3.42 \(\int \frac{a+b x^3+c x^6}{\left (d+e x^3\right )^{9/2}} \, dx\)

Optimal. Leaf size=389 \[ -\frac{2 x \left (-19 a e^2-2 b d e+23 c d^2\right )}{315 d^2 e^2 \left (d+e x^3\right )^{5/2}}+\frac{2 x \left (a e^2-b d e+c d^2\right )}{21 d e^2 \left (d+e x^3\right )^{7/2}}+\frac{2 x \left (247 a e^2+26 b d e+16 c d^2\right )}{1215 d^4 e^2 \sqrt{d+e x^3}}+\frac{2 x \left (247 a e^2+26 b d e+16 c d^2\right )}{2835 d^3 e^2 \left (d+e x^3\right )^{3/2}}+\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (247 a e^2+26 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{1215 \sqrt [4]{3} d^4 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}} \]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)*x)/(21*d*e^2*(d + e*x^3)^(7/2)) - (2*(23*c*d^2 - 2*b*
d*e - 19*a*e^2)*x)/(315*d^2*e^2*(d + e*x^3)^(5/2)) + (2*(16*c*d^2 + 26*b*d*e + 2
47*a*e^2)*x)/(2835*d^3*e^2*(d + e*x^3)^(3/2)) + (2*(16*c*d^2 + 26*b*d*e + 247*a*
e^2)*x)/(1215*d^4*e^2*Sqrt[d + e*x^3]) + (2*Sqrt[2 + Sqrt[3]]*(16*c*d^2 + 26*b*d
*e + 247*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]])/(1215*
3^(1/4)*d^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 = 0.737356, antiderivative size = 389, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 x \left (-19 a e^2-2 b d e+23 c d^2\right )}{315 d^2 e^2 \left (d+e x^3\right )^{5/2}}+\frac{2 x \left (a e^2-b d e+c d^2\right )}{21 d e^2 \left (d+e x^3\right )^{7/2}}+\frac{2 x \left (247 a e^2+26 b d e+16 c d^2\right )}{1215 d^4 e^2 \sqrt{d+e x^3}}+\frac{2 x \left (247 a e^2+26 b d e+16 c d^2\right )}{2835 d^3 e^2 \left (d+e x^3\right )^{3/2}}+\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (247 a e^2+26 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{1215 \sqrt [4]{3} d^4 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3 + c*x^6)/(d + e*x^3)^(9/2),x]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)*x)/(21*d*e^2*(d + e*x^3)^(7/2)) - (2*(23*c*d^2 - 2*b*
d*e - 19*a*e^2)*x)/(315*d^2*e^2*(d + e*x^3)^(5/2)) + (2*(16*c*d^2 + 26*b*d*e + 2
47*a*e^2)*x)/(2835*d^3*e^2*(d + e*x^3)^(3/2)) + (2*(16*c*d^2 + 26*b*d*e + 247*a*
e^2)*x)/(1215*d^4*e^2*Sqrt[d + e*x^3]) + (2*Sqrt[2 + Sqrt[3]]*(16*c*d^2 + 26*b*d
*e + 247*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]])/(1215*
3^(1/4)*d^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 = 57.4961, size = 371, normalized size = 0.95 \[ \frac{2 x \left (a e^{2} - b d e + c d^{2}\right )}{21 d e^{2} \left (d + e x^{3}\right )^{\frac{7}{2}}} + \frac{2 x \left (19 a e^{2} + 2 b d e - 23 c d^{2}\right )}{315 d^{2} e^{2} \left (d + e x^{3}\right )^{\frac{5}{2}}} + \frac{2 x \left (247 a e^{2} + 26 b d e + 16 c d^{2}\right )}{2835 d^{3} e^{2} \left (d + e x^{3}\right )^{\frac{3}{2}}} + \frac{2 x \left (247 a e^{2} + 26 b d e + 16 c d^{2}\right )}{1215 d^{4} e^{2} \sqrt{d + e x^{3}}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{d^{\frac{2}{3}} - \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right ) \left (247 a e^{2} + 26 b d e + 16 c d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{d} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{e} x}{\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3645 d^{4} e^{\frac{7}{3}} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right )}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{d + e x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**6+b*x**3+a)/(e*x**3+d)**(9/2),x)

[Out]

2*x*(a*e**2 - b*d*e + c*d**2)/(21*d*e**2*(d + e*x**3)**(7/2)) + 2*x*(19*a*e**2 +
2*b*d*e - 23*c*d**2)/(315*d**2*e**2*(d + e*x**3)**(5/2)) + 2*x*(247*a*e**2 + 26
*b*d*e + 16*c*d**2)/(2835*d**3*e**2*(d + e*x**3)**(3/2)) + 2*x*(247*a*e**2 + 26*
b*d*e + 16*c*d**2)/(1215*d**4*e**2*sqrt(d + e*x**3)) + 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)*(247*a*e**2 + 26*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))/(3645*d**4*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 = 1.03152, size = 296, normalized size = 0.76 \[ \frac{2 \left (3 \sqrt [3]{-e} x \left (-27 d^2 \left (d+e x^3\right ) \left (23 c d^2-e (19 a e+2 b d)\right )+3 d \left (d+e x^3\right )^2 \left (13 e (19 a e+2 b d)+16 c d^2\right )+7 \left (d+e x^3\right )^3 \left (13 e (19 a e+2 b d)+16 c d^2\right )+405 d^3 \left (e (a e-b d)+c d^2\right )\right )+7 i 3^{3/4} \sqrt [3]{d} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}-1\right )} \sqrt{\frac{(-e)^{2/3} x^2}{d^{2/3}}+\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}+1} \left (d+e x^3\right )^3 \left (13 e (19 a e+2 b d)+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-e} x}{\sqrt [3]{d}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{25515 d^4 (-e)^{7/3} \left (d+e x^3\right )^{7/2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^(9/2),x]

[Out]

(2*(3*(-e)^(1/3)*x*(405*d^3*(c*d^2 + e*(-(b*d) + a*e)) - 27*d^2*(23*c*d^2 - e*(2
*b*d + 19*a*e))*(d + e*x^3) + 3*d*(16*c*d^2 + 13*e*(2*b*d + 19*a*e))*(d + e*x^3)
^2 + 7*(16*c*d^2 + 13*e*(2*b*d + 19*a*e))*(d + e*x^3)^3) + (7*I)*3^(3/4)*d^(1/3)
*(16*c*d^2 + 13*e*(2*b*d + 19*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)^3*Ell
ipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-e)^(1/3)*x)/d^(1/3)]/3^(1/4)], (-1)^(1/3)]
))/(25515*d^4*(-e)^(7/3)*(d + e*x^3)^(7/2))

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Maple [B]  time = 0.078, size = 1182, normalized size = 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)^(9/2),x)

[Out]

a*(2/21/d*x/e^4*(e*x^3+d)^(1/2)/(x^3+d/e)^4+38/315/d^2*x/e^3*(e*x^3+d)^(1/2)/(x^
3+d/e)^3+494/2835/d^3*x/e^2*(e*x^3+d)^(1/2)/(x^3+d/e)^2+494/1215/d^4*x/((x^3+d/e
)*e)^(1/2)-494/3645*I/d^4*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*(-2/21*x/e^5*(e*x
^3+d)^(1/2)/(x^3+d/e)^4+4/315/d*x/e^4*(e*x^3+d)^(1/2)/(x^3+d/e)^3+52/2835/d^2*x/
e^3*(e*x^3+d)^(1/2)/(x^3+d/e)^2+52/1215/e/d^3*x/((x^3+d/e)*e)^(1/2)-52/3645*I/d^
3/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)*Ellipt
icF(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*(2/21*d*x/e^6*(e*x^3+d)^(1/2)/(x^3+d/e)
^4-46/315*x/e^5*(e*x^3+d)^(1/2)/(x^3+d/e)^3+32/2835/d*x/e^4*(e*x^3+d)^(1/2)/(x^3
+d/e)^2+32/1215/e^2/d^2*x/((x^3+d/e)*e)^(1/2)-32/3645*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. \[ \int \frac{c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{c x^{6} + b x^{3} + a}{{\left (e^{4} x^{12} + 4 \, d e^{3} x^{9} + 6 \, d^{2} e^{2} x^{6} + 4 \, d^{3} e x^{3} + d^{4}\right )} \sqrt{e x^{3} + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(9/2),x, algorithm="fricas")

[Out]

integral((c*x^6 + b*x^3 + a)/((e^4*x^12 + 4*d*e^3*x^9 + 6*d^2*e^2*x^6 + 4*d^3*e*
x^3 + d^4)*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)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{c x^{6} + b x^{3} + a}{{\left (e x^{3} + d\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(9/2),x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^(9/2), x)