3.1 \(\int \left (a^2+2 a b x^2+b^2 x^4\right )^{3/4} \, dx\)

Optimal. Leaf size=128 \[ \frac{3 a x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/4}}{8 \left (a+b x^2\right )}+\frac{1}{4} x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/4}+\frac{3 \sqrt{a} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/4} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/2}} \]

[Out]

(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4))/4 + (3*a*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/
4))/(8*(a + b*x^2)) + (3*Sqrt[a]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4)*ArcSinh[(Sqrt
[b]*x)/Sqrt[a]])/(8*Sqrt[b]*(1 + (b*x^2)/a)^(3/2))

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Rubi [A]  time = 0.0843731, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{3 a x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/4}}{8 \left (a+b x^2\right )}+\frac{1}{4} x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/4}+\frac{3 \sqrt{a} \left (a^2+2 a b x^2+b^2 x^4\right )^{3/4} \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4))/4 + (3*a*x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/
4))/(8*(a + b*x^2)) + (3*Sqrt[a]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/4)*ArcSinh[(Sqrt
[b]*x)/Sqrt[a]])/(8*Sqrt[b]*(1 + (b*x^2)/a)^(3/2))

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Rubi in Sympy [A]  time = 24.6035, size = 124, normalized size = 0.97 \[ \frac{3 a^{2} b \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{4}} \operatorname{atanh}{\left (\frac{b x}{\sqrt{a b + b^{2} x^{2}}} \right )}}{8 \left (a b + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{3 a x \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{4}}}{8 \left (a + b x^{2}\right )} + \frac{x \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{4}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/4),x)

[Out]

3*a**2*b*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/4)*atanh(b*x/sqrt(a*b + b**2*x**2))
/(8*(a*b + b**2*x**2)**(3/2)) + 3*a*x*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/4)/(8*
(a + b*x**2)) + x*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/4)/4

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Mathematica [A]  time = 0.0785485, size = 89, normalized size = 0.7 \[ \frac{\left (\left (a+b x^2\right )^2\right )^{3/4} \left (3 a^2 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\sqrt{b} x \sqrt{a+b x^2} \left (5 a+2 b x^2\right )\right )}{8 \sqrt{b} \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(((a + b*x^2)^2)^(3/4)*(Sqrt[b]*x*Sqrt[a + b*x^2]*(5*a + 2*b*x^2) + 3*a^2*Log[b*
x + Sqrt[b]*Sqrt[a + b*x^2]]))/(8*Sqrt[b]*(a + b*x^2)^(3/2))

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Maple [A]  time = 0.056, size = 77, normalized size = 0.6 \[{\frac{x \left ( 2\,b{x}^{2}+5\,a \right ) \left ( b{x}^{2}+a \right ) }{8}{\frac{1}{\sqrt [4]{ \left ( b{x}^{2}+a \right ) ^{2}}}}}+{\frac{3\,{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \sqrt{b{x}^{2}+a}{\frac{1}{\sqrt{b}}}{\frac{1}{\sqrt [4]{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x*(2*b*x^2+5*a)*(b*x^2+a)/((b*x^2+a)^2)^(1/4)+3/8*a^2*ln(x*b^(1/2)+(b*x^2+a)
^(1/2))/b^(1/2)/((b*x^2+a)^2)^(1/4)*(b*x^2+a)^(1/2)

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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^4 + 2*a*b*x^2 + a^2)^(3/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290218, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \sqrt{b} \log \left (-2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}}{\left (2 \, b^{2} x^{3} + 5 \, a b x\right )}}{16 \, b}, -\frac{3 \, a^{2} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}}}\right ) -{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}}{\left (2 \, b^{2} x^{3} + 5 \, a b x\right )}}{8 \, b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(3*a^2*sqrt(b)*log(-2*(b^2*x^4 + 2*a*b*x^2 + a^2)^(1/4)*b*x - (2*b*x^2 + a
)*sqrt(b)) + 2*(b^2*x^4 + 2*a*b*x^2 + a^2)^(1/4)*(2*b^2*x^3 + 5*a*b*x))/b, -1/8*
(3*a^2*sqrt(-b)*arctan(sqrt(-b)*x/(b^2*x^4 + 2*a*b*x^2 + a^2)^(1/4)) - (b^2*x^4
+ 2*a*b*x^2 + a^2)^(1/4)*(2*b^2*x^3 + 5*a*b*x))/b]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.284736, size = 117, normalized size = 0.91 \[ -\frac{1}{8} \,{\left (\frac{x^{4}{\left (\frac{5 \, \sqrt{-b x^{2} - a}{\left (b + \frac{a}{x^{2}}\right )}{\left | x \right |}}{x^{2}} - \frac{3 \, \sqrt{-b x^{2} - a} b{\left | x \right |}}{x^{2}}\right )}}{a^{2}} + \frac{3 \, \arctan \left (\frac{\sqrt{-b x^{2} - a}{\left | x \right |}}{\sqrt{b} x^{2}}\right )}{\sqrt{b}}\right )} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(x^4*(5*sqrt(-b*x^2 - a)*(b + a/x^2)*abs(x)/x^2 - 3*sqrt(-b*x^2 - a)*b*abs(
x)/x^2)/a^2 + 3*arctan(sqrt(-b*x^2 - a)*abs(x)/(sqrt(b)*x^2))/sqrt(b))*a^2