^3+a)*x)^(1/2)*a*b+I*A*(1/b^2*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)*
((b*x^3+a)*x)^(1/2)*3^(1/2)*x^3*b^2-18*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))*(-a*b^2)^(2/3)*3^(1/2)*
x^5*a-18*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))*3^(1/2)*x^7*a*b^2-36*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))*(-a*b^2)^(1/3)*3^(1/2)*x^6*a*b+18*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))*3^(1/2)*x^7*a*b^2-18
*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))*x^7*a*b^2+3*
I*B*(1/b^2*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)*((b*x^3+a)*x)^(1/2)*3^
(1/2)*x^3*a*b+18*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))*x^7*a*b^2+36*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))*(-a*b^2)^(1/3)*x^6*a*b-36*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))*(-a*b
^2)^(1/3)*x^6*a*b-18*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))*(-a*b^2)^(2/3)*x^5*a+18*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))*(-a*b^2)^(2/3)*x^5*a+18*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))*(-a*b^2)^(2/3)*3^(1
/2)*x^5*a-9*B*(1/b^2*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)*((b*x^3+a)*x
)^(1/2)*x^3*a*b)/((b*x^3+a)*x)^(1/2)/a/(I*3^(1/2)-3)/(1/b^2*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)
_______________________________________________________________________________________
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)*sqrt(b*x^3 + a)/x^(11/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.372831, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, B a \sqrt{b} x^{5} \log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} - 4 \,{\left (2 \, b x^{4} + a x\right )} \sqrt{b x^{3} + a} \sqrt{b} \sqrt{x} - a^{2}\right ) - 4 \,{\left ({\left (3 \, B a + A b\right )} x^{3} + A a\right )} \sqrt{b x^{3} + a} \sqrt{x}}{18 \, a x^{5}}, \frac{3 \, B a \sqrt{-b} x^{5} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} b x^{\frac{3}{2}}}{{\left (2 \, b x^{3} + a\right )} \sqrt{-b}}\right ) - 2 \,{\left ({\left (3 \, B a + A b\right )} x^{3} + A a\right )} \sqrt{b x^{3} + a} \sqrt{x}}{9 \, a x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^(11/2),x, algorithm="fricas")
[Out]
[1/18*(3*B*a*sqrt(b)*x^5*log(-8*b^2*x^6 - 8*a*b*x^3 - 4*(2*b*x^4 + a*x)*sqrt(b*x
^3 + a)*sqrt(b)*sqrt(x) - a^2) - 4*((3*B*a + A*b)*x^3 + A*a)*sqrt(b*x^3 + a)*sqr
t(x))/(a*x^5), 1/9*(3*B*a*sqrt(-b)*x^5*arctan(2*sqrt(b*x^3 + a)*b*x^(3/2)/((2*b*
x^3 + a)*sqrt(-b))) - 2*((3*B*a + A*b)*x^3 + A*a)*sqrt(b*x^3 + a)*sqrt(x))/(a*x^
5)]
_______________________________________________________________________________________
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)*(b*x**3+a)**(1/2)/x**(11/2),x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.230913, size = 147, normalized size = 1.86 \[ -\frac{2 \, B b \arctan \left (\frac{\sqrt{b + \frac{a}{x^{3}}}}{\sqrt{-b}}\right )}{3 \, \sqrt{-b}} + \frac{2 \,{\left (3 \, B a b \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 3 \, B a \sqrt{-b} \sqrt{b} + A \sqrt{-b} b^{\frac{3}{2}}\right )}}{9 \, a \sqrt{-b}} - \frac{2 \,{\left (3 \, B a^{3} \sqrt{b + \frac{a}{x^{3}}} + A a^{2}{\left (b + \frac{a}{x^{3}}\right )}^{\frac{3}{2}}\right )}}{9 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(b*x^3 + a)/x^(11/2),x, algorithm="giac")
[Out]
-2/3*B*b*arctan(sqrt(b + a/x^3)/sqrt(-b))/sqrt(-b) + 2/9*(3*B*a*b*arctan(sqrt(b)
/sqrt(-b)) + 3*B*a*sqrt(-b)*sqrt(b) + A*sqrt(-b)*b^(3/2))/(a*sqrt(-b)) - 2/9*(3*
B*a^3*sqrt(b + a/x^3) + A*a^2*(b + a/x^3)^(3/2))/a^3