3.107 \(\int \frac{x^6}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx\)

Optimal. Leaf size=359 \[ \frac{x}{162 a b^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{5 x}{486 a^2 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x}{27 b^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x^4}{12 b \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{5 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{8/3} b^{7/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{5 \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{8/3} b^{7/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{5 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{8/3} b^{7/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

(5*x)/(486*a^2*b^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - x^4/(12*b*(a + b*x^3)^3*Sq
rt[a^2 + 2*a*b*x^3 + b^2*x^6]) - x/(27*b^2*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6]) + x/(162*a*b^2*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (5*(a +
b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a^(8/3)*b
^(7/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (5*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x
])/(729*a^(8/3)*b^(7/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (5*(a + b*x^3)*Log[a^
(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1458*a^(8/3)*b^(7/3)*Sqrt[a^2 + 2*a*b
*x^3 + b^2*x^6])

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Rubi [A]  time = 0.411326, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346 \[ \frac{x}{162 a b^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{5 x}{486 a^2 b^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x}{27 b^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x^4}{12 b \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{5 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{8/3} b^{7/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{5 \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{8/3} b^{7/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{5 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{8/3} b^{7/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(5*x)/(486*a^2*b^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - x^4/(12*b*(a + b*x^3)^3*Sq
rt[a^2 + 2*a*b*x^3 + b^2*x^6]) - x/(27*b^2*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6]) + x/(162*a*b^2*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (5*(a +
b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a^(8/3)*b
^(7/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (5*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x
])/(729*a^(8/3)*b^(7/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (5*(a + b*x^3)*Log[a^
(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1458*a^(8/3)*b^(7/3)*Sqrt[a^2 + 2*a*b
*x^3 + b^2*x^6])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.265722, size = 218, normalized size = 0.61 \[ \frac{\left (a+b x^3\right ) \left (-\frac{10 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{8/3}}+\frac{20 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{8/3}}+\frac{20 \sqrt{3} \left (a+b x^3\right )^4 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{8/3}}+\frac{30 \sqrt [3]{b} x \left (a+b x^3\right )^3}{a^2}+\frac{18 \sqrt [3]{b} x \left (a+b x^3\right )^2}{a}-351 \sqrt [3]{b} x \left (a+b x^3\right )+243 a \sqrt [3]{b} x\right )}{2916 b^{7/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

((a + b*x^3)*(243*a*b^(1/3)*x - 351*b^(1/3)*x*(a + b*x^3) + (18*b^(1/3)*x*(a + b
*x^3)^2)/a + (30*b^(1/3)*x*(a + b*x^3)^3)/a^2 + (20*Sqrt[3]*(a + b*x^3)^4*ArcTan
[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/a^(8/3) + (20*(a + b*x^3)^4*Log[a^
(1/3) + b^(1/3)*x])/a^(8/3) - (10*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2])/a^(8/3)))/(2916*b^(7/3)*((a + b*x^3)^2)^(5/2))

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Maple [B]  time = 0.028, size = 519, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

1/2916*(-20*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^12*b^4+
20*ln(x+(a/b)^(1/3))*x^12*b^4-10*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^12*b^4+30*(
a/b)^(2/3)*x^10*b^4-80*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2
)*x^9*a*b^3+80*ln(x+(a/b)^(1/3))*x^9*a*b^3-40*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*
x^9*a*b^3+108*(a/b)^(2/3)*x^7*a*b^3-120*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a
/b)^(1/3))*3^(1/2)*x^6*a^2*b^2+120*ln(x+(a/b)^(1/3))*x^6*a^2*b^2-60*ln(x^2-x*(a/
b)^(1/3)+(a/b)^(2/3))*x^6*a^2*b^2-225*(a/b)^(2/3)*x^4*a^2*b^2-80*arctan(1/3*(-2*
x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^3*a^3*b+80*ln(x+(a/b)^(1/3))*x^3*a
^3*b-40*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^3*a^3*b-60*(a/b)^(2/3)*x*a^3*b-20*ar
ctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*a^4+20*ln(x+(a/b)^(1/3)
)*a^4-10*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^4)*(b*x^3+a)/(a/b)^(2/3)/b^3/a^2/((
b*x^3+a)^2)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274164, size = 416, normalized size = 1.16 \[ -\frac{\sqrt{3}{\left (10 \, \sqrt{3}{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 20 \, \sqrt{3}{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 60 \,{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (10 \, b^{3} x^{10} + 36 \, a b^{2} x^{7} - 75 \, a^{2} b x^{4} - 20 \, a^{3} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{8748 \,{\left (a^{2} b^{6} x^{12} + 4 \, a^{3} b^{5} x^{9} + 6 \, a^{4} b^{4} x^{6} + 4 \, a^{5} b^{3} x^{3} + a^{6} b^{2}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/8748*sqrt(3)*(10*sqrt(3)*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^
3 + a^4)*log((a^2*b)^(2/3)*x^2 - (a^2*b)^(1/3)*a*x + a^2) - 20*sqrt(3)*(b^4*x^12
+ 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*log((a^2*b)^(1/3)*x + a) - 6
0*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*arctan(1/3*(2*sqr
t(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)/a) - 3*sqrt(3)*(10*b^3*x^10 + 36*a*b^2*x^7 - 7
5*a^2*b*x^4 - 20*a^3*x)*(a^2*b)^(1/3))/((a^2*b^6*x^12 + 4*a^3*b^5*x^9 + 6*a^4*b^
4*x^6 + 4*a^5*b^3*x^3 + a^6*b^2)*(a^2*b)^(1/3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(x**6/((a + b*x**3)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 1.10826, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x