3.1123 \(\int \frac{x^{10}}{\left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{21 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{11/4}}+\frac{21 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{11/4}}-\frac{7 a x^3 \sqrt [4]{a+b x^4}}{32 b^2}+\frac{x^7 \sqrt [4]{a+b x^4}}{8 b} \]

[Out]

(-7*a*x^3*(a + b*x^4)^(1/4))/(32*b^2) + (x^7*(a + b*x^4)^(1/4))/(8*b) - (21*a^2*
ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(11/4)) + (21*a^2*ArcTanh[(b^(1/4)*
x)/(a + b*x^4)^(1/4)])/(64*b^(11/4))

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Rubi [A]  time = 0.11014, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{21 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{11/4}}+\frac{21 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{11/4}}-\frac{7 a x^3 \sqrt [4]{a+b x^4}}{32 b^2}+\frac{x^7 \sqrt [4]{a+b x^4}}{8 b} \]

Antiderivative was successfully verified.

[In]  Int[x^10/(a + b*x^4)^(3/4),x]

[Out]

(-7*a*x^3*(a + b*x^4)^(1/4))/(32*b^2) + (x^7*(a + b*x^4)^(1/4))/(8*b) - (21*a^2*
ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(11/4)) + (21*a^2*ArcTanh[(b^(1/4)*
x)/(a + b*x^4)^(1/4)])/(64*b^(11/4))

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Rubi in Sympy [A]  time = 14.1684, size = 99, normalized size = 0.93 \[ - \frac{21 a^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 b^{\frac{11}{4}}} + \frac{21 a^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 b^{\frac{11}{4}}} - \frac{7 a x^{3} \sqrt [4]{a + b x^{4}}}{32 b^{2}} + \frac{x^{7} \sqrt [4]{a + b x^{4}}}{8 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10/(b*x**4+a)**(3/4),x)

[Out]

-21*a**2*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(64*b**(11/4)) + 21*a**2*atanh(b**
(1/4)*x/(a + b*x**4)**(1/4))/(64*b**(11/4)) - 7*a*x**3*(a + b*x**4)**(1/4)/(32*b
**2) + x**7*(a + b*x**4)**(1/4)/(8*b)

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Mathematica [C]  time = 0.0636981, size = 80, normalized size = 0.75 \[ \frac{x^3 \left (7 a^2 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-7 a^2-3 a b x^4+4 b^2 x^8\right )}{32 b^2 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^10/(a + b*x^4)^(3/4),x]

[Out]

(x^3*(-7*a^2 - 3*a*b*x^4 + 4*b^2*x^8 + 7*a^2*(1 + (b*x^4)/a)^(3/4)*Hypergeometri
c2F1[3/4, 3/4, 7/4, -((b*x^4)/a)]))/(32*b^2*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{{x}^{10} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10/(b*x^4+a)^(3/4),x)

[Out]

int(x^10/(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(x^10/(b*x^4 + a)^(3/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252819, size = 284, normalized size = 2.68 \[ -\frac{84 \, b^{2} \left (\frac{a^{8}}{b^{11}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{3} x \left (\frac{a^{8}}{b^{11}}\right )^{\frac{1}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} + x \sqrt{\frac{b^{6} x^{2} \sqrt{\frac{a^{8}}{b^{11}}} + \sqrt{b x^{4} + a} a^{4}}{x^{2}}}}\right ) - 21 \, b^{2} \left (\frac{a^{8}}{b^{11}}\right )^{\frac{1}{4}} \log \left (\frac{21 \,{\left (b^{3} x \left (\frac{a^{8}}{b^{11}}\right )^{\frac{1}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}\right )}}{x}\right ) + 21 \, b^{2} \left (\frac{a^{8}}{b^{11}}\right )^{\frac{1}{4}} \log \left (-\frac{21 \,{\left (b^{3} x \left (\frac{a^{8}}{b^{11}}\right )^{\frac{1}{4}} -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2}\right )}}{x}\right ) - 4 \,{\left (4 \, b x^{7} - 7 \, a x^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{128 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(b*x^4 + a)^(3/4),x, algorithm="fricas")

[Out]

-1/128*(84*b^2*(a^8/b^11)^(1/4)*arctan(b^3*x*(a^8/b^11)^(1/4)/((b*x^4 + a)^(1/4)
*a^2 + x*sqrt((b^6*x^2*sqrt(a^8/b^11) + sqrt(b*x^4 + a)*a^4)/x^2))) - 21*b^2*(a^
8/b^11)^(1/4)*log(21*(b^3*x*(a^8/b^11)^(1/4) + (b*x^4 + a)^(1/4)*a^2)/x) + 21*b^
2*(a^8/b^11)^(1/4)*log(-21*(b^3*x*(a^8/b^11)^(1/4) - (b*x^4 + a)^(1/4)*a^2)/x) -
4*(4*b*x^7 - 7*a*x^3)*(b*x^4 + a)^(1/4))/b^2

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Sympy [A]  time = 7.29836, size = 37, normalized size = 0.35 \[ \frac{x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{15}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10/(b*x**4+a)**(3/4),x)

[Out]

x**11*gamma(11/4)*hyper((3/4, 11/4), (15/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3
/4)*gamma(15/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{10}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/(b*x^4 + a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^10/(b*x^4 + a)^(3/4), x)