3.224 \(\int \frac{x \left (A+B x^3\right )}{\sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=517 \[ \frac{2 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (7 A b-4 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (7 A b-4 a B) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3} (7 A b-4 a B)}{7 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 B x^2 \sqrt{a+b x^3}}{7 b} \]

[Out]

(2*B*x^2*Sqrt[a + b*x^3])/(7*b) + (2*(7*A*b - 4*a*B)*Sqrt[a + b*x^3])/(7*b^(5/3)
*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(7*A*
b - 4*a*B)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2
)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3)
+ b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(7*b^(5/3)*
Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt
[a + b*x^3]) + (2*Sqrt[2]*a^(1/3)*(7*A*b - 4*a*B)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^
(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*
EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^
(1/3)*x)], -7 - 4*Sqrt[3]])/(7*3^(1/4)*b^(5/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*
x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi [A]  time = 0.557967, antiderivative size = 517, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (7 A b-4 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (7 A b-4 a B) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 \sqrt{a+b x^3} (7 A b-4 a B)}{7 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{2 B x^2 \sqrt{a+b x^3}}{7 b} \]

Antiderivative was successfully verified.

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

[Out]

(2*B*x^2*Sqrt[a + b*x^3])/(7*b) + (2*(7*A*b - 4*a*B)*Sqrt[a + b*x^3])/(7*b^(5/3)
*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(7*A*
b - 4*a*B)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2
)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3)
+ b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(7*b^(5/3)*
Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt
[a + b*x^3]) + (2*Sqrt[2]*a^(1/3)*(7*A*b - 4*a*B)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^
(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*
EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^
(1/3)*x)], -7 - 4*Sqrt[3]])/(7*3^(1/4)*b^(5/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*
x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi in Sympy [A]  time = 37.6063, size = 461, normalized size = 0.89 \[ \frac{2 B x^{2} \sqrt{a + b x^{3}}}{7 b} - \frac{\sqrt [4]{3} \sqrt [3]{a} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (7 A b - 4 B a\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{7 b^{\frac{5}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} + \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [3]{a} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (7 A b - 4 B a\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{21 b^{\frac{5}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} + \frac{2 \sqrt{a + b x^{3}} \left (7 A b - 4 B a\right )}{7 b^{\frac{5}{3}} \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*x**2*sqrt(a + b*x**3)/(7*b) - 3**(1/4)*a**(1/3)*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)*sqrt(-sqrt(3
) + 2)*(a**(1/3) + b**(1/3)*x)*(7*A*b - 4*B*a)*elliptic_e(asin((-a**(1/3)*(-1 +
sqrt(3)) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)), -7 - 4*sqrt(3))/(
7*b**(5/3)*sqrt(a**(1/3)*(a**(1/3) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**(1
/3)*x)**2)*sqrt(a + b*x**3)) + 2*sqrt(2)*3**(3/4)*a**(1/3)*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)*(a**(
1/3) + b**(1/3)*x)*(7*A*b - 4*B*a)*elliptic_f(asin((-a**(1/3)*(-1 + sqrt(3)) + b
**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)), -7 - 4*sqrt(3))/(21*b**(5/3)*
sqrt(a**(1/3)*(a**(1/3) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2)*
sqrt(a + b*x**3)) + 2*sqrt(a + b*x**3)*(7*A*b - 4*B*a)/(7*b**(5/3)*(a**(1/3)*(1
+ sqrt(3)) + b**(1/3)*x))

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Mathematica [C]  time = 0.577171, size = 231, normalized size = 0.45 \[ \frac{2 B x^2 \sqrt{a+b x^3}}{7 b}-\frac{2 \sqrt [6]{-1} a^{2/3} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}-1\right )} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} (7 A b-4 a B) \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{7 \sqrt [4]{3} (-b)^{5/3} \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*B*x^2*Sqrt[a + b*x^3])/(7*b) - (2*(-1)^(1/6)*a^(2/3)*(7*A*b - 4*a*B)*Sqrt[(-1
)^(5/6)*(-1 + ((-b)^(1/3)*x)/a^(1/3))]*Sqrt[1 + ((-b)^(1/3)*x)/a^(1/3) + ((-b)^(
2/3)*x^2)/a^(2/3)]*((-I)*Sqrt[3]*EllipticE[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-b)^(1/
3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)] + (-1)^(1/3)*EllipticF[ArcSin[Sqrt[-(-1)^(5
/6) - (I*(-b)^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)]))/(7*3^(1/4)*(-b)^(5/3)*Sq
rt[a + b*x^3])

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Maple [B]  time = 0.01, size = 892, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*I*A*3^(1/2)/b*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a
*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a
*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1
/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*(
(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*
(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3)
)^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b
^2)^(1/3)))^(1/2))+1/b*(-a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)
^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2
)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)
))+B*(2/7/b*x^2*(b*x^3+a)^(1/2)+8/21*I*a/b^2*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*
(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*(
(x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(
1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b
^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2
)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^
2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-
a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/b*(-a*b^2)^(1/3)*Elliptic
F(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)
*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*
I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{4} + A x}{\sqrt{b x^{3} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^4 + A*x)/sqrt(b*x^3 + a), x)

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Sympy [A]  time = 4.73715, size = 80, normalized size = 0.15 \[ \frac{A x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{5}{3}\right )} + \frac{B x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{8}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*x**2*gamma(2/3)*hyper((1/2, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*sqrt(a)
*gamma(5/3)) + B*x**5*gamma(5/3)*hyper((1/2, 5/3), (8/3,), b*x**3*exp_polar(I*pi
)/a)/(3*sqrt(a)*gamma(8/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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