3.37 \(\int \sqrt{d+e x^3} \left (a+b x^3+c x^6\right ) \, dx\)

Optimal. Leaf size=316 \[ \frac{2 x \sqrt{d+e x^3} \left (187 a e^2-34 b d e+16 c d^2\right )}{935 e^2}+\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} d \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (187 a e^2-34 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{935 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}-\frac{2 x \left (d+e x^3\right )^{3/2} (8 c d-17 b e)}{187 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e} \]

[Out]

(2*(16*c*d^2 - 34*b*d*e + 187*a*e^2)*x*Sqrt[d + e*x^3])/(935*e^2) - (2*(8*c*d -
17*b*e)*x*(d + e*x^3)^(3/2))/(187*e^2) + (2*c*x^4*(d + e*x^3)^(3/2))/(17*e) + (2
*3^(3/4)*Sqrt[2 + Sqrt[3]]*d*(16*c*d^2 - 34*b*d*e + 187*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]])/(935*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 = 0.504449, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 x \sqrt{d+e x^3} \left (187 a e^2-34 b d e+16 c d^2\right )}{935 e^2}+\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} d \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (187 a e^2-34 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{935 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}-\frac{2 x \left (d+e x^3\right )^{3/2} (8 c d-17 b e)}{187 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{3/2}}{17 e} \]

Antiderivative was successfully verified.

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

[Out]

(2*(16*c*d^2 - 34*b*d*e + 187*a*e^2)*x*Sqrt[d + e*x^3])/(935*e^2) - (2*(8*c*d -
17*b*e)*x*(d + e*x^3)^(3/2))/(187*e^2) + (2*c*x^4*(d + e*x^3)^(3/2))/(17*e) + (2
*3^(3/4)*Sqrt[2 + Sqrt[3]]*d*(16*c*d^2 - 34*b*d*e + 187*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]])/(935*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 = 33.4453, size = 294, normalized size = 0.93 \[ \frac{2 c x^{4} \left (d + e x^{3}\right )^{\frac{3}{2}}}{17 e} + \frac{2 \cdot 3^{\frac{3}{4}} d \sqrt{\frac{d^{\frac{2}{3}} - \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right ) \left (187 a e^{2} - 34 b d e + 16 c d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{d} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{e} x}{\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{935 e^{\frac{7}{3}} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right )}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{d + e x^{3}}} + \frac{2 x \left (d + e x^{3}\right )^{\frac{3}{2}} \left (17 b e - 8 c d\right )}{187 e^{2}} + \frac{2 x \sqrt{d + e x^{3}} \left (187 a e^{2} - 34 b d e + 16 c d^{2}\right )}{935 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*x**4*(d + e*x**3)**(3/2)/(17*e) + 2*3**(3/4)*d*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)*(187*a*e**2 - 34*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))/(935*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)) + 2*x*(d + e*x**3)**(3/2)*(17*b
*e - 8*c*d)/(187*e**2) + 2*x*sqrt(d + e*x**3)*(187*a*e**2 - 34*b*d*e + 16*c*d**2
)/(935*e**2)

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Mathematica [C]  time = 0.47553, size = 219, normalized size = 0.69 \[ -\frac{2 \left (\sqrt [3]{-e} x \left (d+e x^3\right ) \left (c \left (24 d^2-15 d e x^3-55 e^2 x^6\right )-17 e \left (11 a e+3 b d+5 b e x^3\right )\right )-i 3^{3/4} d^{4/3} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}-1\right )} \sqrt{\frac{(-e)^{2/3} x^2}{d^{2/3}}+\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}+1} \left (17 e (11 a e-2 b d)+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-e} x}{\sqrt [3]{d}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{935 (-e)^{7/3} \sqrt{d+e x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*((-e)^(1/3)*x*(d + e*x^3)*(-17*e*(3*b*d + 11*a*e + 5*b*e*x^3) + c*(24*d^2 -
15*d*e*x^3 - 55*e^2*x^6)) - I*3^(3/4)*d^(4/3)*(16*c*d^2 + 17*e*(-2*b*d + 11*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[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-e)^(1/3)*x
)/d^(1/3)]/3^(1/4)], (-1)^(1/3)]))/(935*(-e)^(7/3)*Sqrt[d + e*x^3])

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Maple [B]  time = 0.044, size = 956, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*(2/5*x*(e*x^3+d)^(1/2)-2/5*I*d*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)*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)))+b*(2/11*x^4*
(e*x^3+d)^(1/2)+6/55*d/e*x*(e*x^3+d)^(1/2)+4/55*I*d^2/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*(2/17*x^7*(e*x^3+d)^(1/2)+6/187*d/e*x^4*(e*x^3+d)^(1/2)-48/935*d^2/
e^2*x*(e*x^3+d)^(1/2)-32/935*I*d^3/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. \[ \int{\left (c x^{6} + b x^{3} + a\right )} \sqrt{e x^{3} + d}\,{d x} \]

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. \[{\rm integral}\left ({\left (c x^{6} + b x^{3} + a\right )} \sqrt{e x^{3} + d}, x\right ) \]

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 = 6.816, size = 124, normalized size = 0.39 \[ \frac{a \sqrt{d} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} + \frac{b \sqrt{d} x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{c \sqrt{d} x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*sqrt(d)*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), e*x**3*exp_polar(I*pi)/d)/(3*g
amma(4/3)) + b*sqrt(d)*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), e*x**3*exp_pol
ar(I*pi)/d)/(3*gamma(7/3)) + c*sqrt(d)*x**7*gamma(7/3)*hyper((-1/2, 7/3), (10/3,
), e*x**3*exp_polar(I*pi)/d)/(3*gamma(10/3))

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

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)