3.1029 \(\int x^{12} \left (a+b x^4\right )^{3/4} \, dx\)

Optimal. Leaf size=149 \[ -\frac{45 a^4 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}-\frac{45 a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}+\frac{45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac{9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac{1}{16} x^{13} \left (a+b x^4\right )^{3/4}+\frac{a x^9 \left (a+b x^4\right )^{3/4}}{64 b} \]

[Out]

(45*a^3*x*(a + b*x^4)^(3/4))/(2048*b^3) - (9*a^2*x^5*(a + b*x^4)^(3/4))/(512*b^2
) + (a*x^9*(a + b*x^4)^(3/4))/(64*b) + (x^13*(a + b*x^4)^(3/4))/16 - (45*a^4*Arc
Tan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(4096*b^(13/4)) - (45*a^4*ArcTanh[(b^(1/4)*x
)/(a + b*x^4)^(1/4)])/(4096*b^(13/4))

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Rubi [A]  time = 0.162419, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{45 a^4 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}-\frac{45 a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{13/4}}+\frac{45 a^3 x \left (a+b x^4\right )^{3/4}}{2048 b^3}-\frac{9 a^2 x^5 \left (a+b x^4\right )^{3/4}}{512 b^2}+\frac{1}{16} x^{13} \left (a+b x^4\right )^{3/4}+\frac{a x^9 \left (a+b x^4\right )^{3/4}}{64 b} \]

Antiderivative was successfully verified.

[In]  Int[x^12*(a + b*x^4)^(3/4),x]

[Out]

(45*a^3*x*(a + b*x^4)^(3/4))/(2048*b^3) - (9*a^2*x^5*(a + b*x^4)^(3/4))/(512*b^2
) + (a*x^9*(a + b*x^4)^(3/4))/(64*b) + (x^13*(a + b*x^4)^(3/4))/16 - (45*a^4*Arc
Tan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(4096*b^(13/4)) - (45*a^4*ArcTanh[(b^(1/4)*x
)/(a + b*x^4)^(1/4)])/(4096*b^(13/4))

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Rubi in Sympy [A]  time = 19.8731, size = 139, normalized size = 0.93 \[ - \frac{45 a^{4} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{4096 b^{\frac{13}{4}}} - \frac{45 a^{4} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{4096 b^{\frac{13}{4}}} + \frac{45 a^{3} x \left (a + b x^{4}\right )^{\frac{3}{4}}}{2048 b^{3}} - \frac{9 a^{2} x^{5} \left (a + b x^{4}\right )^{\frac{3}{4}}}{512 b^{2}} + \frac{a x^{9} \left (a + b x^{4}\right )^{\frac{3}{4}}}{64 b} + \frac{x^{13} \left (a + b x^{4}\right )^{\frac{3}{4}}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**12*(b*x**4+a)**(3/4),x)

[Out]

-45*a**4*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(4096*b**(13/4)) - 45*a**4*atanh(b
**(1/4)*x/(a + b*x**4)**(1/4))/(4096*b**(13/4)) + 45*a**3*x*(a + b*x**4)**(3/4)/
(2048*b**3) - 9*a**2*x**5*(a + b*x**4)**(3/4)/(512*b**2) + a*x**9*(a + b*x**4)**
(3/4)/(64*b) + x**13*(a + b*x**4)**(3/4)/16

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Mathematica [A]  time = 0.16511, size = 135, normalized size = 0.91 \[ \left (a+b x^4\right )^{3/4} \left (\frac{45 a^3 x}{2048 b^3}-\frac{9 a^2 x^5}{512 b^2}+\frac{a x^9}{64 b}+\frac{x^{13}}{16}\right )-\frac{45 a^4 \left (-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right )}{8192 b^{13/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^12*(a + b*x^4)^(3/4),x]

[Out]

(a + b*x^4)^(3/4)*((45*a^3*x)/(2048*b^3) - (9*a^2*x^5)/(512*b^2) + (a*x^9)/(64*b
) + x^13/16) - (45*a^4*(2*ArcTan[(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)]))/(8192*b^(13/
4))

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{x}^{12} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12*(b*x^4+a)^(3/4),x)

[Out]

int(x^12*(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^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.297731, size = 315, normalized size = 2.11 \[ -\frac{180 \, \left (\frac{a^{16}}{b^{13}}\right )^{\frac{1}{4}} b^{3} \arctan \left (\frac{\left (\frac{a^{16}}{b^{13}}\right )^{\frac{3}{4}} b^{10} x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{12} + x \sqrt{\frac{\sqrt{\frac{a^{16}}{b^{13}}} a^{16} b^{7} x^{2} + \sqrt{b x^{4} + a} a^{24}}{x^{2}}}}\right ) + 45 \, \left (\frac{a^{16}}{b^{13}}\right )^{\frac{1}{4}} b^{3} \log \left (\frac{91125 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{12} + \left (\frac{a^{16}}{b^{13}}\right )^{\frac{3}{4}} b^{10} x\right )}}{x}\right ) - 45 \, \left (\frac{a^{16}}{b^{13}}\right )^{\frac{1}{4}} b^{3} \log \left (\frac{91125 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{12} - \left (\frac{a^{16}}{b^{13}}\right )^{\frac{3}{4}} b^{10} x\right )}}{x}\right ) - 4 \,{\left (128 \, b^{3} x^{13} + 32 \, a b^{2} x^{9} - 36 \, a^{2} b x^{5} + 45 \, a^{3} x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{8192 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^12,x, algorithm="fricas")

[Out]

-1/8192*(180*(a^16/b^13)^(1/4)*b^3*arctan((a^16/b^13)^(3/4)*b^10*x/((b*x^4 + a)^
(1/4)*a^12 + x*sqrt((sqrt(a^16/b^13)*a^16*b^7*x^2 + sqrt(b*x^4 + a)*a^24)/x^2)))
+ 45*(a^16/b^13)^(1/4)*b^3*log(91125*((b*x^4 + a)^(1/4)*a^12 + (a^16/b^13)^(3/4
)*b^10*x)/x) - 45*(a^16/b^13)^(1/4)*b^3*log(91125*((b*x^4 + a)^(1/4)*a^12 - (a^1
6/b^13)^(3/4)*b^10*x)/x) - 4*(128*b^3*x^13 + 32*a*b^2*x^9 - 36*a^2*b*x^5 + 45*a^
3*x)*(b*x^4 + a)^(3/4))/b^3

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Sympy [A]  time = 23.4742, size = 39, normalized size = 0.26 \[ \frac{a^{\frac{3}{4}} x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{17}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**12*(b*x**4+a)**(3/4),x)

[Out]

a**(3/4)*x**13*gamma(13/4)*hyper((-3/4, 13/4), (17/4,), b*x**4*exp_polar(I*pi)/a
)/(4*gamma(17/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{12}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^12,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)*x^12, x)