3.1061 \(\int x^6 \left (a+b x^4\right )^{5/4} \, dx\)

Optimal. Leaf size=124 \[ \frac{5 a^3 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{7/4}}-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{7/4}}+\frac{5 a^2 x^3 \sqrt [4]{a+b x^4}}{384 b}+\frac{1}{12} x^7 \left (a+b x^4\right )^{5/4}+\frac{5}{96} a x^7 \sqrt [4]{a+b x^4} \]

[Out]

(5*a^2*x^3*(a + b*x^4)^(1/4))/(384*b) + (5*a*x^7*(a + b*x^4)^(1/4))/96 + (x^7*(a
+ b*x^4)^(5/4))/12 + (5*a^3*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(256*b^(7/4)
) - (5*a^3*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(256*b^(7/4))

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Rubi [A]  time = 0.137653, antiderivative size = 124, 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 \[ \frac{5 a^3 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{7/4}}-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{256 b^{7/4}}+\frac{5 a^2 x^3 \sqrt [4]{a+b x^4}}{384 b}+\frac{1}{12} x^7 \left (a+b x^4\right )^{5/4}+\frac{5}{96} a x^7 \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^6*(a + b*x^4)^(5/4),x]

[Out]

(5*a^2*x^3*(a + b*x^4)^(1/4))/(384*b) + (5*a*x^7*(a + b*x^4)^(1/4))/96 + (x^7*(a
+ b*x^4)^(5/4))/12 + (5*a^3*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(256*b^(7/4)
) - (5*a^3*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(256*b^(7/4))

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Rubi in Sympy [A]  time = 17.6029, size = 116, normalized size = 0.94 \[ \frac{5 a^{3} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{256 b^{\frac{7}{4}}} - \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{256 b^{\frac{7}{4}}} + \frac{5 a^{2} x^{3} \sqrt [4]{a + b x^{4}}}{384 b} + \frac{5 a x^{7} \sqrt [4]{a + b x^{4}}}{96} + \frac{x^{7} \left (a + b x^{4}\right )^{\frac{5}{4}}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(b*x**4+a)**(5/4),x)

[Out]

5*a**3*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(256*b**(7/4)) - 5*a**3*atanh(b**(1/
4)*x/(a + b*x**4)**(1/4))/(256*b**(7/4)) + 5*a**2*x**3*(a + b*x**4)**(1/4)/(384*
b) + 5*a*x**7*(a + b*x**4)**(1/4)/96 + x**7*(a + b*x**4)**(5/4)/12

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Mathematica [C]  time = 0.0620172, size = 91, normalized size = 0.73 \[ \frac{x^3 \left (-5 a^3 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+5 a^3+57 a^2 b x^4+84 a b^2 x^8+32 b^3 x^{12}\right )}{384 b \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(5*a^3 + 57*a^2*b*x^4 + 84*a*b^2*x^8 + 32*b^3*x^12 - 5*a^3*(1 + (b*x^4)/a)^
(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, -((b*x^4)/a)]))/(384*b*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{x}^{6} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(b*x^4+a)^(5/4),x)

[Out]

int(x^6*(b*x^4+a)^(5/4),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)^(5/4)*x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278895, size = 290, normalized size = 2.34 \[ \frac{60 \, \left (\frac{a^{12}}{b^{7}}\right )^{\frac{1}{4}} b \arctan \left (\frac{\left (\frac{a^{12}}{b^{7}}\right )^{\frac{1}{4}} b^{2} x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} + x \sqrt{\frac{\sqrt{b x^{4} + a} a^{6} + \sqrt{\frac{a^{12}}{b^{7}}} b^{4} x^{2}}{x^{2}}}}\right ) - 15 \, \left (\frac{a^{12}}{b^{7}}\right )^{\frac{1}{4}} b \log \left (\frac{5 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} + \left (\frac{a^{12}}{b^{7}}\right )^{\frac{1}{4}} b^{2} x\right )}}{x}\right ) + 15 \, \left (\frac{a^{12}}{b^{7}}\right )^{\frac{1}{4}} b \log \left (\frac{5 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} - \left (\frac{a^{12}}{b^{7}}\right )^{\frac{1}{4}} b^{2} x\right )}}{x}\right ) + 4 \,{\left (32 \, b^{2} x^{11} + 52 \, a b x^{7} + 5 \, a^{2} x^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{1536 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1536*(60*(a^12/b^7)^(1/4)*b*arctan((a^12/b^7)^(1/4)*b^2*x/((b*x^4 + a)^(1/4)*a
^3 + x*sqrt((sqrt(b*x^4 + a)*a^6 + sqrt(a^12/b^7)*b^4*x^2)/x^2))) - 15*(a^12/b^7
)^(1/4)*b*log(5*((b*x^4 + a)^(1/4)*a^3 + (a^12/b^7)^(1/4)*b^2*x)/x) + 15*(a^12/b
^7)^(1/4)*b*log(5*((b*x^4 + a)^(1/4)*a^3 - (a^12/b^7)^(1/4)*b^2*x)/x) + 4*(32*b^
2*x^11 + 52*a*b*x^7 + 5*a^2*x^3)*(b*x^4 + a)^(1/4))/b

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Sympy [A]  time = 17.2165, size = 39, normalized size = 0.31 \[ \frac{a^{\frac{5}{4}} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(5/4)*x**7*gamma(7/4)*hyper((-5/4, 7/4), (11/4,), b*x**4*exp_polar(I*pi)/a)/(
4*gamma(11/4))

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GIAC/XCAS [A]  time = 0.239202, size = 397, normalized size = 3.2 \[ \frac{1}{3072} \,{\left (\frac{8 \,{\left (\frac{42 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )} b}{x} - \frac{15 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2}}{x} + \frac{5 \,{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{9}}\right )} x^{12}}{a^{3} b} - \frac{30 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b^{2}} - \frac{30 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b^{2}} - \frac{15 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b^{2}} + \frac{15 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b^{2}}\right )} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3072*(8*(42*(b*x^4 + a)^(1/4)*(b + a/x^4)*b/x - 15*(b*x^4 + a)^(1/4)*b^2/x + 5
*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4 + a)^(1/4)/x^9)*x^12/(a^3*b) - 30*sqrt(2)*(-
b)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4) + 2*(b*x^4 + a)^(1/4)/x)/(-b)^(1
/4))/b^2 - 30*sqrt(2)*(-b)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4) - 2*(b*
x^4 + a)^(1/4)/x)/(-b)^(1/4))/b^2 - 15*sqrt(2)*(-b)^(1/4)*ln(sqrt(-b) + sqrt(2)*
(b*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*x^4 + a)/x^2)/b^2 + 15*sqrt(2)*(-b)^(1/4
)*ln(sqrt(-b) - sqrt(2)*(b*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*x^4 + a)/x^2)/b^
2)*a^3