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

Optimal. Leaf size=364 \[ \frac{11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac{110 \left (a+b x^3\right )^5 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac{55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \]

[Out]

(x*(a + b*x^3))/(12*a*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) + (11*x*(a + b*x^3)^2)/
(108*a^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) + (11*x*(a + b*x^3)^3)/(81*a^3*(a^2
+ 2*a*b*x^3 + b^2*x^6)^(5/2)) + (55*x*(a + b*x^3)^4)/(243*a^4*(a^2 + 2*a*b*x^3 +
b^2*x^6)^(5/2)) - (110*(a + b*x^3)^5*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^
(1/3))])/(243*Sqrt[3]*a^(14/3)*b^(1/3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) + (110
*(a + b*x^3)^5*Log[a^(1/3) + b^(1/3)*x])/(729*a^(14/3)*b^(1/3)*(a^2 + 2*a*b*x^3
+ b^2*x^6)^(5/2)) - (55*(a + b*x^3)^5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*
x^2])/(729*a^(14/3)*b^(1/3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))

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Rubi [A]  time = 0.406254, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac{110 \left (a+b x^3\right )^5 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac{55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac{11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^3))/(12*a*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) + (11*x*(a + b*x^3)^2)/
(108*a^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) + (11*x*(a + b*x^3)^3)/(81*a^3*(a^2
+ 2*a*b*x^3 + b^2*x^6)^(5/2)) + (55*x*(a + b*x^3)^4)/(243*a^4*(a^2 + 2*a*b*x^3 +
b^2*x^6)^(5/2)) - (110*(a + b*x^3)^5*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^
(1/3))])/(243*Sqrt[3]*a^(14/3)*b^(1/3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)) + (110
*(a + b*x^3)^5*Log[a^(1/3) + b^(1/3)*x])/(729*a^(14/3)*b^(1/3)*(a^2 + 2*a*b*x^3
+ b^2*x^6)^(5/2)) - (55*(a + b*x^3)^5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*
x^2])/(729*a^(14/3)*b^(1/3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))

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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/(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 = 0.23057, size = 211, normalized size = 0.58 \[ \frac{\left (a+b x^3\right ) \left (-\frac{220 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+660 a^{2/3} x \left (a+b x^3\right )^3+396 a^{5/3} x \left (a+b x^3\right )^2+297 a^{8/3} x \left (a+b x^3\right )+243 a^{11/3} x+\frac{440 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{440 \sqrt{3} \left (a+b x^3\right )^4 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}\right )}{2916 a^{14/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^3)*(243*a^(11/3)*x + 297*a^(8/3)*x*(a + b*x^3) + 396*a^(5/3)*x*(a + b*
x^3)^2 + 660*a^(2/3)*x*(a + b*x^3)^3 + (440*Sqrt[3]*(a + b*x^3)^4*ArcTan[(-a^(1/
3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/b^(1/3) + (440*(a + b*x^3)^4*Log[a^(1/3) +
b^(1/3)*x])/b^(1/3) - (220*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2
/3)*x^2])/b^(1/3)))/(2916*a^(14/3)*((a + b*x^3)^2)^(5/2))

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Maple [A]  time = 0.013, size = 519, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2916*(-440*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^12*b^4
+440*ln(x+(a/b)^(1/3))*x^12*b^4-220*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^12*b^4+6
60*(a/b)^(2/3)*x^10*b^4-1760*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*
3^(1/2)*x^9*a*b^3+1760*ln(x+(a/b)^(1/3))*x^9*a*b^3-880*ln(x^2-x*(a/b)^(1/3)+(a/b
)^(2/3))*x^9*a*b^3+2376*(a/b)^(2/3)*x^7*a*b^3-2640*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+2640*ln(x+(a/b)^(1/3))*x^6*a^2*b^2-132
0*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^6*a^2*b^2+3069*(a/b)^(2/3)*x^4*a^2*b^2-176
0*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^3*a^3*b+1760*ln(x
+(a/b)^(1/3))*x^3*a^3*b-880*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^3*a^3*b+1596*(a/
b)^(2/3)*x*a^3*b-440*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*
a^4+440*ln(x+(a/b)^(1/3))*a^4-220*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^4)*(b*x^3+
a)/(a/b)^(2/3)/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((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 = 0.27057, size = 408, normalized size = 1.12 \[ -\frac{\sqrt{3}{\left (220 \, \sqrt{3}{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 440 \, \sqrt{3}{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 1320 \,{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (220 \, b^{3} x^{10} + 792 \, a b^{2} x^{7} + 1023 \, a^{2} b x^{4} + 532 \, a^{3} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{8748 \,{\left (a^{4} b^{4} x^{12} + 4 \, a^{5} b^{3} x^{9} + 6 \, a^{6} b^{2} x^{6} + 4 \, a^{7} b x^{3} + a^{8}\right )} \left (a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8748*sqrt(3)*(220*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) - 440*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) -
1320*(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
*sqrt(3)*(a^2*b)^(1/3)*x - sqrt(3)*a)/a) - 3*sqrt(3)*(220*b^3*x^10 + 792*a*b^2*x
^7 + 1023*a^2*b*x^4 + 532*a^3*x)*(a^2*b)^(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^2*b)^(1/3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}\, dx \]

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)

[Out]

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

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GIAC/XCAS [A]  time = 0.659759, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x