3.38 \(\int \frac{a+b x^3+c x^6}{\sqrt{d+e x^3}} \, dx\)

Optimal. Leaf size=278 \[ \frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (55 a e^2-22 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}-\frac{2 x \sqrt{d+e x^3} (8 c d-11 b e)}{55 e^2}+\frac{2 c x^4 \sqrt{d+e x^3}}{11 e} \]

[Out]

(-2*(8*c*d - 11*b*e)*x*Sqrt[d + e*x^3])/(55*e^2) + (2*c*x^4*Sqrt[d + e*x^3])/(11
*e) + (2*Sqrt[2 + Sqrt[3]]*(16*c*d^2 - 22*b*d*e + 55*a*e^2)*(d^(1/3) + e^(1/3)*x
)*Sqrt[(d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2)/((1 + Sqrt[3])*d^(1/3) + e^(1
/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*d^(1/3) + e^(1/3)*x)/((1 + Sqrt[3])*d^
(1/3) + e^(1/3)*x)], -7 - 4*Sqrt[3]])/(55*3^(1/4)*e^(7/3)*Sqrt[(d^(1/3)*(d^(1/3)
+ e^(1/3)*x))/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)^2]*Sqrt[d + e*x^3])

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Rubi [A]  time = 0.370475, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (55 a e^2-22 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}-\frac{2 x \sqrt{d+e x^3} (8 c d-11 b e)}{55 e^2}+\frac{2 c x^4 \sqrt{d+e x^3}}{11 e} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3 + c*x^6)/Sqrt[d + e*x^3],x]

[Out]

(-2*(8*c*d - 11*b*e)*x*Sqrt[d + e*x^3])/(55*e^2) + (2*c*x^4*Sqrt[d + e*x^3])/(11
*e) + (2*Sqrt[2 + Sqrt[3]]*(16*c*d^2 - 22*b*d*e + 55*a*e^2)*(d^(1/3) + e^(1/3)*x
)*Sqrt[(d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2)/((1 + Sqrt[3])*d^(1/3) + e^(1
/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*d^(1/3) + e^(1/3)*x)/((1 + Sqrt[3])*d^
(1/3) + e^(1/3)*x)], -7 - 4*Sqrt[3]])/(55*3^(1/4)*e^(7/3)*Sqrt[(d^(1/3)*(d^(1/3)
+ e^(1/3)*x))/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)^2]*Sqrt[d + e*x^3])

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Rubi in Sympy [A]  time = 27.4319, size = 253, normalized size = 0.91 \[ \frac{2 c x^{4} \sqrt{d + e x^{3}}}{11 e} + \frac{2 x \sqrt{d + e x^{3}} \left (11 b e - 8 c d\right )}{55 e^{2}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{d^{\frac{2}{3}} - \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right ) \left (55 a e^{2} - 22 b d e + 16 c d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{d} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{e} x}{\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{165 e^{\frac{7}{3}} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right )}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{d + e x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**6+b*x**3+a)/(e*x**3+d)**(1/2),x)

[Out]

2*c*x**4*sqrt(d + e*x**3)/(11*e) + 2*x*sqrt(d + e*x**3)*(11*b*e - 8*c*d)/(55*e**
2) + 2*3**(3/4)*sqrt((d**(2/3) - d**(1/3)*e**(1/3)*x + e**(2/3)*x**2)/(d**(1/3)*
(1 + sqrt(3)) + e**(1/3)*x)**2)*sqrt(sqrt(3) + 2)*(d**(1/3) + e**(1/3)*x)*(55*a*
e**2 - 22*b*d*e + 16*c*d**2)*elliptic_f(asin((-d**(1/3)*(-1 + sqrt(3)) + e**(1/3
)*x)/(d**(1/3)*(1 + sqrt(3)) + e**(1/3)*x)), -7 - 4*sqrt(3))/(165*e**(7/3)*sqrt(
d**(1/3)*(d**(1/3) + e**(1/3)*x)/(d**(1/3)*(1 + sqrt(3)) + e**(1/3)*x)**2)*sqrt(
d + e*x**3))

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Mathematica [C]  time = 0.606227, size = 194, normalized size = 0.7 \[ \frac{2 \sqrt{d+e x^3} \left (11 b e x-8 c d x+5 c e x^4\right )}{55 e^2}+\frac{2 i \sqrt [3]{d} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}-1\right )} \sqrt{\frac{(-e)^{2/3} x^2}{d^{2/3}}+\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}+1} \left (11 e (5 a e-2 b d)+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-e} x}{\sqrt [3]{d}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{55 \sqrt [4]{3} (-e)^{7/3} \sqrt{d+e x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^3 + c*x^6)/Sqrt[d + e*x^3],x]

[Out]

(2*Sqrt[d + e*x^3]*(-8*c*d*x + 11*b*e*x + 5*c*e*x^4))/(55*e^2) + (((2*I)/55)*d^(
1/3)*(16*c*d^2 + 11*e*(-2*b*d + 5*a*e))*Sqrt[(-1)^(5/6)*(-1 + ((-e)^(1/3)*x)/d^(
1/3))]*Sqrt[1 + ((-e)^(1/3)*x)/d^(1/3) + ((-e)^(2/3)*x^2)/d^(2/3)]*EllipticF[Arc
Sin[Sqrt[-(-1)^(5/6) - (I*(-e)^(1/3)*x)/d^(1/3)]/3^(1/4)], (-1)^(1/3)])/(3^(1/4)
*(-e)^(7/3)*Sqrt[d + e*x^3])

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Maple [B]  time = 0.044, size = 907, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^6+b*x^3+a)/(e*x^3+d)^(1/2),x)

[Out]

-2/3*I*a*3^(1/2)/e*(-e^2*d)^(1/3)*(I*(x+1/2/e*(-e^2*d)^(1/3)-1/2*I*3^(1/2)/e*(-e
^2*d)^(1/3))*3^(1/2)*e/(-e^2*d)^(1/3))^(1/2)*((x-1/e*(-e^2*d)^(1/3))/(-3/2/e*(-e
^2*d)^(1/3)+1/2*I*3^(1/2)/e*(-e^2*d)^(1/3)))^(1/2)*(-I*(x+1/2/e*(-e^2*d)^(1/3)+1
/2*I*3^(1/2)/e*(-e^2*d)^(1/3))*3^(1/2)*e/(-e^2*d)^(1/3))^(1/2)/(e*x^3+d)^(1/2)*E
llipticF(1/3*3^(1/2)*(I*(x+1/2/e*(-e^2*d)^(1/3)-1/2*I*3^(1/2)/e*(-e^2*d)^(1/3))*
3^(1/2)*e/(-e^2*d)^(1/3))^(1/2),(I*3^(1/2)/e*(-e^2*d)^(1/3)/(-3/2/e*(-e^2*d)^(1/
3)+1/2*I*3^(1/2)/e*(-e^2*d)^(1/3)))^(1/2))+b*(2/5/e*x*(e*x^3+d)^(1/2)+4/15*I*d/e
^2*3^(1/2)*(-e^2*d)^(1/3)*(I*(x+1/2/e*(-e^2*d)^(1/3)-1/2*I*3^(1/2)/e*(-e^2*d)^(1
/3))*3^(1/2)*e/(-e^2*d)^(1/3))^(1/2)*((x-1/e*(-e^2*d)^(1/3))/(-3/2/e*(-e^2*d)^(1
/3)+1/2*I*3^(1/2)/e*(-e^2*d)^(1/3)))^(1/2)*(-I*(x+1/2/e*(-e^2*d)^(1/3)+1/2*I*3^(
1/2)/e*(-e^2*d)^(1/3))*3^(1/2)*e/(-e^2*d)^(1/3))^(1/2)/(e*x^3+d)^(1/2)*EllipticF
(1/3*3^(1/2)*(I*(x+1/2/e*(-e^2*d)^(1/3)-1/2*I*3^(1/2)/e*(-e^2*d)^(1/3))*3^(1/2)*
e/(-e^2*d)^(1/3))^(1/2),(I*3^(1/2)/e*(-e^2*d)^(1/3)/(-3/2/e*(-e^2*d)^(1/3)+1/2*I
*3^(1/2)/e*(-e^2*d)^(1/3)))^(1/2)))+c*(2/11/e*x^4*(e*x^3+d)^(1/2)-16/55*d/e^2*x*
(e*x^3+d)^(1/2)-32/165*I*d^2/e^3*3^(1/2)*(-e^2*d)^(1/3)*(I*(x+1/2/e*(-e^2*d)^(1/
3)-1/2*I*3^(1/2)/e*(-e^2*d)^(1/3))*3^(1/2)*e/(-e^2*d)^(1/3))^(1/2)*((x-1/e*(-e^2
*d)^(1/3))/(-3/2/e*(-e^2*d)^(1/3)+1/2*I*3^(1/2)/e*(-e^2*d)^(1/3)))^(1/2)*(-I*(x+
1/2/e*(-e^2*d)^(1/3)+1/2*I*3^(1/2)/e*(-e^2*d)^(1/3))*3^(1/2)*e/(-e^2*d)^(1/3))^(
1/2)/(e*x^3+d)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/e*(-e^2*d)^(1/3)-1/2*I*3^(1
/2)/e*(-e^2*d)^(1/3))*3^(1/2)*e/(-e^2*d)^(1/3))^(1/2),(I*3^(1/2)/e*(-e^2*d)^(1/3
)/(-3/2/e*(-e^2*d)^(1/3)+1/2*I*3^(1/2)/e*(-e^2*d)^(1/3)))^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{c x^{6} + b x^{3} + a}{\sqrt{e x^{3} + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)/sqrt(e*x^3 + d),x, algorithm="maxima")

[Out]

integrate((c*x^6 + b*x^3 + a)/sqrt(e*x^3 + d), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{c x^{6} + b x^{3} + a}{\sqrt{e x^{3} + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)/sqrt(e*x^3 + d),x, algorithm="fricas")

[Out]

integral((c*x^6 + b*x^3 + a)/sqrt(e*x^3 + d), x)

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Sympy [A]  time = 6.30519, size = 119, normalized size = 0.43 \[ \frac{a x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt{d} \Gamma \left (\frac{4}{3}\right )} + \frac{b x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt{d} \Gamma \left (\frac{7}{3}\right )} + \frac{c x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \sqrt{d} \Gamma \left (\frac{10}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**6+b*x**3+a)/(e*x**3+d)**(1/2),x)

[Out]

a*x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), e*x**3*exp_polar(I*pi)/d)/(3*sqrt(d)*ga
mma(4/3)) + b*x**4*gamma(4/3)*hyper((1/2, 4/3), (7/3,), e*x**3*exp_polar(I*pi)/d
)/(3*sqrt(d)*gamma(7/3)) + c*x**7*gamma(7/3)*hyper((1/2, 7/3), (10/3,), e*x**3*e
xp_polar(I*pi)/d)/(3*sqrt(d)*gamma(10/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{c x^{6} + b x^{3} + a}{\sqrt{e x^{3} + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)/sqrt(e*x^3 + d),x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)/sqrt(e*x^3 + d), x)