∖int√ ____ a+bx3 ∖left(A+Bx3∖right) (ex)7∕2 ∖,dx

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

Optimal. Leaf size=283 \[ \frac{3^{3/4} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (5 a B+4 A b) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{20 \sqrt [3]{a} e^4 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{\sqrt{e x} \sqrt{a+b x^3} (5 a B+4 A b)}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{3/2}}{5 a e (e x)^{5/2}} \]

[Out]

((4*A*b + 5*a*B)*Sqrt[e*x]*Sqrt[a + b*x^3])/(10*a*e^4) - (2*A*(a + b*x^3)^(3/2))

/(5*a*e*(e*x)^(5/2)) + (3^(3/4)*(4*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]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqr

t[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(20*a^(1/3)*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])

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Rubi [A] time = 0.561793, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3^{3/4} \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (5 a B+4 A b) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{20 \sqrt [3]{a} e^4 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{\sqrt{e x} \sqrt{a+b x^3} (5 a B+4 A b)}{10 a e^4}-\frac{2 A \left (a+b x^3\right )^{3/2}}{5 a e (e x)^{5/2}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

((4*A*b + 5*a*B)*Sqrt[e*x]*Sqrt[a + b*x^3])/(10*a*e^4) - (2*A*(a + b*x^3)^(3/2))

/(5*a*e*(e*x)^(5/2)) + (3^(3/4)*(4*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]*EllipticF[ArcCos[(a^(1/3) + (1 - Sqrt[3])*b^(1/3)*x)/(a^(1/3) + (1 + Sqr

t[3])*b^(1/3)*x)], (2 + Sqrt[3])/4])/(20*a^(1/3)*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])

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Rubi in Sympy [A] time = 29.3372, size = 253, normalized size = 0.89 \[ - \frac{2 A \left (a + b x^{3}\right )^{\frac{3}{2}}}{5 a e \left (e x\right )^{\frac{5}{2}}} + \frac{\sqrt{e x} \sqrt{a + b x^{3}} \left (4 A b + 5 B a\right )}{10 a e^{4}} + \frac{3^{\frac{3}{4}} \sqrt{e x} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (4 A b + 5 B a\right ) F\left (\operatorname{acos}{\left (\frac{\sqrt [3]{a} + \sqrt [3]{b} x \left (- \sqrt{3} + 1\right )}{\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{20 \sqrt [3]{a} e^{4} \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} + \sqrt [3]{b} x \left (1 + \sqrt{3}\right )\right )^{2}}} \sqrt{a + b x^{3}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate((B*x**3+A)*(b*x**3+a)**(1/2)/(e*x)**(7/2),x)

[Out]

-2*A*(a + b*x**3)**(3/2)/(5*a*e*(e*x)**(5/2)) + sqrt(e*x)*sqrt(a + b*x**3)*(4*A*

b + 5*B*a)/(10*a*e**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)*(a**(1/3) + b**(1/3)

*x)*(4*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)/(20*a**(1/3)*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))

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Mathematica [C] time = 0.714999, size = 199, normalized size = 0.7 \[ \frac{x \left (\sqrt [3]{-a} \left (a+b x^3\right ) \left (5 B x^3-4 A\right )-i 3^{3/4} \sqrt [3]{b} x^4 \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{-a}-\sqrt [3]{b} x\right )}{\sqrt [3]{b} x}} \sqrt{\frac{\frac{(-a)^{2/3}}{b^{2/3}}+\frac{\sqrt [3]{-a} x}{\sqrt [3]{b}}+x^2}{x^2}} (5 a B+4 A b) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-a}}{\sqrt [3]{b} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{10 \sqrt [3]{-a} (e x)^{7/2} \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((-a)^(1/3)*(a + b*x^3)*(-4*A + 5*B*x^3) - I*3^(3/4)*b^(1/3)*(4*A*b + 5*a*B)*

x^4*Sqrt[((-1)^(5/6)*((-a)^(1/3) - b^(1/3)*x))/(b^(1/3)*x)]*Sqrt[((-a)^(2/3)/b^(

2/3) + ((-a)^(1/3)*x)/b^(1/3) + x^2)/x^2]*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I

*(-a)^(1/3))/(b^(1/3)*x)]/3^(1/4)], (-1)^(1/3)]))/(10*(-a)^(1/3)*(e*x)^(7/2)*Sqr

t[a + b*x^3])

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Maple [C] time = 0.067, size = 3512, normalized size = 12.4 \[ \text{output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((B*x^3+A)*(b*x^3+a)^(1/2)/(e*x)^(7/2),x)

[Out]

-1/10*(b*x^3+a)^(1/2)/x^2/(-a*b^2)^(1/3)/b*(-48*I*A*(-a*b^2)^(1/3)*3^(1/2)*(-(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^4*b^2*e-5*I*B*((b*

x^3+a)*e*x)^(1/2)*(-a*b^2)^(1/3)*3^(1/2)*(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^3*b+24*I*A*(-a*b^2)^(2/3)*3^(1/2)*(-(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^3*b*e-24*A*(-(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^5*b^3*e+48*A*(-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)*Ellipti

cF((-(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^4*b^2*e-24*A*(-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)*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^3*b*e+30*I*B

*3^(1/2)*(-(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^5*a*

b^2*e-12*A*((b*x^3+a)*e*x)^(1/2)*(-a*b^2)^(1/3)*(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)*b-60*I*B*(-a*b^2)^(1/3)*3^(1/2)*(-(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^4*a*b*e+24*I*A*3^(1/2)*(-(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)*Ellipti

cF((-(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^5*b^3*e+4*I*A*((b*x^3+a)*e*

x)^(1/2)*(-a*b^2)^(1/3)*3^(1/2)*(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)*b-30*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^5*a*b^2*e+60*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))*x^4*a*b*e-30*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)*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^3*a*e+30*I*B*(-a*b^2)^(2/3)*3^(1/2)*(-(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))*x^3*a*e+15*B*((b*x^3+a)*e

*x)^(1/2)*(-a*b^2)^(1/3)*(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)

*x^3*b)/e^3/(e*x)^(1/2)/((b*x^3+a)*e*x)^(1/2)/(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)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^3 + A)*sqrt(b*x^3 + a)/(sqrt(e*x)*e^3*x^3), 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)*(b*x**3+a)**(1/2)/(e*x)**(7/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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