3.109 \(\int \frac{x^4}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx\)

Optimal. Leaf size=368 \[ \frac{7 x^2}{324 a^2 b \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{x^2}{54 a b \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x^2}{12 b \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{7 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{10/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{7 \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{10/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{10/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7 x^2}{243 a^3 b \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

(7*x^2)/(243*a^3*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - x^2/(12*b*(a + b*x^3)^3*Sq
rt[a^2 + 2*a*b*x^3 + b^2*x^6]) + x^2/(54*a*b*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6]) + (7*x^2)/(324*a^2*b*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) -
(7*(a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a
^(10/3)*b^(5/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (7*(a + b*x^3)*Log[a^(1/3) +
b^(1/3)*x])/(729*a^(10/3)*b^(5/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (7*(a + b*x
^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1458*a^(10/3)*b^(5/3)*Sqrt[
a^2 + 2*a*b*x^3 + b^2*x^6])

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Rubi [A]  time = 0.424529, antiderivative size = 368, 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 \[ \frac{7 x^2}{324 a^2 b \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{x^2}{54 a b \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{x^2}{12 b \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{7 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{10/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{7 \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{10/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{10/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7 x^2}{243 a^3 b \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(7*x^2)/(243*a^3*b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - x^2/(12*b*(a + b*x^3)^3*Sq
rt[a^2 + 2*a*b*x^3 + b^2*x^6]) + x^2/(54*a*b*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6]) + (7*x^2)/(324*a^2*b*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) -
(7*(a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a
^(10/3)*b^(5/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (7*(a + b*x^3)*Log[a^(1/3) +
b^(1/3)*x])/(729*a^(10/3)*b^(5/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (7*(a + b*x
^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1458*a^(10/3)*b^(5/3)*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**4/(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 = 0.269733, size = 229, normalized size = 0.62 \[ \frac{\left (a+b x^3\right ) \left (-243 a^{10/3} b^{2/3} x^2+63 a^{4/3} b^{2/3} x^2 \left (a+b x^3\right )^2+54 a^{7/3} b^{2/3} x^2 \left (a+b x^3\right )+14 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+84 \sqrt [3]{a} b^{2/3} x^2 \left (a+b x^3\right )^3-28 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+28 \sqrt{3} \left (a+b x^3\right )^4 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )\right )}{2916 a^{10/3} b^{5/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

((a + b*x^3)*(-243*a^(10/3)*b^(2/3)*x^2 + 54*a^(7/3)*b^(2/3)*x^2*(a + b*x^3) + 6
3*a^(4/3)*b^(2/3)*x^2*(a + b*x^3)^2 + 84*a^(1/3)*b^(2/3)*x^2*(a + b*x^3)^3 + 28*
Sqrt[3]*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] - 28*(a
+ b*x^3)^4*Log[a^(1/3) + b^(1/3)*x] + 14*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^
(1/3)*x + b^(2/3)*x^2]))/(2916*a^(10/3)*b^(5/3)*((a + b*x^3)^2)^(5/2))

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Maple [B]  time = 0.028, size = 521, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

1/2916*(-28*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^12*b^4-
28*ln(x+(a/b)^(1/3))*x^12*b^4+14*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^12*b^4+84*(
a/b)^(1/3)*x^11*b^4-112*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/
2)*x^9*a*b^3-112*ln(x+(a/b)^(1/3))*x^9*a*b^3+56*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3)
)*x^9*a*b^3+315*(a/b)^(1/3)*x^8*a*b^3-168*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/
(a/b)^(1/3))*3^(1/2)*x^6*a^2*b^2-168*ln(x+(a/b)^(1/3))*x^6*a^2*b^2+84*ln(x^2-x*(
a/b)^(1/3)+(a/b)^(2/3))*x^6*a^2*b^2+432*(a/b)^(1/3)*x^5*a^2*b^2-112*arctan(1/3*(
-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^3*a^3*b-112*ln(x+(a/b)^(1/3))*x
^3*a^3*b+56*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^3*a^3*b-42*(a/b)^(1/3)*x^2*a^3*b
-28*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*a^4-28*ln(x+(a/b)
^(1/3))*a^4+14*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*a^4)*(b*x^3+a)/(a/b)^(1/3)/b^2/
a^3/((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^4/(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 = 0.280486, size = 431, normalized size = 1.17 \[ -\frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 28 \, \sqrt{3}{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 84 \,{\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (28 \, b^{3} x^{11} + 105 \, a b^{2} x^{8} + 144 \, a^{2} b x^{5} - 14 \, a^{3} x^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{8748 \,{\left (a^{3} b^{5} x^{12} + 4 \, a^{4} b^{4} x^{9} + 6 \, a^{5} b^{3} x^{6} + 4 \, a^{6} b^{2} x^{3} + a^{7} b\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/8748*sqrt(3)*(14*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) - 28*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
) + 84*(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)*(28*b^3*x^11 + 105
*a*b^2*x^8 + 144*a^2*b*x^5 - 14*a^3*x^2)*(-a*b^2)^(1/3))/((a^3*b^5*x^12 + 4*a^4*
b^4*x^9 + 6*a^5*b^3*x^6 + 4*a^6*b^2*x^3 + a^7*b)*(-a*b^2)^(1/3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(x**4/((a + b*x**3)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.710137, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x