3.1030 \(\int x^8 \left (a+b x^4\right )^{3/4} \, dx\)

Optimal. Leaf size=125 \[ \frac{5 a^3 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}-\frac{5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac{1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac{a x^5 \left (a+b x^4\right )^{3/4}}{32 b} \]

[Out]

(-5*a^2*x*(a + b*x^4)^(3/4))/(128*b^2) + (a*x^5*(a + b*x^4)^(3/4))/(32*b) + (x^9
*(a + b*x^4)^(3/4))/12 + (5*a^3*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(256*b^(9
/4)) + (5*a^3*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(256*b^(9/4))

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Rubi [A]  time = 0.123861, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{5 a^3 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{9/4}}-\frac{5 a^2 x \left (a+b x^4\right )^{3/4}}{128 b^2}+\frac{1}{12} x^9 \left (a+b x^4\right )^{3/4}+\frac{a x^5 \left (a+b x^4\right )^{3/4}}{32 b} \]

Antiderivative was successfully verified.

[In]  Int[x^8*(a + b*x^4)^(3/4),x]

[Out]

(-5*a^2*x*(a + b*x^4)^(3/4))/(128*b^2) + (a*x^5*(a + b*x^4)^(3/4))/(32*b) + (x^9
*(a + b*x^4)^(3/4))/12 + (5*a^3*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(256*b^(9
/4)) + (5*a^3*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(256*b^(9/4))

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Rubi in Sympy [A]  time = 15.0444, size = 116, normalized size = 0.93 \[ \frac{5 a^{3} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{256 b^{\frac{9}{4}}} + \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{256 b^{\frac{9}{4}}} - \frac{5 a^{2} x \left (a + b x^{4}\right )^{\frac{3}{4}}}{128 b^{2}} + \frac{a x^{5} \left (a + b x^{4}\right )^{\frac{3}{4}}}{32 b} + \frac{x^{9} \left (a + b x^{4}\right )^{\frac{3}{4}}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(b*x**4+a)**(3/4),x)

[Out]

5*a**3*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(256*b**(9/4)) + 5*a**3*atanh(b**(1/
4)*x/(a + b*x**4)**(1/4))/(256*b**(9/4)) - 5*a**2*x*(a + b*x**4)**(3/4)/(128*b**
2) + a*x**5*(a + b*x**4)**(3/4)/(32*b) + x**9*(a + b*x**4)**(3/4)/12

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Mathematica [A]  time = 0.121862, size = 122, normalized size = 0.98 \[ \frac{5 a^3 \left (-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right )}{512 b^{9/4}}+\left (a+b x^4\right )^{3/4} \left (-\frac{5 a^2 x}{128 b^2}+\frac{a x^5}{32 b}+\frac{x^9}{12}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*(a + b*x^4)^(3/4),x]

[Out]

(a + b*x^4)^(3/4)*((-5*a^2*x)/(128*b^2) + (a*x^5)/(32*b) + x^9/12) + (5*a^3*(2*A
rcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)] - Log[1 - (b^(1/4)*x)/(a + b*x^4)^(1/4)] +
Log[1 + (b^(1/4)*x)/(a + b*x^4)^(1/4)]))/(512*b^(9/4))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{x}^{8} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(b*x^4+a)^(3/4),x)

[Out]

int(x^8*(b*x^4+a)^(3/4),x)

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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*x^4 + a)^(3/4)*x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271732, size = 300, normalized size = 2.4 \[ \frac{60 \, \left (\frac{a^{12}}{b^{9}}\right )^{\frac{1}{4}} b^{2} \arctan \left (\frac{\left (\frac{a^{12}}{b^{9}}\right )^{\frac{3}{4}} b^{7} x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{9} + x \sqrt{\frac{\sqrt{\frac{a^{12}}{b^{9}}} a^{12} b^{5} x^{2} + \sqrt{b x^{4} + a} a^{18}}{x^{2}}}}\right ) + 15 \, \left (\frac{a^{12}}{b^{9}}\right )^{\frac{1}{4}} b^{2} \log \left (\frac{125 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{9} + \left (\frac{a^{12}}{b^{9}}\right )^{\frac{3}{4}} b^{7} x\right )}}{x}\right ) - 15 \, \left (\frac{a^{12}}{b^{9}}\right )^{\frac{1}{4}} b^{2} \log \left (\frac{125 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{9} - \left (\frac{a^{12}}{b^{9}}\right )^{\frac{3}{4}} b^{7} x\right )}}{x}\right ) + 4 \,{\left (32 \, b^{2} x^{9} + 12 \, a b x^{5} - 15 \, a^{2} x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{1536 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^8,x, algorithm="fricas")

[Out]

1/1536*(60*(a^12/b^9)^(1/4)*b^2*arctan((a^12/b^9)^(3/4)*b^7*x/((b*x^4 + a)^(1/4)
*a^9 + x*sqrt((sqrt(a^12/b^9)*a^12*b^5*x^2 + sqrt(b*x^4 + a)*a^18)/x^2))) + 15*(
a^12/b^9)^(1/4)*b^2*log(125*((b*x^4 + a)^(1/4)*a^9 + (a^12/b^9)^(3/4)*b^7*x)/x)
- 15*(a^12/b^9)^(1/4)*b^2*log(125*((b*x^4 + a)^(1/4)*a^9 - (a^12/b^9)^(3/4)*b^7*
x)/x) + 4*(32*b^2*x^9 + 12*a*b*x^5 - 15*a^2*x)*(b*x^4 + a)^(3/4))/b^2

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Sympy [A]  time = 12.6381, size = 39, normalized size = 0.31 \[ \frac{a^{\frac{3}{4}} x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(b*x**4+a)**(3/4),x)

[Out]

a**(3/4)*x**9*gamma(9/4)*hyper((-3/4, 9/4), (13/4,), b*x**4*exp_polar(I*pi)/a)/(
4*gamma(13/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{8}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^8,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)*x^8, x)