3.394 \(\int x^6 \left (a+b x^3\right )^{3/2} \, dx\)

Optimal. Leaf size=296 \[ -\frac{432 a^3 x \sqrt{a+b x^3}}{21505 b^2}+\frac{54 a^2 x^4 \sqrt{a+b x^3}}{4301 b}+\frac{288\ 3^{3/4} \sqrt{2+\sqrt{3}} a^4 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{21505 b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2}{23} x^7 \left (a+b x^3\right )^{3/2}+\frac{18}{391} a x^7 \sqrt{a+b x^3} \]

[Out]

(-432*a^3*x*Sqrt[a + b*x^3])/(21505*b^2) + (54*a^2*x^4*Sqrt[a + b*x^3])/(4301*b)
+ (18*a*x^7*Sqrt[a + b*x^3])/391 + (2*x^7*(a + b*x^3)^(3/2))/23 + (288*3^(3/4)*
Sqrt[2 + Sqrt[3]]*a^4*(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]])/
(21505*b^(7/3)*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 = 0.322909, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{432 a^3 x \sqrt{a+b x^3}}{21505 b^2}+\frac{54 a^2 x^4 \sqrt{a+b x^3}}{4301 b}+\frac{288\ 3^{3/4} \sqrt{2+\sqrt{3}} a^4 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{21505 b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2}{23} x^7 \left (a+b x^3\right )^{3/2}+\frac{18}{391} a x^7 \sqrt{a+b x^3} \]

Antiderivative was successfully verified.

[In]  Int[x^6*(a + b*x^3)^(3/2),x]

[Out]

(-432*a^3*x*Sqrt[a + b*x^3])/(21505*b^2) + (54*a^2*x^4*Sqrt[a + b*x^3])/(4301*b)
+ (18*a*x^7*Sqrt[a + b*x^3])/391 + (2*x^7*(a + b*x^3)^(3/2))/23 + (288*3^(3/4)*
Sqrt[2 + Sqrt[3]]*a^4*(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]])/
(21505*b^(7/3)*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 = 29.3998, size = 267, normalized size = 0.9 \[ \frac{288 \cdot 3^{\frac{3}{4}} a^{4} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{21505 b^{\frac{7}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} - \frac{432 a^{3} x \sqrt{a + b x^{3}}}{21505 b^{2}} + \frac{54 a^{2} x^{4} \sqrt{a + b x^{3}}}{4301 b} + \frac{18 a x^{7} \sqrt{a + b x^{3}}}{391} + \frac{2 x^{7} \left (a + b x^{3}\right )^{\frac{3}{2}}}{23} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

288*3**(3/4)*a**4*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)*elli
ptic_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))/(21505*b**(7/3)*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)) - 432*a**3*x*sqrt(
a + b*x**3)/(21505*b**2) + 54*a**2*x**4*sqrt(a + b*x**3)/(4301*b) + 18*a*x**7*sq
rt(a + b*x**3)/391 + 2*x**7*(a + b*x**3)**(3/2)/23

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Mathematica [C]  time = 0.423289, size = 195, normalized size = 0.66 \[ \sqrt{a+b x^3} \left (-\frac{432 a^3 x}{21505 b^2}+\frac{54 a^2 x^4}{4301 b}+\frac{52 a x^7}{391}+\frac{2 b x^{10}}{23}\right )+\frac{288 i 3^{3/4} a^{13/3} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}-1\right )} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{21505 \sqrt [3]{-b} b^2 \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

Sqrt[a + b*x^3]*((-432*a^3*x)/(21505*b^2) + (54*a^2*x^4)/(4301*b) + (52*a*x^7)/3
91 + (2*b*x^10)/23) + (((288*I)/21505)*3^(3/4)*a^(13/3)*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
)]*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-b)^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^
(1/3)])/((-b)^(1/3)*b^2*Sqrt[a + b*x^3])

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Maple [A]  time = 0.024, size = 355, normalized size = 1.2 \[{\frac{2\,b{x}^{10}}{23}\sqrt{b{x}^{3}+a}}+{\frac{52\,a{x}^{7}}{391}\sqrt{b{x}^{3}+a}}+{\frac{54\,{x}^{4}{a}^{2}}{4301\,b}\sqrt{b{x}^{3}+a}}-{\frac{432\,{a}^{3}x}{21505\,{b}^{2}}\sqrt{b{x}^{3}+a}}-{\frac{{\frac{288\,i}{21505}}{a}^{4}\sqrt{3}}{{b}^{3}}\sqrt [3]{-a{b}^{2}}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}\sqrt{{1 \left ( x-{\frac{1}{b}\sqrt [3]{-a{b}^{2}}} \right ) \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}}\sqrt{{-i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-a{b}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ){\frac{1}{\sqrt [3]{-a{b}^{2}}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}} \left ( -{\frac{3}{2\,b}\sqrt [3]{-a{b}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-a{b}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{b{x}^{3}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(b*x^3+a)^(3/2),x)

[Out]

2/23*b*x^10*(b*x^3+a)^(1/2)+52/391*a*x^7*(b*x^3+a)^(1/2)+54/4301*a^2*x^4*(b*x^3+
a)^(1/2)/b-432/21505*a^3*x*(b*x^3+a)^(1/2)/b^2-288/21505*I*a^4/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)*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. \[ \int{\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(3/2)*x^6, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.35204, size = 39, normalized size = 0.13 \[ \frac{a^{\frac{3}{2}} x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*x**7*gamma(7/3)*hyper((-3/2, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(
3*gamma(10/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(3/2)*x^6, x)