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

Optimal. Leaf size=398 \[ \frac{7}{54 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^2}+\frac{1}{12 a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^3}-\frac{770 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{770 b^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{385 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{154}{243 a^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{77}{324 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )} \]

[Out]

154/(243*a^4*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(12*a*x^2*(a + b*x^3)^3*Sq
rt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 7/(54*a^2*x^2*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6]) + 77/(324*a^3*x^2*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (
385*(a + b*x^3))/(243*a^5*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (770*b^(2/3)*(a
+ b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a^(17/
3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (770*b^(2/3)*(a + b*x^3)*Log[a^(1/3) + b^(
1/3)*x])/(729*a^(17/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (385*b^(2/3)*(a + b*x^
3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(729*a^(17/3)*Sqrt[a^2 + 2*a*
b*x^3 + b^2*x^6])

_______________________________________________________________________________________

Rubi [A]  time = 0.487287, 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 \[ \frac{7}{54 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^2}+\frac{1}{12 a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^3}-\frac{770 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{770 b^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{385 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{154}{243 a^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{77}{324 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

154/(243*a^4*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(12*a*x^2*(a + b*x^3)^3*Sq
rt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 7/(54*a^2*x^2*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6]) + 77/(324*a^3*x^2*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (
385*(a + b*x^3))/(243*a^5*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (770*b^(2/3)*(a
+ b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a^(17/
3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (770*b^(2/3)*(a + b*x^3)*Log[a^(1/3) + b^(
1/3)*x])/(729*a^(17/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (385*b^(2/3)*(a + b*x^
3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(729*a^(17/3)*Sqrt[a^2 + 2*a*
b*x^3 + b^2*x^6])

_______________________________________________________________________________________

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

[Out]

Exception raised: RecursionError

_______________________________________________________________________________________

Mathematica [A]  time = 0.278541, size = 234, normalized size = 0.59 \[ \frac{\left (a+b x^3\right ) \left (1540 b^{2/3} \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-3162 a^{2/3} b x \left (a+b x^3\right )^3-1314 a^{5/3} b x \left (a+b x^3\right )^2-621 a^{8/3} b x \left (a+b x^3\right )-\frac{1458 a^{2/3} \left (a+b x^3\right )^4}{x^2}-243 a^{11/3} b x-3080 b^{2/3} \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-3080 \sqrt{3} b^{2/3} \left (a+b x^3\right )^4 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )\right )}{2916 a^{17/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^3)*(-243*a^(11/3)*b*x - 621*a^(8/3)*b*x*(a + b*x^3) - 1314*a^(5/3)*b*x
*(a + b*x^3)^2 - 3162*a^(2/3)*b*x*(a + b*x^3)^3 - (1458*a^(2/3)*(a + b*x^3)^4)/x
^2 - 3080*Sqrt[3]*b^(2/3)*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]
*a^(1/3))] - 3080*b^(2/3)*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x] + 1540*b^(2/3)*
(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(2916*a^(17/3)*((
a + b*x^3)^2)^(5/2))

_______________________________________________________________________________________

Maple [B]  time = 0.033, size = 542, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2916*(-3080*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^14*b
^4+3080*ln(x+(a/b)^(1/3))*x^14*b^4-1540*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^14*b
^4+4620*(a/b)^(2/3)*x^12*b^4-12320*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(
1/3))*3^(1/2)*x^11*a*b^3+12320*ln(x+(a/b)^(1/3))*x^11*a*b^3-6160*ln(x^2-x*(a/b)^
(1/3)+(a/b)^(2/3))*x^11*a*b^3+16632*(a/b)^(2/3)*x^9*a*b^3-18480*arctan(1/3*(-2*x
+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^8*a^2*b^2+18480*ln(x+(a/b)^(1/3))*x
^8*a^2*b^2-9240*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^8*a^2*b^2+21483*(a/b)^(2/3)*
x^6*a^2*b^2-12320*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^5
*a^3*b+12320*ln(x+(a/b)^(1/3))*x^5*a^3*b-6160*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*
x^5*a^3*b+11172*(a/b)^(2/3)*x^3*a^3*b-3080*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)
/(a/b)^(1/3))*3^(1/2)*x^2*a^4+3080*ln(x+(a/b)^(1/3))*x^2*a^4-1540*ln(x^2-x*(a/b)
^(1/3)+(a/b)^(2/3))*x^2*a^4+1458*(a/b)^(2/3)*a^4)*(b*x^3+a)/x^2/(a/b)^(2/3)/a^5/
((b*x^3+a)^2)^(5/2)

_______________________________________________________________________________________

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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.276656, size = 504, normalized size = 1.27 \[ -\frac{\sqrt{3}{\left (1540 \, \sqrt{3}{\left (b^{4} x^{14} + 4 \, a b^{3} x^{11} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{5} + a^{4} x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 3080 \, \sqrt{3}{\left (b^{4} x^{14} + 4 \, a b^{3} x^{11} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{5} + a^{4} x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 9240 \,{\left (b^{4} x^{14} + 4 \, a b^{3} x^{11} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{5} + a^{4} x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (1540 \, b^{4} x^{12} + 5544 \, a b^{3} x^{9} + 7161 \, a^{2} b^{2} x^{6} + 3724 \, a^{3} b x^{3} + 486 \, a^{4}\right )}\right )}}{8748 \,{\left (a^{5} b^{4} x^{14} + 4 \, a^{6} b^{3} x^{11} + 6 \, a^{7} b^{2} x^{8} + 4 \, a^{8} b x^{5} + a^{9} x^{2}\right )}} \]

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^3),x, algorithm="fricas")

[Out]

-1/8748*sqrt(3)*(1540*sqrt(3)*(b^4*x^14 + 4*a*b^3*x^11 + 6*a^2*b^2*x^8 + 4*a^3*b
*x^5 + a^4*x^2)*(-b^2/a^2)^(1/3)*log(b^2*x^2 + a*b*x*(-b^2/a^2)^(1/3) + a^2*(-b^
2/a^2)^(2/3)) - 3080*sqrt(3)*(b^4*x^14 + 4*a*b^3*x^11 + 6*a^2*b^2*x^8 + 4*a^3*b*
x^5 + a^4*x^2)*(-b^2/a^2)^(1/3)*log(b*x - a*(-b^2/a^2)^(1/3)) + 9240*(b^4*x^14 +
4*a*b^3*x^11 + 6*a^2*b^2*x^8 + 4*a^3*b*x^5 + a^4*x^2)*(-b^2/a^2)^(1/3)*arctan(1
/3*(2*sqrt(3)*b*x + sqrt(3)*a*(-b^2/a^2)^(1/3))/(a*(-b^2/a^2)^(1/3))) + 3*sqrt(3
)*(1540*b^4*x^12 + 5544*a*b^3*x^9 + 7161*a^2*b^2*x^6 + 3724*a^3*b*x^3 + 486*a^4)
)/(a^5*b^4*x^14 + 4*a^6*b^3*x^11 + 6*a^7*b^2*x^8 + 4*a^8*b*x^5 + a^9*x^2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.699327, 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^3),x, algorithm="giac")

[Out]

sage0*x