3.1449 \(\int \frac{x^9}{a+b x^8} \, dx\)

Optimal. Leaf size=203 \[ \frac{\sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{5/4}}-\frac{\sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{5/4}}+\frac{\sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} b^{5/4}}-\frac{\sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} b^{5/4}}+\frac{x^2}{2 b} \]

[Out]

x^2/(2*b) + (a^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*b^(5/
4)) - (a^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*b^(5/4)) +
(a^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(8*Sqrt[2]*b^
(5/4)) - (a^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(8*S
qrt[2]*b^(5/4))

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Rubi [A]  time = 0.431605, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ \frac{\sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{5/4}}-\frac{\sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{5/4}}+\frac{\sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} b^{5/4}}-\frac{\sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} b^{5/4}}+\frac{x^2}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + b*x^8),x]

[Out]

x^2/(2*b) + (a^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*b^(5/
4)) - (a^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*b^(5/4)) +
(a^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(8*Sqrt[2]*b^
(5/4)) - (a^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(8*S
qrt[2]*b^(5/4))

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Rubi in Sympy [A]  time = 60.8123, size = 185, normalized size = 0.91 \[ \frac{\sqrt{2} \sqrt [4]{a} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{2} + \sqrt{a} + \sqrt{b} x^{4} \right )}}{16 b^{\frac{5}{4}}} - \frac{\sqrt{2} \sqrt [4]{a} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{2} + \sqrt{a} + \sqrt{b} x^{4} \right )}}{16 b^{\frac{5}{4}}} + \frac{\sqrt{2} \sqrt [4]{a} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x^{2}}{\sqrt [4]{a}} \right )}}{8 b^{\frac{5}{4}}} - \frac{\sqrt{2} \sqrt [4]{a} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x^{2}}{\sqrt [4]{a}} \right )}}{8 b^{\frac{5}{4}}} + \frac{x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(b*x**8+a),x)

[Out]

sqrt(2)*a**(1/4)*log(-sqrt(2)*a**(1/4)*b**(1/4)*x**2 + sqrt(a) + sqrt(b)*x**4)/(
16*b**(5/4)) - sqrt(2)*a**(1/4)*log(sqrt(2)*a**(1/4)*b**(1/4)*x**2 + sqrt(a) + s
qrt(b)*x**4)/(16*b**(5/4)) + sqrt(2)*a**(1/4)*atan(1 - sqrt(2)*b**(1/4)*x**2/a**
(1/4))/(8*b**(5/4)) - sqrt(2)*a**(1/4)*atan(1 + sqrt(2)*b**(1/4)*x**2/a**(1/4))/
(8*b**(5/4)) + x**2/(2*b)

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Mathematica [A]  time = 0.456489, size = 361, normalized size = 1.78 \[ \frac{-\sqrt{2} \sqrt [4]{a} \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sqrt{2} \sqrt [4]{a} \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sqrt{2} \sqrt [4]{a} \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sqrt{2} \sqrt [4]{a} \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-2 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )+8 \sqrt [4]{b} x^2}{16 b^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + b*x^8),x]

[Out]

(8*b^(1/4)*x^2 + 2*Sqrt[2]*a^(1/4)*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1
/8)] + 2*Sqrt[2]*a^(1/4)*ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)] - 2*S
qrt[2]*a^(1/4)*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]] + 2*Sqrt[2]*a^(
1/4)*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]] + Sqrt[2]*a^(1/4)*Log[a^(
1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + Sqrt[2]*a^(1/4)*Log[a^(1/4
) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] - Sqrt[2]*a^(1/4)*Log[a^(1/4) +
b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]] - Sqrt[2]*a^(1/4)*Log[a^(1/4) + b^
(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]])/(16*b^(5/4))

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Maple [A]  time = 0.008, size = 144, normalized size = 0.7 \[{\frac{{x}^{2}}{2\,b}}-{\frac{\sqrt{2}}{16\,b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{4}+\sqrt [4]{{\frac{a}{b}}}{x}^{2}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{4}-\sqrt [4]{{\frac{a}{b}}}{x}^{2}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{8\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({{x}^{2}\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{\sqrt{2}}{8\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({{x}^{2}\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b*x^8+a),x)

[Out]

1/2*x^2/b-1/16/b*(a/b)^(1/4)*2^(1/2)*ln((x^4+(a/b)^(1/4)*x^2*2^(1/2)+(a/b)^(1/2)
)/(x^4-(a/b)^(1/4)*x^2*2^(1/2)+(a/b)^(1/2)))-1/8/b*(a/b)^(1/4)*2^(1/2)*arctan(2^
(1/2)/(a/b)^(1/4)*x^2+1)-1/8/b*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^
2-1)

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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^9/(b*x^8 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233384, size = 147, normalized size = 0.72 \[ \frac{4 \, b \left (-\frac{a}{b^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{b \left (-\frac{a}{b^{5}}\right )^{\frac{1}{4}}}{x^{2} + \sqrt{x^{4} + b^{2} \sqrt{-\frac{a}{b^{5}}}}}\right ) - b \left (-\frac{a}{b^{5}}\right )^{\frac{1}{4}} \log \left (x^{2} + b \left (-\frac{a}{b^{5}}\right )^{\frac{1}{4}}\right ) + b \left (-\frac{a}{b^{5}}\right )^{\frac{1}{4}} \log \left (x^{2} - b \left (-\frac{a}{b^{5}}\right )^{\frac{1}{4}}\right ) + 4 \, x^{2}}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^8 + a),x, algorithm="fricas")

[Out]

1/8*(4*b*(-a/b^5)^(1/4)*arctan(b*(-a/b^5)^(1/4)/(x^2 + sqrt(x^4 + b^2*sqrt(-a/b^
5)))) - b*(-a/b^5)^(1/4)*log(x^2 + b*(-a/b^5)^(1/4)) + b*(-a/b^5)^(1/4)*log(x^2
- b*(-a/b^5)^(1/4)) + 4*x^2)/b

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Sympy [A]  time = 1.50471, size = 27, normalized size = 0.13 \[ \operatorname{RootSum}{\left (4096 t^{4} b^{5} + a, \left ( t \mapsto t \log{\left (- 8 t b + x^{2} \right )} \right )\right )} + \frac{x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b*x**8+a),x)

[Out]

RootSum(4096*_t**4*b**5 + a, Lambda(_t, _t*log(-8*_t*b + x**2))) + x**2/(2*b)

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GIAC/XCAS [A]  time = 0.231077, size = 247, normalized size = 1.22 \[ \frac{x^{2}}{2 \, b} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x^{2} + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, b^{2}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x^{2} - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, b^{2}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{4} + \sqrt{2} x^{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{16 \, b^{2}} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{4} - \sqrt{2} x^{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{16 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^8 + a),x, algorithm="giac")

[Out]

1/2*x^2/b - 1/8*sqrt(2)*(a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(2*x^2 + sqrt(2)*(a/b)^
(1/4))/(a/b)^(1/4))/b^2 - 1/8*sqrt(2)*(a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(2*x^2 -
sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/b^2 - 1/16*sqrt(2)*(a*b^3)^(1/4)*ln(x^4 + sqrt
(2)*x^2*(a/b)^(1/4) + sqrt(a/b))/b^2 + 1/16*sqrt(2)*(a*b^3)^(1/4)*ln(x^4 - sqrt(
2)*x^2*(a/b)^(1/4) + sqrt(a/b))/b^2