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

Optimal. Leaf size=396 \[ \frac{2 x \left (d+e x^3\right )^{5/2} \left (667 a e^2-58 b d e+16 c d^2\right )}{11339 e^2}+\frac{30 d x \left (d+e x^3\right )^{3/2} \left (667 a e^2-58 b d e+16 c d^2\right )}{124729 e^2}+\frac{54 d^2 x \sqrt{d+e x^3} \left (667 a e^2-58 b d e+16 c d^2\right )}{124729 e^2}+\frac{54\ 3^{3/4} \sqrt{2+\sqrt{3}} d^3 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (667 a e^2-58 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{124729 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}-\frac{2 x \left (d+e x^3\right )^{7/2} (8 c d-29 b e)}{667 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e} \]

[Out]

(54*d^2*(16*c*d^2 - 58*b*d*e + 667*a*e^2)*x*Sqrt[d + e*x^3])/(124729*e^2) + (30*
d*(16*c*d^2 - 58*b*d*e + 667*a*e^2)*x*(d + e*x^3)^(3/2))/(124729*e^2) + (2*(16*c
*d^2 - 58*b*d*e + 667*a*e^2)*x*(d + e*x^3)^(5/2))/(11339*e^2) - (2*(8*c*d - 29*b
*e)*x*(d + e*x^3)^(7/2))/(667*e^2) + (2*c*x^4*(d + e*x^3)^(7/2))/(29*e) + (54*3^
(3/4)*Sqrt[2 + Sqrt[3]]*d^3*(16*c*d^2 - 58*b*d*e + 667*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]])/(124729*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 = 0.774189, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 x \left (d+e x^3\right )^{5/2} \left (667 a e^2-58 b d e+16 c d^2\right )}{11339 e^2}+\frac{30 d x \left (d+e x^3\right )^{3/2} \left (667 a e^2-58 b d e+16 c d^2\right )}{124729 e^2}+\frac{54 d^2 x \sqrt{d+e x^3} \left (667 a e^2-58 b d e+16 c d^2\right )}{124729 e^2}+\frac{54\ 3^{3/4} \sqrt{2+\sqrt{3}} d^3 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (667 a e^2-58 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{124729 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}-\frac{2 x \left (d+e x^3\right )^{7/2} (8 c d-29 b e)}{667 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{7/2}}{29 e} \]

Antiderivative was successfully verified.

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

[Out]

(54*d^2*(16*c*d^2 - 58*b*d*e + 667*a*e^2)*x*Sqrt[d + e*x^3])/(124729*e^2) + (30*
d*(16*c*d^2 - 58*b*d*e + 667*a*e^2)*x*(d + e*x^3)^(3/2))/(124729*e^2) + (2*(16*c
*d^2 - 58*b*d*e + 667*a*e^2)*x*(d + e*x^3)^(5/2))/(11339*e^2) - (2*(8*c*d - 29*b
*e)*x*(d + e*x^3)^(7/2))/(667*e^2) + (2*c*x^4*(d + e*x^3)^(7/2))/(29*e) + (54*3^
(3/4)*Sqrt[2 + Sqrt[3]]*d^3*(16*c*d^2 - 58*b*d*e + 667*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]])/(124729*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 = 46.9048, size = 379, normalized size = 0.96 \[ \frac{2 c x^{4} \left (d + e x^{3}\right )^{\frac{7}{2}}}{29 e} + \frac{54 \cdot 3^{\frac{3}{4}} d^{3} \sqrt{\frac{d^{\frac{2}{3}} - \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right ) \left (667 a e^{2} - 58 b d e + 16 c d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{d} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{e} x}{\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{124729 e^{\frac{7}{3}} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right )}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{d + e x^{3}}} + \frac{54 d^{2} x \sqrt{d + e x^{3}} \left (667 a e^{2} - 58 b d e + 16 c d^{2}\right )}{124729 e^{2}} + \frac{30 d x \left (d + e x^{3}\right )^{\frac{3}{2}} \left (667 a e^{2} - 58 b d e + 16 c d^{2}\right )}{124729 e^{2}} + \frac{2 x \left (d + e x^{3}\right )^{\frac{7}{2}} \left (29 b e - 8 c d\right )}{667 e^{2}} + \frac{2 x \left (d + e x^{3}\right )^{\frac{5}{2}} \left (667 a e^{2} - 58 b d e + 16 c d^{2}\right )}{11339 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*x**4*(d + e*x**3)**(7/2)/(29*e) + 54*3**(3/4)*d**3*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)*(667*a*e**2 - 58*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))/(124729*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)) + 54*d**2*x*sqrt(d + e*x
**3)*(667*a*e**2 - 58*b*d*e + 16*c*d**2)/(124729*e**2) + 30*d*x*(d + e*x**3)**(3
/2)*(667*a*e**2 - 58*b*d*e + 16*c*d**2)/(124729*e**2) + 2*x*(d + e*x**3)**(7/2)*
(29*b*e - 8*c*d)/(667*e**2) + 2*x*(d + e*x**3)**(5/2)*(667*a*e**2 - 58*b*d*e + 1
6*c*d**2)/(11339*e**2)

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Mathematica [C]  time = 0.581605, size = 279, normalized size = 0.7 \[ -\frac{2 \left (\sqrt [3]{-e} \left (d+e x^3\right ) \left (-11 e^2 x^7 \left (29 e (23 a e+49 b d)+781 c d^2\right )-d e x^4 \left (29 e (851 a e+487 b d)+405 c d^2\right )+d^2 x \left (648 c d^2-29 e (1219 a e+81 b d)\right )-187 e^3 x^{10} (29 b e+61 c d)-4301 c e^4 x^{13}\right )-27 i 3^{3/4} d^{10/3} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}-1\right )} \sqrt{\frac{(-e)^{2/3} x^2}{d^{2/3}}+\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}+1} \left (29 e (23 a e-2 b d)+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-e} x}{\sqrt [3]{d}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{124729 (-e)^{7/3} \sqrt{d+e x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*((-e)^(1/3)*(d + e*x^3)*(d^2*(648*c*d^2 - 29*e*(81*b*d + 1219*a*e))*x - d*e*
(405*c*d^2 + 29*e*(487*b*d + 851*a*e))*x^4 - 11*e^2*(781*c*d^2 + 29*e*(49*b*d +
23*a*e))*x^7 - 187*e^3*(61*c*d + 29*b*e)*x^10 - 4301*c*e^4*x^13) - (27*I)*3^(3/4
)*d^(10/3)*(16*c*d^2 + 29*e*(-2*b*d + 23*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)]*Ellipt
icF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-e)^(1/3)*x)/d^(1/3)]/3^(1/4)], (-1)^(1/3)]))/
(124729*(-e)^(7/3)*Sqrt[d + e*x^3])

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Maple [B]  time = 0.203, size = 1070, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*(2/17*e^2*x^7*(e*x^3+d)^(1/2)+74/187*d*e*x^4*(e*x^3+d)^(1/2)+106/187*d^2*x*(e*
x^3+d)^(1/2)-54/187*I*d^3*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*(2/23*e^2*x^10*(e
*x^3+d)^(1/2)+98/391*d*e*x^7*(e*x^3+d)^(1/2)+974/4301*d^2*x^4*(e*x^3+d)^(1/2)+16
2/4301*d^3/e*x*(e*x^3+d)^(1/2)+108/4301*I*d^4/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*(2/29*e^2*x^13*(e*x^3+d)^(1/2)+122/667*d*e*x^10*(e*x^3+d)^(1/2)+1562/11339*
d^2*x^7*(e*x^3+d)^(1/2)+810/124729*d^3/e*x^4*(e*x^3+d)^(1/2)-1296/124729*d^4/e^2
*x*(e*x^3+d)^(1/2)-864/124729*I*d^5/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. \[ \int{\left (c x^{6} + b x^{3} + a\right )}{\left (e x^{3} + d\right )}^{\frac{5}{2}}\,{d x} \]

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. \[{\rm integral}\left ({\left (c e^{2} x^{12} +{\left (2 \, c d e + b e^{2}\right )} x^{9} +{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{6} +{\left (b d^{2} + 2 \, a d e\right )} x^{3} + a d^{2}\right )} \sqrt{e x^{3} + d}, x\right ) \]

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

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Sympy [A]  time = 25.6834, size = 400, normalized size = 1.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*d**(5/2)*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), e*x**3*exp_polar(I*pi)/d)/(3*
gamma(4/3)) + 2*a*d**(3/2)*e*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), e*x**3*e
xp_polar(I*pi)/d)/(3*gamma(7/3)) + a*sqrt(d)*e**2*x**7*gamma(7/3)*hyper((-1/2, 7
/3), (10/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(10/3)) + b*d**(5/2)*x**4*gamma(
4/3)*hyper((-1/2, 4/3), (7/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(7/3)) + 2*b*d
**(3/2)*e*x**7*gamma(7/3)*hyper((-1/2, 7/3), (10/3,), e*x**3*exp_polar(I*pi)/d)/
(3*gamma(10/3)) + b*sqrt(d)*e**2*x**10*gamma(10/3)*hyper((-1/2, 10/3), (13/3,),
e*x**3*exp_polar(I*pi)/d)/(3*gamma(13/3)) + c*d**(5/2)*x**7*gamma(7/3)*hyper((-1
/2, 7/3), (10/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(10/3)) + 2*c*d**(3/2)*e*x*
*10*gamma(10/3)*hyper((-1/2, 10/3), (13/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(
13/3)) + c*sqrt(d)*e**2*x**13*gamma(13/3)*hyper((-1/2, 13/3), (16/3,), e*x**3*ex
p_polar(I*pi)/d)/(3*gamma(16/3))

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

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)