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

Optimal. Leaf size=356 \[ \frac{2 x \left (d+e x^3\right )^{3/2} \left (391 a e^2-46 b d e+16 c d^2\right )}{4301 e^2}+\frac{18 d x \sqrt{d+e x^3} \left (391 a e^2-46 b d e+16 c d^2\right )}{21505 e^2}+\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} d^2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (391 a e^2-46 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{21505 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}-\frac{2 x \left (d+e x^3\right )^{5/2} (8 c d-23 b e)}{391 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e} \]

[Out]

(18*d*(16*c*d^2 - 46*b*d*e + 391*a*e^2)*x*Sqrt[d + e*x^3])/(21505*e^2) + (2*(16*
c*d^2 - 46*b*d*e + 391*a*e^2)*x*(d + e*x^3)^(3/2))/(4301*e^2) - (2*(8*c*d - 23*b
*e)*x*(d + e*x^3)^(5/2))/(391*e^2) + (2*c*x^4*(d + e*x^3)^(5/2))/(23*e) + (18*3^
(3/4)*Sqrt[2 + Sqrt[3]]*d^2*(16*c*d^2 - 46*b*d*e + 391*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]])/(21505*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])

_______________________________________________________________________________________

Rubi [A]  time = 0.601849, antiderivative size = 356, 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 \[ \frac{2 x \left (d+e x^3\right )^{3/2} \left (391 a e^2-46 b d e+16 c d^2\right )}{4301 e^2}+\frac{18 d x \sqrt{d+e x^3} \left (391 a e^2-46 b d e+16 c d^2\right )}{21505 e^2}+\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} d^2 \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt{\frac{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \left (391 a e^2-46 b d e+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{e} x+\left (1-\sqrt{3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt{3}\right ) \sqrt [3]{d}}\right )|-7-4 \sqrt{3}\right )}{21505 e^{7/3} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt{d+e x^3}}-\frac{2 x \left (d+e x^3\right )^{5/2} (8 c d-23 b e)}{391 e^2}+\frac{2 c x^4 \left (d+e x^3\right )^{5/2}}{23 e} \]

Antiderivative was successfully verified.

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

[Out]

(18*d*(16*c*d^2 - 46*b*d*e + 391*a*e^2)*x*Sqrt[d + e*x^3])/(21505*e^2) + (2*(16*
c*d^2 - 46*b*d*e + 391*a*e^2)*x*(d + e*x^3)^(3/2))/(4301*e^2) - (2*(8*c*d - 23*b
*e)*x*(d + e*x^3)^(5/2))/(391*e^2) + (2*c*x^4*(d + e*x^3)^(5/2))/(23*e) + (18*3^
(3/4)*Sqrt[2 + Sqrt[3]]*d^2*(16*c*d^2 - 46*b*d*e + 391*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]])/(21505*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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 39.8086, size = 337, normalized size = 0.95 \[ \frac{2 c x^{4} \left (d + e x^{3}\right )^{\frac{5}{2}}}{23 e} + \frac{18 \cdot 3^{\frac{3}{4}} d^{2} \sqrt{\frac{d^{\frac{2}{3}} - \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right ) \left (391 a e^{2} - 46 b d e + 16 c d^{2}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{d} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{e} x}{\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{21505 e^{\frac{7}{3}} \sqrt{\frac{\sqrt [3]{d} \left (\sqrt [3]{d} + \sqrt [3]{e} x\right )}{\left (\sqrt [3]{d} \left (1 + \sqrt{3}\right ) + \sqrt [3]{e} x\right )^{2}}} \sqrt{d + e x^{3}}} + \frac{18 d x \sqrt{d + e x^{3}} \left (391 a e^{2} - 46 b d e + 16 c d^{2}\right )}{21505 e^{2}} + \frac{2 x \left (d + e x^{3}\right )^{\frac{5}{2}} \left (23 b e - 8 c d\right )}{391 e^{2}} + \frac{2 x \left (d + e x^{3}\right )^{\frac{3}{2}} \left (391 a e^{2} - 46 b d e + 16 c d^{2}\right )}{4301 e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*x**4*(d + e*x**3)**(5/2)/(23*e) + 18*3**(3/4)*d**2*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)*(391*a*e**2 - 46*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))/(21505*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)) + 18*d*x*sqrt(d + e*x**3)
*(391*a*e**2 - 46*b*d*e + 16*c*d**2)/(21505*e**2) + 2*x*(d + e*x**3)**(5/2)*(23*
b*e - 8*c*d)/(391*e**2) + 2*x*(d + e*x**3)**(3/2)*(391*a*e**2 - 46*b*d*e + 16*c*
d**2)/(4301*e**2)

_______________________________________________________________________________________

Mathematica [C]  time = 0.523604, size = 249, normalized size = 0.7 \[ -\frac{2 \left (\sqrt [3]{-e} \left (d+e x^3\right ) \left (-5 e x^4 \left (23 e (17 a e+20 b d)+27 c d^2\right )+d x \left (216 c d^2-23 e (238 a e+27 b d)\right )-55 e^2 x^7 (23 b e+26 c d)-935 c e^3 x^{10}\right )-9 i 3^{3/4} d^{7/3} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}-1\right )} \sqrt{\frac{(-e)^{2/3} x^2}{d^{2/3}}+\frac{\sqrt [3]{-e} x}{\sqrt [3]{d}}+1} \left (23 e (17 a e-2 b d)+16 c d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-e} x}{\sqrt [3]{d}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{21505 (-e)^{7/3} \sqrt{d+e x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*((-e)^(1/3)*(d + e*x^3)*(d*(216*c*d^2 - 23*e*(27*b*d + 238*a*e))*x - 5*e*(27
*c*d^2 + 23*e*(20*b*d + 17*a*e))*x^4 - 55*e^2*(26*c*d + 23*b*e)*x^7 - 935*c*e^3*
x^10) - (9*I)*3^(3/4)*d^(7/3)*(16*c*d^2 + 23*e*(-2*b*d + 17*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)]))/(21505*(-e)^(7/3)*Sqrt[d + e*x^3])

_______________________________________________________________________________________

Maple [B]  time = 0.045, size = 1010, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*(2/11*e*x^4*(e*x^3+d)^(1/2)+28/55*d*x*(e*x^3+d)^(1/2)-18/55*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)*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*(2/17*e*x^7*(e*x^3+d)^(1/2)+40/187*d*x^4*(e*x^3+d)^(1/2
)+54/935*d^2/e*x*(e*x^3+d)^(1/2)+36/935*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)*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*(2/23*e*x^10*(e*x^3+d)^(1/2)+52/391*d*x^7*(e*x^3+d)^(1/2)+54/4301*d^2/e*x^4
*(e*x^3+d)^(1/2)-432/21505*d^3/e^2*x*(e*x^3+d)^(1/2)-288/21505*I*d^4/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)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 13.5335, size = 257, normalized size = 0.72 \[ \frac{a d^{\frac{3}{2}} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} + \frac{a \sqrt{d} e x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{b d^{\frac{3}{2}} x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{b \sqrt{d} e x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} + \frac{c d^{\frac{3}{2}} x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} + \frac{c \sqrt{d} e x^{10} \Gamma \left (\frac{10}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{10}{3} \\ \frac{13}{3} \end{matrix}\middle |{\frac{e x^{3} e^{i \pi }}{d}} \right )}}{3 \Gamma \left (\frac{13}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*d**(3/2)*x*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), e*x**3*exp_polar(I*pi)/d)/(3*
gamma(4/3)) + a*sqrt(d)*e*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7/3,), e*x**3*exp_
polar(I*pi)/d)/(3*gamma(7/3)) + b*d**(3/2)*x**4*gamma(4/3)*hyper((-1/2, 4/3), (7
/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(7/3)) + b*sqrt(d)*e*x**7*gamma(7/3)*hyp
er((-1/2, 7/3), (10/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(10/3)) + c*d**(3/2)*
x**7*gamma(7/3)*hyper((-1/2, 7/3), (10/3,), e*x**3*exp_polar(I*pi)/d)/(3*gamma(1
0/3)) + c*sqrt(d)*e*x**10*gamma(10/3)*hyper((-1/2, 10/3), (13/3,), e*x**3*exp_po
lar(I*pi)/d)/(3*gamma(13/3))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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