3.115 \(\int \frac{1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx\)

Optimal. Leaf size=398 \[ \frac{13}{108 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^2}+\frac{1}{12 a x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^3}+\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \left (a+b x^3\right )}{243 a^5 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455}{972 a^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{65}{324 a^3 x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )} \]

[Out]

455/(972*a^4*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(12*a*x*(a + b*x^3)^3*Sqrt[a
^2 + 2*a*b*x^3 + b^2*x^6]) + 13/(108*a^2*x*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6]) + 65/(324*a^3*x*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (455*(a
+ b*x^3))/(243*a^5*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (455*b^(1/3)*(a + b*x^3
)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a^(16/3)*Sqrt[
a^2 + 2*a*b*x^3 + b^2*x^6]) + (455*b^(1/3)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])
/(729*a^(16/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (455*b^(1/3)*(a + b*x^3)*Log[a
^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1458*a^(16/3)*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6])

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Rubi [A]  time = 0.477252, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346 \[ \frac{13}{108 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^2}+\frac{1}{12 a x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^3}+\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{455 \left (a+b x^3\right )}{243 a^5 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{455}{972 a^4 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{65}{324 a^3 x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]

[Out]

455/(972*a^4*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(12*a*x*(a + b*x^3)^3*Sqrt[a
^2 + 2*a*b*x^3 + b^2*x^6]) + 13/(108*a^2*x*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3 +
b^2*x^6]) + 65/(324*a^3*x*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (455*(a
+ b*x^3))/(243*a^5*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (455*b^(1/3)*(a + b*x^3
)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a^(16/3)*Sqrt[
a^2 + 2*a*b*x^3 + b^2*x^6]) + (455*b^(1/3)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])
/(729*a^(16/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (455*b^(1/3)*(a + b*x^3)*Log[a
^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1458*a^(16/3)*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(1/x**2/(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 = 0.276389, size = 242, normalized size = 0.61 \[ \frac{\left (a+b x^3\right ) \left (-910 \sqrt [3]{b} \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-243 a^{10/3} b x^2-1179 a^{4/3} b x^2 \left (a+b x^3\right )^2-594 a^{7/3} b x^2 \left (a+b x^3\right )-\frac{2916 \sqrt [3]{a} \left (a+b x^3\right )^4}{x}+1820 \sqrt [3]{b} \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-1820 \sqrt{3} \sqrt [3]{b} \left (a+b x^3\right )^4 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-2544 \sqrt [3]{a} b x^2 \left (a+b x^3\right )^3\right )}{2916 a^{16/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]

[Out]

((a + b*x^3)*(-243*a^(10/3)*b*x^2 - 594*a^(7/3)*b*x^2*(a + b*x^3) - 1179*a^(4/3)
*b*x^2*(a + b*x^3)^2 - 2544*a^(1/3)*b*x^2*(a + b*x^3)^3 - (2916*a^(1/3)*(a + b*x
^3)^4)/x - 1820*Sqrt[3]*b^(1/3)*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(S
qrt[3]*a^(1/3))] + 1820*b^(1/3)*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x] - 910*b^(
1/3)*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(2916*a^(16/
3)*((a + b*x^3)^2)^(5/2))

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Maple [B]  time = 0.033, size = 536, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

-1/2916*(-1820*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^13*b
^4-1820*ln(x+(a/b)^(1/3))*x^13*b^4+910*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^13*b^
4+5460*(a/b)^(1/3)*x^12*b^4-7280*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/
3))*3^(1/2)*x^10*a*b^3-7280*ln(x+(a/b)^(1/3))*x^10*a*b^3+3640*ln(x^2-x*(a/b)^(1/
3)+(a/b)^(2/3))*x^10*a*b^3+20475*(a/b)^(1/3)*x^9*a*b^3-10920*arctan(1/3*(-2*x+(a
/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^7*a^2*b^2-10920*ln(x+(a/b)^(1/3))*x^7*
a^2*b^2+5460*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^7*a^2*b^2+28080*(a/b)^(1/3)*x^6
*a^2*b^2-7280*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^4*a^3
*b-7280*ln(x+(a/b)^(1/3))*x^4*a^3*b+3640*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^4*a
^3*b+16224*(a/b)^(1/3)*x^3*a^3*b-1820*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b
)^(1/3))*3^(1/2)*x*a^4-1820*ln(x+(a/b)^(1/3))*x*a^4+910*ln(x^2-x*(a/b)^(1/3)+(a/
b)^(2/3))*x*a^4+2916*(a/b)^(1/3)*a^4)*(b*x^3+a)/x/(a/b)^(1/3)/a^5/((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(1/((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.268049, size = 455, normalized size = 1.14 \[ -\frac{\sqrt{3}{\left (910 \, \sqrt{3}{\left (b^{4} x^{13} + 4 \, a b^{3} x^{10} + 6 \, a^{2} b^{2} x^{7} + 4 \, a^{3} b x^{4} + a^{4} x\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 1820 \, \sqrt{3}{\left (b^{4} x^{13} + 4 \, a b^{3} x^{10} + 6 \, a^{2} b^{2} x^{7} + 4 \, a^{3} b x^{4} + a^{4} x\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 5460 \,{\left (b^{4} x^{13} + 4 \, a b^{3} x^{10} + 6 \, a^{2} b^{2} x^{7} + 4 \, a^{3} b x^{4} + a^{4} x\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (1820 \, b^{4} x^{12} + 6825 \, a b^{3} x^{9} + 9360 \, a^{2} b^{2} x^{6} + 5408 \, a^{3} b x^{3} + 972 \, a^{4}\right )}\right )}}{8748 \,{\left (a^{5} b^{4} x^{13} + 4 \, a^{6} b^{3} x^{10} + 6 \, a^{7} b^{2} x^{7} + 4 \, a^{8} b x^{4} + a^{9} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*x^2),x, algorithm="fricas")

[Out]

-1/8748*sqrt(3)*(910*sqrt(3)*(b^4*x^13 + 4*a*b^3*x^10 + 6*a^2*b^2*x^7 + 4*a^3*b*
x^4 + a^4*x)*(b/a)^(1/3)*log(b*x^2 - a*x*(b/a)^(2/3) + a*(b/a)^(1/3)) - 1820*sqr
t(3)*(b^4*x^13 + 4*a*b^3*x^10 + 6*a^2*b^2*x^7 + 4*a^3*b*x^4 + a^4*x)*(b/a)^(1/3)
*log(b*x + a*(b/a)^(2/3)) - 5460*(b^4*x^13 + 4*a*b^3*x^10 + 6*a^2*b^2*x^7 + 4*a^
3*b*x^4 + a^4*x)*(b/a)^(1/3)*arctan(-1/3*(2*sqrt(3)*b*x - sqrt(3)*a*(b/a)^(2/3))
/(a*(b/a)^(2/3))) + 3*sqrt(3)*(1820*b^4*x^12 + 6825*a*b^3*x^9 + 9360*a^2*b^2*x^6
+ 5408*a^3*b*x^3 + 972*a^4))/(a^5*b^4*x^13 + 4*a^6*b^3*x^10 + 6*a^7*b^2*x^7 + 4
*a^8*b*x^4 + a^9*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(1/(x**2*((a + b*x**3)**2)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.657459, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*x^2),x, algorithm="giac")

[Out]

sage0*x