3.1248 \(\int \frac{x^6}{\left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=235 \[ -\frac{3 a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{7/4}}+\frac{3 a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{8 \sqrt{2} b^{7/4}}+\frac{3 a \log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{16 \sqrt{2} b^{7/4}}-\frac{3 a \log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{16 \sqrt{2} b^{7/4}}-\frac{x^3 \sqrt [4]{a-b x^4}}{4 b} \]

[Out]

-(x^3*(a - b*x^4)^(1/4))/(4*b) - (3*a*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)
^(1/4)])/(8*Sqrt[2]*b^(7/4)) + (3*a*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(
1/4)])/(8*Sqrt[2]*b^(7/4)) + (3*a*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[
2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(16*Sqrt[2]*b^(7/4)) - (3*a*Log[1 + (Sqrt[b]*x
^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(16*Sqrt[2]*b^(7/4
))

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Rubi [A]  time = 0.263036, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{3 a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{7/4}}+\frac{3 a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{8 \sqrt{2} b^{7/4}}+\frac{3 a \log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{16 \sqrt{2} b^{7/4}}-\frac{3 a \log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{16 \sqrt{2} b^{7/4}}-\frac{x^3 \sqrt [4]{a-b x^4}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a - b*x^4)^(3/4),x]

[Out]

-(x^3*(a - b*x^4)^(1/4))/(4*b) - (3*a*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)
^(1/4)])/(8*Sqrt[2]*b^(7/4)) + (3*a*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(
1/4)])/(8*Sqrt[2]*b^(7/4)) + (3*a*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[
2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(16*Sqrt[2]*b^(7/4)) - (3*a*Log[1 + (Sqrt[b]*x
^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(16*Sqrt[2]*b^(7/4
))

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Rubi in Sympy [A]  time = 37.0921, size = 216, normalized size = 0.92 \[ \frac{3 \sqrt{2} a \log{\left (- \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{32 b^{\frac{7}{4}}} - \frac{3 \sqrt{2} a \log{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{32 b^{\frac{7}{4}}} + \frac{3 \sqrt{2} a \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} - 1 \right )}}{16 b^{\frac{7}{4}}} + \frac{3 \sqrt{2} a \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + 1 \right )}}{16 b^{\frac{7}{4}}} - \frac{x^{3} \sqrt [4]{a - b x^{4}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(-b*x**4+a)**(3/4),x)

[Out]

3*sqrt(2)*a*log(-sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4) + sqrt(b)*x**2/sqrt(a -
b*x**4) + 1)/(32*b**(7/4)) - 3*sqrt(2)*a*log(sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1
/4) + sqrt(b)*x**2/sqrt(a - b*x**4) + 1)/(32*b**(7/4)) + 3*sqrt(2)*a*atan(sqrt(2
)*b**(1/4)*x/(a - b*x**4)**(1/4) - 1)/(16*b**(7/4)) + 3*sqrt(2)*a*atan(sqrt(2)*b
**(1/4)*x/(a - b*x**4)**(1/4) + 1)/(16*b**(7/4)) - x**3*(a - b*x**4)**(1/4)/(4*b
)

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Mathematica [C]  time = 0.0515701, size = 66, normalized size = 0.28 \[ \frac{x^3 \left (a \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )-a+b x^4\right )}{4 b \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a - b*x^4)^(3/4),x]

[Out]

(x^3*(-a + b*x^4 + a*(1 - (b*x^4)/a)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, (b*x
^4)/a]))/(4*b*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{{x}^{6} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(-b*x^4+a)^(3/4),x)

[Out]

int(x^6/(-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^6/(-b*x^4 + a)^(3/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243334, size = 270, normalized size = 1.15 \[ -\frac{4 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{3} - 12 \, b \left (-\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{2} x \left (-\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}}}{x \sqrt{\frac{b^{4} x^{2} \sqrt{-\frac{a^{4}}{b^{7}}} + \sqrt{-b x^{4} + a} a^{2}}{x^{2}}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a}\right ) + 3 \, b \left (-\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} \log \left (\frac{3 \,{\left (b^{2} x \left (-\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a\right )}}{x}\right ) - 3 \, b \left (-\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} \log \left (-\frac{3 \,{\left (b^{2} x \left (-\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} -{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a\right )}}{x}\right )}{16 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-b*x^4 + a)^(3/4),x, algorithm="fricas")

[Out]

-1/16*(4*(-b*x^4 + a)^(1/4)*x^3 - 12*b*(-a^4/b^7)^(1/4)*arctan(b^2*x*(-a^4/b^7)^
(1/4)/(x*sqrt((b^4*x^2*sqrt(-a^4/b^7) + sqrt(-b*x^4 + a)*a^2)/x^2) + (-b*x^4 + a
)^(1/4)*a)) + 3*b*(-a^4/b^7)^(1/4)*log(3*(b^2*x*(-a^4/b^7)^(1/4) + (-b*x^4 + a)^
(1/4)*a)/x) - 3*b*(-a^4/b^7)^(1/4)*log(-3*(b^2*x*(-a^4/b^7)^(1/4) - (-b*x^4 + a)
^(1/4)*a)/x))/b

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Sympy [A]  time = 4.94367, size = 39, normalized size = 0.17 \[ \frac{x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(-b*x**4+a)**(3/4),x)

[Out]

x**7*gamma(7/4)*hyper((3/4, 7/4), (11/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**(3/
4)*gamma(11/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(-b*x^4 + a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^6/(-b*x^4 + a)^(3/4), x)