3.382 \(\int x^7 \sqrt{a+b x^3} \, dx\)

Optimal. Leaf size=535 \[ \frac{80 \sqrt{2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{120 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{240 a^3 \sqrt{a+b x^3}}{1729 b^{8/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{60 a^2 x^2 \sqrt{a+b x^3}}{1729 b^2}+\frac{2}{19} x^8 \sqrt{a+b x^3}+\frac{6 a x^5 \sqrt{a+b x^3}}{247 b} \]

[Out]

(-60*a^2*x^2*Sqrt[a + b*x^3])/(1729*b^2) + (6*a*x^5*Sqrt[a + b*x^3])/(247*b) + (
2*x^8*Sqrt[a + b*x^3])/19 + (240*a^3*Sqrt[a + b*x^3])/(1729*b^(8/3)*((1 + Sqrt[3
])*a^(1/3) + b^(1/3)*x)) - (120*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(10/3)*(a^(1/3) + b^
(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3)
+ b^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt
[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(1729*b^(8/3)*Sqrt[(a^(1/3)*(a^(1/3
) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3]) + (80*Sq
rt[2]*3^(3/4)*a^(10/3)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt
[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])
/(1729*b^(8/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(
1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi [A]  time = 0.653251, antiderivative size = 535, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{80 \sqrt{2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{120 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{1729 b^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{240 a^3 \sqrt{a+b x^3}}{1729 b^{8/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{60 a^2 x^2 \sqrt{a+b x^3}}{1729 b^2}+\frac{2}{19} x^8 \sqrt{a+b x^3}+\frac{6 a x^5 \sqrt{a+b x^3}}{247 b} \]

Antiderivative was successfully verified.

[In]  Int[x^7*Sqrt[a + b*x^3],x]

[Out]

(-60*a^2*x^2*Sqrt[a + b*x^3])/(1729*b^2) + (6*a*x^5*Sqrt[a + b*x^3])/(247*b) + (
2*x^8*Sqrt[a + b*x^3])/19 + (240*a^3*Sqrt[a + b*x^3])/(1729*b^(8/3)*((1 + Sqrt[3
])*a^(1/3) + b^(1/3)*x)) - (120*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(10/3)*(a^(1/3) + b^
(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3)
+ b^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt
[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(1729*b^(8/3)*Sqrt[(a^(1/3)*(a^(1/3
) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3]) + (80*Sq
rt[2]*3^(3/4)*a^(10/3)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x +
b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt
[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])
/(1729*b^(8/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(
1/3)*x)^2]*Sqrt[a + b*x^3])

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Rubi in Sympy [A]  time = 55.7303, size = 476, normalized size = 0.89 \[ - \frac{120 \sqrt [4]{3} a^{\frac{10}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{1729 b^{\frac{8}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} + \frac{80 \sqrt{2} \cdot 3^{\frac{3}{4}} a^{\frac{10}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{1729 b^{\frac{8}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} + \frac{240 a^{3} \sqrt{a + b x^{3}}}{1729 b^{\frac{8}{3}} \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )} - \frac{60 a^{2} x^{2} \sqrt{a + b x^{3}}}{1729 b^{2}} + \frac{6 a x^{5} \sqrt{a + b x^{3}}}{247 b} + \frac{2 x^{8} \sqrt{a + b x^{3}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b*x**3+a)**(1/2),x)

[Out]

-120*3**(1/4)*a**(10/3)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a
**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2)*sqrt(-sqrt(3) + 2)*(a**(1/3) + b**(1/3)*
x)*elliptic_e(asin((-a**(1/3)*(-1 + sqrt(3)) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3
)) + b**(1/3)*x)), -7 - 4*sqrt(3))/(1729*b**(8/3)*sqrt(a**(1/3)*(a**(1/3) + b**(
1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2)*sqrt(a + b*x**3)) + 80*sqrt(2)
*3**(3/4)*a**(10/3)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(a**(1
/3)*(1 + sqrt(3)) + b**(1/3)*x)**2)*(a**(1/3) + b**(1/3)*x)*elliptic_f(asin((-a*
*(1/3)*(-1 + sqrt(3)) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)), -7 -
4*sqrt(3))/(1729*b**(8/3)*sqrt(a**(1/3)*(a**(1/3) + b**(1/3)*x)/(a**(1/3)*(1 +
sqrt(3)) + b**(1/3)*x)**2)*sqrt(a + b*x**3)) + 240*a**3*sqrt(a + b*x**3)/(1729*b
**(8/3)*(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)) - 60*a**2*x**2*sqrt(a + b*x**3)/(
1729*b**2) + 6*a*x**5*sqrt(a + b*x**3)/(247*b) + 2*x**8*sqrt(a + b*x**3)/19

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Mathematica [C]  time = 1.38066, size = 238, normalized size = 0.44 \[ -\frac{2 \left ((-b)^{2/3} \left (a+b x^3\right ) \left (30 a^2 x^2-21 a b x^5-91 b^2 x^8\right )+40 (-1)^{2/3} 3^{3/4} a^{11/3} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}-1\right )} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} \left ((-1)^{5/6} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+\sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )\right )}{1729 (-b)^{8/3} \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^7*Sqrt[a + b*x^3],x]

[Out]

(-2*((-b)^(2/3)*(a + b*x^3)*(30*a^2*x^2 - 21*a*b*x^5 - 91*b^2*x^8) + 40*(-1)^(2/
3)*3^(3/4)*a^(11/3)*Sqrt[(-1)^(5/6)*(-1 + ((-b)^(1/3)*x)/a^(1/3))]*Sqrt[1 + ((-b
)^(1/3)*x)/a^(1/3) + ((-b)^(2/3)*x^2)/a^(2/3)]*(Sqrt[3]*EllipticE[ArcSin[Sqrt[-(
-1)^(5/6) - (I*(-b)^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)] + (-1)^(5/6)*Ellipti
cF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-b)^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)])))/
(1729*(-b)^(8/3)*Sqrt[a + b*x^3])

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Maple [A]  time = 0.025, size = 491, normalized size = 0.9 \[{\frac{2\,{x}^{8}}{19}\sqrt{b{x}^{3}+a}}+{\frac{6\,a{x}^{5}}{247\,b}\sqrt{b{x}^{3}+a}}-{\frac{60\,{a}^{2}{x}^{2}}{1729\,{b}^{2}}\sqrt{b{x}^{3}+a}}-{\frac{{\frac{80\,i}{1729}}{a}^{3}\sqrt{3}}{{b}^{3}}\sqrt [3]{-a{b}^{2}}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}\sqrt{{1 \left ( x-{\frac{1}{b}\sqrt [3]{-a{b}^{2}}} \right ) \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}}\sqrt{{-i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}} \left ( \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}} \right ) +{\frac{1}{b}\sqrt [3]{-a{b}^{2}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}} \right ) } \right ){\frac{1}{\sqrt{b{x}^{3}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b*x^3+a)^(1/2),x)

[Out]

2/19*x^8*(b*x^3+a)^(1/2)+6/247*a*x^5*(b*x^3+a)^(1/2)/b-60/1729*a^2*x^2*(b*x^3+a)
^(1/2)/b^2-80/1729*I*a^3/b^3*3^(1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1
/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^
(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/
b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)
/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*Ellipti
cE(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2
)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2
*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/b*(-a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I
*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3
))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*
b^2)^(1/3)))^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{3} + a} x^{7}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a)*x^7,x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^3 + a)*x^7, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{b x^{3} + a} x^{7}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a)*x^7,x, algorithm="fricas")

[Out]

integral(sqrt(b*x^3 + a)*x^7, x)

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Sympy [A]  time = 3.16944, size = 39, normalized size = 0.07 \[ \frac{\sqrt{a} x^{8} \Gamma \left (\frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{8}{3} \\ \frac{11}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{11}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(b*x**3+a)**(1/2),x)

[Out]

sqrt(a)*x**8*gamma(8/3)*hyper((-1/2, 8/3), (11/3,), b*x**3*exp_polar(I*pi)/a)/(3
*gamma(11/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{3} + a} x^{7}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a)*x^7,x, algorithm="giac")

[Out]

integrate(sqrt(b*x^3 + a)*x^7, x)