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

Optimal. Leaf size=304 \[ \frac{7 \sqrt{a+b x^3} (13 A b-10 a B)}{60 a^3 x^2}-\frac{13 A b-10 a B}{15 a^2 x^2 \sqrt{a+b x^3}}+\frac{7 \sqrt{2+\sqrt{3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (13 A b-10 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{60 \sqrt [4]{3} a^3 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{A}{5 a x^5 \sqrt{a+b x^3}} \]

[Out]

-A/(5*a*x^5*Sqrt[a + b*x^3]) - (13*A*b - 10*a*B)/(15*a^2*x^2*Sqrt[a + b*x^3]) +
(7*(13*A*b - 10*a*B)*Sqrt[a + b*x^3])/(60*a^3*x^2) + (7*Sqrt[2 + Sqrt[3]]*b^(2/3
)*(13*A*b - 10*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 - Sqrt[3]
)*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(6
0*3^(1/4)*a^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 = 0.383128, antiderivative size = 304, 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 \[ \frac{7 \sqrt{a+b x^3} (13 A b-10 a B)}{60 a^3 x^2}-\frac{13 A b-10 a B}{15 a^2 x^2 \sqrt{a+b x^3}}+\frac{7 \sqrt{2+\sqrt{3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (13 A b-10 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{60 \sqrt [4]{3} a^3 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{A}{5 a x^5 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

-A/(5*a*x^5*Sqrt[a + b*x^3]) - (13*A*b - 10*a*B)/(15*a^2*x^2*Sqrt[a + b*x^3]) +
(7*(13*A*b - 10*a*B)*Sqrt[a + b*x^3])/(60*a^3*x^2) + (7*Sqrt[2 + Sqrt[3]]*b^(2/3
)*(13*A*b - 10*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 - Sqrt[3]
)*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(6
0*3^(1/4)*a^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 = 26.3821, size = 275, normalized size = 0.9 \[ - \frac{A}{5 a x^{5} \sqrt{a + b x^{3}}} - \frac{13 A b - 10 B a}{15 a^{2} x^{2} \sqrt{a + b x^{3}}} + \frac{7 \cdot 3^{\frac{3}{4}} b^{\frac{2}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \left (13 A b - 10 B a\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{180 a^{3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{a + b x^{3}}} + \frac{7 \sqrt{a + b x^{3}} \left (13 A b - 10 B a\right )}{60 a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(5*a*x**5*sqrt(a + b*x**3)) - (13*A*b - 10*B*a)/(15*a**2*x**2*sqrt(a + b*x**3
)) + 7*3**(3/4)*b**(2/3)*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)*(13*A*b - 10*B*a)*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))/(180*a**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))
+ 7*sqrt(a + b*x**3)*(13*A*b - 10*B*a)/(60*a**3*x**2)

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Mathematica [C]  time = 0.767826, size = 218, normalized size = 0.72 \[ \sqrt{a+b x^3} \left (-\frac{2 b x (a B-A b)}{3 a^3 \left (a+b x^3\right )}+\frac{17 A b-10 a B}{20 a^3 x^2}-\frac{A}{5 a^2 x^5}\right )-\frac{7 i b \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}-1\right )} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} (10 a B-13 A b) F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{60 \sqrt [4]{3} a^{8/3} \sqrt [3]{-b} \sqrt{a+b x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

Sqrt[a + b*x^3]*(-A/(5*a^2*x^5) + (17*A*b - 10*a*B)/(20*a^3*x^2) - (2*b*(-(A*b)
+ a*B)*x)/(3*a^3*(a + b*x^3))) - (((7*I)/60)*b*(-13*A*b + 10*a*B)*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)])/(3^(1/4)*a^(8/3)*(-b)^(1/3)*Sqrt[a + b*x^3])

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Maple [B]  time = 0.013, size = 667, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(2/3*b^2/a^3*x/((x^3+a/b)*b)^(1/2)-1/5/a^2*(b*x^3+a)^(1/2)/x^5+17/20/a^3*b*(b*
x^3+a)^(1/2)/x^2-91/180*I/a^3*b*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/a^2*x
/((x^3+a/b)*b)^(1/2)-1/2/a^2*(b*x^3+a)^(1/2)/x^2+7/18*I/a^2*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. \[ \int \frac{B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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