3.234 \(\int \frac{x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=300 \[ -\frac{32 \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (11 A b-14 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{165 \sqrt [4]{3} b^{10/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{16 x \sqrt{a+b x^3} (11 A b-14 a B)}{165 b^3}-\frac{2 x^4 (11 A b-14 a B)}{33 b^2 \sqrt{a+b x^3}}+\frac{2 B x^7}{11 b \sqrt{a+b x^3}} \]

[Out]

(-2*(11*A*b - 14*a*B)*x^4)/(33*b^2*Sqrt[a + b*x^3]) + (2*B*x^7)/(11*b*Sqrt[a + b
*x^3]) + (16*(11*A*b - 14*a*B)*x*Sqrt[a + b*x^3])/(165*b^3) - (32*Sqrt[2 + Sqrt[
3]]*a*(11*A*b - 14*a*B)*(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 - Sqr
t[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]]
)/(165*3^(1/4)*b^(10/3)*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 = 0.411705, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{32 \sqrt{2+\sqrt{3}} a \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (11 A b-14 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{165 \sqrt [4]{3} b^{10/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{16 x \sqrt{a+b x^3} (11 A b-14 a B)}{165 b^3}-\frac{2 x^4 (11 A b-14 a B)}{33 b^2 \sqrt{a+b x^3}}+\frac{2 B x^7}{11 b \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(11*A*b - 14*a*B)*x^4)/(33*b^2*Sqrt[a + b*x^3]) + (2*B*x^7)/(11*b*Sqrt[a + b
*x^3]) + (16*(11*A*b - 14*a*B)*x*Sqrt[a + b*x^3])/(165*b^3) - (32*Sqrt[2 + Sqrt[
3]]*a*(11*A*b - 14*a*B)*(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 - Sqr
t[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]]
)/(165*3^(1/4)*b^(10/3)*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 = 25.9981, size = 275, normalized size = 0.92 \[ \frac{2 B x^{7}}{11 b \sqrt{a + b x^{3}}} - \frac{32 \cdot 3^{\frac{3}{4}} a \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (11 A b - 14 B a\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{495 b^{\frac{10}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} - \frac{2 x^{4} \left (11 A b - 14 B a\right )}{33 b^{2} \sqrt{a + b x^{3}}} + \frac{16 x \sqrt{a + b x^{3}} \left (11 A b - 14 B a\right )}{165 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*x**7/(11*b*sqrt(a + b*x**3)) - 32*3**(3/4)*a*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)*(11*A*b - 14*B*a)*elliptic_f(asin((-a**(1/3)*(-1 + sq
rt(3)) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)), -7 - 4*sqrt(3))/(49
5*b**(10/3)*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)) - 2*x**4*(11*A*b - 14*B*a)/(33*b**2*sqrt(a + b*x**
3)) + 16*x*sqrt(a + b*x**3)*(11*A*b - 14*B*a)/(165*b**3)

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Mathematica [C]  time = 0.909361, size = 205, normalized size = 0.68 \[ \frac{-6 \sqrt [3]{-b} x \left (-112 a^2 B+a \left (88 A b-42 b B x^3\right )+3 b^2 x^3 \left (11 A+5 B x^3\right )\right )+32 i 3^{3/4} a^{4/3} \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{-b} x-\sqrt [3]{a}\right )}{\sqrt [3]{a}}} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} (11 A b-14 a B) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{495 (-b)^{10/3} \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-6*(-b)^(1/3)*x*(-112*a^2*B + 3*b^2*x^3*(11*A + 5*B*x^3) + a*(88*A*b - 42*b*B*x
^3)) + (32*I)*3^(3/4)*a^(4/3)*(11*A*b - 14*a*B)*Sqrt[((-1)^(5/6)*(-a^(1/3) + (-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)])/(495*(-b)^(10/3)*Sqrt[a + b*x^3])

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Maple [B]  time = 0.032, size = 666, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(2/3/b^2*a*x/((x^3+a/b)*b)^(1/2)+2/5/b^2*x*(b*x^3+a)^(1/2)+32/45*I*a/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)))+B*(-2/3/b^3*a^2*x/((x^3+a/b)*b)^(1/2)+2/11/b^2*x^4*(
b*x^3+a)^(1/2)-38/55*a/b^3*x*(b*x^3+a)^(1/2)-448/495*I*a^2/b^4*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. \[ \int \frac{{\left (B x^{3} + A\right )} x^{6}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^3 + A)*x^6/(b*x^3 + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{9} + A x^{6}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^9 + A*x^6)/(b*x^3 + a)^(3/2), x)

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Sympy [A]  time = 173.336, size = 80, normalized size = 0.27 \[ \frac{A x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{2}} \Gamma \left (\frac{10}{3}\right )} + \frac{B x^{10} \Gamma \left (\frac{10}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{10}{3} \\ \frac{13}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{2}} \Gamma \left (\frac{13}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*x**7*gamma(7/3)*hyper((3/2, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(3/
2)*gamma(10/3)) + B*x**10*gamma(10/3)*hyper((3/2, 10/3), (13/3,), b*x**3*exp_pol
ar(I*pi)/a)/(3*a**(3/2)*gamma(13/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^3 + A)*x^6/(b*x^3 + a)^(3/2), x)