3.865 \(\int \frac{x^{14}}{\left (a+b x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=282 \[ \frac{77 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{60 b^{15/4} \sqrt{a+b x^4}}-\frac{77 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 b^{15/4} \sqrt{a+b x^4}}+\frac{77 a^2 x \sqrt{a+b x^4}}{30 b^{7/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{77 a x^3 \sqrt{a+b x^4}}{90 b^3}+\frac{11 x^7 \sqrt{a+b x^4}}{18 b^2}-\frac{x^{11}}{2 b \sqrt{a+b x^4}} \]

[Out]

-x^11/(2*b*Sqrt[a + b*x^4]) - (77*a*x^3*Sqrt[a + b*x^4])/(90*b^3) + (11*x^7*Sqrt
[a + b*x^4])/(18*b^2) + (77*a^2*x*Sqrt[a + b*x^4])/(30*b^(7/2)*(Sqrt[a] + Sqrt[b
]*x^2)) - (77*a^(9/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b
]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(30*b^(15/4)*Sqrt[a + b
*x^4]) + (77*a^(9/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]
*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(60*b^(15/4)*Sqrt[a + b*
x^4])

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Rubi [A]  time = 0.303914, antiderivative size = 282, 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 \[ \frac{77 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{60 b^{15/4} \sqrt{a+b x^4}}-\frac{77 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 b^{15/4} \sqrt{a+b x^4}}+\frac{77 a^2 x \sqrt{a+b x^4}}{30 b^{7/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{77 a x^3 \sqrt{a+b x^4}}{90 b^3}+\frac{11 x^7 \sqrt{a+b x^4}}{18 b^2}-\frac{x^{11}}{2 b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^14/(a + b*x^4)^(3/2),x]

[Out]

-x^11/(2*b*Sqrt[a + b*x^4]) - (77*a*x^3*Sqrt[a + b*x^4])/(90*b^3) + (11*x^7*Sqrt
[a + b*x^4])/(18*b^2) + (77*a^2*x*Sqrt[a + b*x^4])/(30*b^(7/2)*(Sqrt[a] + Sqrt[b
]*x^2)) - (77*a^(9/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b
]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(30*b^(15/4)*Sqrt[a + b
*x^4]) + (77*a^(9/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]
*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(60*b^(15/4)*Sqrt[a + b*
x^4])

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Rubi in Sympy [A]  time = 35.9243, size = 258, normalized size = 0.91 \[ - \frac{77 a^{\frac{9}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{30 b^{\frac{15}{4}} \sqrt{a + b x^{4}}} + \frac{77 a^{\frac{9}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{60 b^{\frac{15}{4}} \sqrt{a + b x^{4}}} + \frac{77 a^{2} x \sqrt{a + b x^{4}}}{30 b^{\frac{7}{2}} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} - \frac{77 a x^{3} \sqrt{a + b x^{4}}}{90 b^{3}} - \frac{x^{11}}{2 b \sqrt{a + b x^{4}}} + \frac{11 x^{7} \sqrt{a + b x^{4}}}{18 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14/(b*x**4+a)**(3/2),x)

[Out]

-77*a**(9/4)*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x
**2)*elliptic_e(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(30*b**(15/4)*sqrt(a + b*x**4)
) + 77*a**(9/4)*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b
)*x**2)*elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(60*b**(15/4)*sqrt(a + b*x*
*4)) + 77*a**2*x*sqrt(a + b*x**4)/(30*b**(7/2)*(sqrt(a) + sqrt(b)*x**2)) - 77*a*
x**3*sqrt(a + b*x**4)/(90*b**3) - x**11/(2*b*sqrt(a + b*x**4)) + 11*x**7*sqrt(a
+ b*x**4)/(18*b**2)

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Mathematica [C]  time = 0.344506, size = 183, normalized size = 0.65 \[ \frac{-231 a^{5/2} \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+231 a^{5/2} \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{b} x^3 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (-77 a^2-22 a b x^4+10 b^2 x^8\right )}{90 b^{7/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14/(a + b*x^4)^(3/2),x]

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*Sqrt[b]*x^3*(-77*a^2 - 22*a*b*x^4 + 10*b^2*x^8) + 231
*a^(5/2)*Sqrt[1 + (b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -
1] - 231*a^(5/2)*Sqrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a
]]*x], -1])/(90*Sqrt[(I*Sqrt[b])/Sqrt[a]]*b^(7/2)*Sqrt[a + b*x^4])

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Maple [C]  time = 0.018, size = 157, normalized size = 0.6 \[ -{\frac{{x}^{3}{a}^{2}}{2\,{b}^{3}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{{x}^{7}}{9\,{b}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{16\,a{x}^{3}}{45\,{b}^{3}}\sqrt{b{x}^{4}+a}}+{{\frac{77\,i}{30}}{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14/(b*x^4+a)^(3/2),x)

[Out]

-1/2/b^3*a^2*x^3/((x^4+a/b)*b)^(1/2)+1/9*x^7*(b*x^4+a)^(1/2)/b^2-16/45*a*x^3*(b*
x^4+a)^(1/2)/b^3+77/30*I*a^(5/2)/b^(7/2)/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*
b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*(EllipticF(x*
(I/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^4 + a)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^14/(b*x^4 + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{14}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^4 + a)^(3/2),x, algorithm="fricas")

[Out]

integral(x^14/(b*x^4 + a)^(3/2), x)

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Sympy [A]  time = 10.2945, size = 37, normalized size = 0.13 \[ \frac{x^{15} \Gamma \left (\frac{15}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{15}{4} \\ \frac{19}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{19}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14/(b*x**4+a)**(3/2),x)

[Out]

x**15*gamma(15/4)*hyper((3/2, 15/4), (19/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3
/2)*gamma(19/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^4 + a)^(3/2),x, algorithm="giac")

[Out]

integrate(x^14/(b*x^4 + a)^(3/2), x)