∖int√ ____ a+bx3 ∖left(A+Bx3∖right) x11∕2 ∖,dx

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

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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

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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))

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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

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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)

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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

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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)]

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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)*(b*x**3+a)**(1/2)/x**(11/2),x)

[Out]

Timed out

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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

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