3.1039 \(\int x^6 \left (a+b x^4\right )^{3/4} \, dx\)

Optimal. Leaf size=126 \[ -\frac{3 a^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 b^{3/2} \sqrt [4]{a+b x^4}}-\frac{3 a^2 x^3}{40 b \sqrt [4]{a+b x^4}}+\frac{1}{10} x^7 \left (a+b x^4\right )^{3/4}+\frac{a x^3 \left (a+b x^4\right )^{3/4}}{20 b} \]

[Out]

(-3*a^2*x^3)/(40*b*(a + b*x^4)^(1/4)) + (a*x^3*(a + b*x^4)^(3/4))/(20*b) + (x^7*
(a + b*x^4)^(3/4))/10 - (3*a^(5/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqr
t[b]*x^2)/Sqrt[a]]/2, 2])/(40*b^(3/2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.194994, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ -\frac{3 a^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 b^{3/2} \sqrt [4]{a+b x^4}}-\frac{3 a^2 x^3}{40 b \sqrt [4]{a+b x^4}}+\frac{1}{10} x^7 \left (a+b x^4\right )^{3/4}+\frac{a x^3 \left (a+b x^4\right )^{3/4}}{20 b} \]

Antiderivative was successfully verified.

[In]  Int[x^6*(a + b*x^4)^(3/4),x]

[Out]

(-3*a^2*x^3)/(40*b*(a + b*x^4)^(1/4)) + (a*x^3*(a + b*x^4)^(3/4))/(20*b) + (x^7*
(a + b*x^4)^(3/4))/10 - (3*a^(5/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqr
t[b]*x^2)/Sqrt[a]]/2, 2])/(40*b^(3/2)*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{3} x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{80 b^{2} \sqrt [4]{a + b x^{4}}} - \frac{3 a^{3}}{40 b^{2} x \sqrt [4]{a + b x^{4}}} - \frac{3 a^{2} x^{3}}{40 b \sqrt [4]{a + b x^{4}}} + \frac{a x^{3} \left (a + b x^{4}\right )^{\frac{3}{4}}}{20 b} + \frac{x^{7} \left (a + b x^{4}\right )^{\frac{3}{4}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**3*x*(a/(b*x**4) + 1)**(1/4)*Integral((a*x**2/b + 1)**(-1/4), (x, x**(-2)))/
(80*b**2*(a + b*x**4)**(1/4)) - 3*a**3/(40*b**2*x*(a + b*x**4)**(1/4)) - 3*a**2*
x**3/(40*b*(a + b*x**4)**(1/4)) + a*x**3*(a + b*x**4)**(3/4)/(20*b) + x**7*(a +
b*x**4)**(3/4)/10

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Mathematica [C]  time = 0.0529185, size = 78, normalized size = 0.62 \[ \frac{x^3 \left (-a^2 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+a^2+3 a b x^4+2 b^2 x^8\right )}{20 b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(a^2 + 3*a*b*x^4 + 2*b^2*x^8 - a^2*(1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[
1/4, 3/4, 7/4, -((b*x^4)/a)]))/(20*b*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{x}^{6} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}\, dx \]

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. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(3/4)*x^6, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{6}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(3/4)*x^6, x)

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Sympy [A]  time = 6.92388, size = 39, normalized size = 0.31 \[ \frac{a^{\frac{3}{4}} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(3/4)*x^6, x)