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

Optimal. Leaf size=359 \[ \frac{5 x^2}{54 a^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{x^2}{12 a \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{35 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{13/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{35 \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{13/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{35 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{13/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{35 x^2}{243 a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{35 x^2}{324 a^3 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

(35*x^2)/(243*a^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + x^2/(12*a*(a + b*x^3)^3*Sqr
t[a^2 + 2*a*b*x^3 + b^2*x^6]) + (5*x^2)/(54*a^2*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x
^3 + b^2*x^6]) + (35*x^2)/(324*a^3*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])
- (35*(a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3
]*a^(13/3)*b^(2/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (35*(a + b*x^3)*Log[a^(1/3
) + b^(1/3)*x])/(729*a^(13/3)*b^(2/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (35*(a
+ b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1458*a^(13/3)*b^(2/3)*
Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

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Rubi [A]  time = 0.398218, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{5 x^2}{54 a^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{x^2}{12 a \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{35 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{13/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{35 \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{13/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{35 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{13/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{35 x^2}{243 a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{35 x^2}{324 a^3 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

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

[Out]

(35*x^2)/(243*a^4*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + x^2/(12*a*(a + b*x^3)^3*Sqr
t[a^2 + 2*a*b*x^3 + b^2*x^6]) + (5*x^2)/(54*a^2*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x
^3 + b^2*x^6]) + (35*x^2)/(324*a^3*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])
- (35*(a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3
]*a^(13/3)*b^(2/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (35*(a + b*x^3)*Log[a^(1/3
) + b^(1/3)*x])/(729*a^(13/3)*b^(2/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (35*(a
+ b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1458*a^(13/3)*b^(2/3)*
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/(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 = 0.235952, size = 219, normalized size = 0.61 \[ \frac{\left (a+b x^3\right ) \left (\frac{70 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+315 a^{4/3} x^2 \left (a+b x^3\right )^2+270 a^{7/3} x^2 \left (a+b x^3\right )+243 a^{10/3} x^2-\frac{140 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac{140 \sqrt{3} \left (a+b x^3\right )^4 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{b^{2/3}}+420 \sqrt [3]{a} x^2 \left (a+b x^3\right )^3\right )}{2916 a^{13/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^3)*(243*a^(10/3)*x^2 + 270*a^(7/3)*x^2*(a + b*x^3) + 315*a^(4/3)*x^2*(
a + b*x^3)^2 + 420*a^(1/3)*x^2*(a + b*x^3)^3 + (140*Sqrt[3]*(a + b*x^3)^4*ArcTan
[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/b^(2/3) - (140*(a + b*x^3)^4*Log[a
^(1/3) + b^(1/3)*x])/b^(2/3) + (70*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2])/b^(2/3)))/(2916*a^(13/3)*((a + b*x^3)^2)^(5/2))

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Maple [B]  time = 0.015, size = 521, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2916*(-140*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^12*b^4
-140*ln(x+(a/b)^(1/3))*x^12*b^4+70*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^12*b^4+42
0*(a/b)^(1/3)*x^11*b^4-560*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^
(1/2)*x^9*a*b^3-560*ln(x+(a/b)^(1/3))*x^9*a*b^3+280*ln(x^2-x*(a/b)^(1/3)+(a/b)^(
2/3))*x^9*a*b^3+1575*(a/b)^(1/3)*x^8*a*b^3-840*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(
1/2)/(a/b)^(1/3))*3^(1/2)*x^6*a^2*b^2-840*ln(x+(a/b)^(1/3))*x^6*a^2*b^2+420*ln(x
^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^6*a^2*b^2+2160*(a/b)^(1/3)*x^5*a^2*b^2-560*arcta
n(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^3*a^3*b-560*ln(x+(a/b)^(
1/3))*x^3*a^3*b+280*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^3*a^3*b+1248*(a/b)^(1/3)
*x^2*a^3*b-140*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*a^4-14
0*ln(x+(a/b)^(1/3))*a^4+70*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^4)*(b*x^3+a)/(a/b
)^(1/3)/b/a^4/((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/(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 = 0.278379, size = 425, normalized size = 1.18 \[ -\frac{\sqrt{3}{\left (70 \, \sqrt{3}{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 140 \, \sqrt{3}{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 420 \,{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (140 \, b^{3} x^{11} + 525 \, a b^{2} x^{8} + 720 \, a^{2} b x^{5} + 416 \, a^{3} x^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{8748 \,{\left (a^{4} b^{4} x^{12} + 4 \, a^{5} b^{3} x^{9} + 6 \, a^{6} b^{2} x^{6} + 4 \, a^{7} b x^{3} + a^{8}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8748*sqrt(3)*(70*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*b^2)^(1/3)*b*x^2 - a*b + (-a*b^2)^(2/3)*x) - 140*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*b + (-a*b^2)^(2/3)*
x) + 420*(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*(sqrt(3)*a*b - 2*sqrt(3)*(-a*b^2)^(2/3)*x)/(a*b)) - 3*sqrt(3)*(140*b^3*x^11 +
525*a*b^2*x^8 + 720*a^2*b*x^5 + 416*a^3*x^2)*(-a*b^2)^(1/3))/((a^4*b^4*x^12 + 4*
a^5*b^3*x^9 + 6*a^6*b^2*x^6 + 4*a^7*b*x^3 + a^8)*(-a*b^2)^(1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.721539, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x