3.40 \(\int \frac{a+b x^3+c x^6}{\left (d+e x^3\right )^{5/2}} \, dx\)

Optimal. Leaf size=309 \[ -\frac{2 x \left (-7 a e^2-2 b d e+11 c d^2\right )}{27 d^2 e^2 \sqrt{d+e x^3}}+\frac{2 x \left (a e^2-b d e+c d^2\right )}{9 d e^2 \left (d+e x^3\right )^{3/2}}+\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (e (7 a e+2 b d)+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{27 \sqrt [4]{3} d^2 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}} \]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)*x)/(9*d*e^2*(d + e*x^3)^(3/2)) - (2*(11*c*d^2 - 2*b*d
*e - 7*a*e^2)*x)/(27*d^2*e^2*Sqrt[d + e*x^3]) + (2*Sqrt[2 + Sqrt[3]]*(16*c*d^2 +
e*(2*b*d + 7*a*e))*(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]])/(2
7*3^(1/4)*d^2*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 = 0.438426, antiderivative size = 309, 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 \[ -\frac{2 x \left (-7 a e^2-2 b d e+11 c d^2\right )}{27 d^2 e^2 \sqrt{d+e x^3}}+\frac{2 x \left (a e^2-b d e+c d^2\right )}{9 d e^2 \left (d+e x^3\right )^{3/2}}+\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (e (7 a e+2 b d)+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{27 \sqrt [4]{3} d^2 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3 + c*x^6)/(d + e*x^3)^(5/2),x]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)*x)/(9*d*e^2*(d + e*x^3)^(3/2)) - (2*(11*c*d^2 - 2*b*d
*e - 7*a*e^2)*x)/(27*d^2*e^2*Sqrt[d + e*x^3]) + (2*Sqrt[2 + Sqrt[3]]*(16*c*d^2 +
e*(2*b*d + 7*a*e))*(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]])/(2
7*3^(1/4)*d^2*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 = 43.7627, size = 286, normalized size = 0.93 \[ \frac{2 x \left (a e^{2} - b d e + c d^{2}\right )}{9 d e^{2} \left (d + e x^{3}\right )^{\frac{3}{2}}} + \frac{2 x \left (7 a e^{2} + 2 b d e - 11 c d^{2}\right )}{27 d^{2} e^{2} \sqrt{d + e x^{3}}} + \frac{4 \cdot 3^{\frac{3}{4}} \sqrt{\frac{d^{\frac{2}{3}} - \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right ) \left (\frac{7 a e^{2}}{2} + b d e + 8 c d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{d} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{e} x}{\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{81 d^{2} e^{\frac{7}{3}} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right )}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{d + e x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**6+b*x**3+a)/(e*x**3+d)**(5/2),x)

[Out]

2*x*(a*e**2 - b*d*e + c*d**2)/(9*d*e**2*(d + e*x**3)**(3/2)) + 2*x*(7*a*e**2 + 2
*b*d*e - 11*c*d**2)/(27*d**2*e**2*sqrt(d + e*x**3)) + 4*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)*(7*a*e**2/2 + b*d*e + 8*c*d**2)*ellip
tic_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))/(81*d**2*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 = 0.673866, size = 224, normalized size = 0.72 \[ \frac{2 \left (3 \sqrt [3]{-e} x \left (e \left (a e \left (10 d+7 e x^3\right )-b d \left (d-2 e x^3\right )\right )-c d^2 \left (8 d+11 e x^3\right )\right )+i 3^{3/4} \sqrt [3]{d} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}-1\right )} \sqrt{\frac{(-e)^{2/3} x^2}{d^{2/3}}+\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}+1} \left (d+e x^3\right ) \left (e (7 a e+2 b d)+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-e} x}{\sqrt [3]{d}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{81 d^2 (-e)^{7/3} \left (d+e x^3\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^(5/2),x]

[Out]

(2*(3*(-e)^(1/3)*x*(-(c*d^2*(8*d + 11*e*x^3)) + e*(-(b*d*(d - 2*e*x^3)) + a*e*(1
0*d + 7*e*x^3))) + I*3^(3/4)*d^(1/3)*(16*c*d^2 + e*(2*b*d + 7*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)]*(d + e*x^3)*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-e)^(1/3)*x)/
d^(1/3)]/3^(1/4)], (-1)^(1/3)]))/(81*d^2*(-e)^(7/3)*(d + e*x^3)^(3/2))

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Maple [B]  time = 0.07, size = 1005, 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)^(5/2),x)

[Out]

a*(2/9/d*x/e^2*(e*x^3+d)^(1/2)/(x^3+d/e)^2+14/27/d^2*x/((x^3+d/e)*e)^(1/2)-14/81
*I/d^2*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)*Ell
ipticF(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*(-2/9*x/e^3*(e*x^3+d)^(1/2)/(x^3+d/e
)^2+4/27/e/d*x/((x^3+d/e)*e)^(1/2)-4/81*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*(2/9*d*x/e^4*(e*x^3+d)^(1/2)/(x^3+d/e)^2-22/27/e^2*x/((x^3+d/e)*e)^(1/2)-32/8
1*I/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)*Elli
pticF(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. \[ \int \frac{c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(5/2),x, algorithm="maxima")

[Out]

integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{c x^{6} + b x^{3} + a}{{\left (e^{2} x^{6} + 2 \, d e x^{3} + d^{2}\right )} \sqrt{e x^{3} + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(5/2),x, algorithm="fricas")

[Out]

integral((c*x^6 + b*x^3 + a)/((e^2*x^6 + 2*d*e*x^3 + d^2)*sqrt(e*x^3 + d)), 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((c*x**6+b*x**3+a)/(e*x**3+d)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(5/2),x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(5/2), x)