3.1457 \(\int \frac{1}{x^7 \left (a+b x^8\right )} \, dx\)

Optimal. Leaf size=203 \[ \frac{b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4}}-\frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{7/4}}+\frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{7/4}}-\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{7/4}}-\frac{1}{6 a x^6} \]

[Out]

-1/(6*a*x^6) + (b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^
(7/4)) - (b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^(7/4))
+ (b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(8*Sqrt[2]
*a^(7/4)) - (b^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(
8*Sqrt[2]*a^(7/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.383847, 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 \[ \frac{b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4}}-\frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{7/4}}+\frac{b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{7/4}}-\frac{b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{7/4}}-\frac{1}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(a + b*x^8)),x]

[Out]

-1/(6*a*x^6) + (b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^
(7/4)) - (b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^(7/4))
+ (b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(8*Sqrt[2]
*a^(7/4)) - (b^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(
8*Sqrt[2]*a^(7/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 60.5455, size = 187, normalized size = 0.92 \[ - \frac{1}{6 a x^{6}} + \frac{\sqrt{2} b^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{2} + \sqrt{a} + \sqrt{b} x^{4} \right )}}{16 a^{\frac{7}{4}}} - \frac{\sqrt{2} b^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{2} + \sqrt{a} + \sqrt{b} x^{4} \right )}}{16 a^{\frac{7}{4}}} + \frac{\sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x^{2}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}}} - \frac{\sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x^{2}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(b*x**8+a),x)

[Out]

-1/(6*a*x**6) + sqrt(2)*b**(3/4)*log(-sqrt(2)*a**(1/4)*b**(1/4)*x**2 + sqrt(a) +
sqrt(b)*x**4)/(16*a**(7/4)) - sqrt(2)*b**(3/4)*log(sqrt(2)*a**(1/4)*b**(1/4)*x*
*2 + sqrt(a) + sqrt(b)*x**4)/(16*a**(7/4)) + sqrt(2)*b**(3/4)*atan(1 - sqrt(2)*b
**(1/4)*x**2/a**(1/4))/(8*a**(7/4)) - sqrt(2)*b**(3/4)*atan(1 + sqrt(2)*b**(1/4)
*x**2/a**(1/4))/(8*a**(7/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.397707, size = 387, normalized size = 1.91 \[ \frac{-8 a^{3/4}-6 \sqrt{2} b^{3/4} x^6 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+6 \sqrt{2} b^{3/4} x^6 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+6 \sqrt{2} b^{3/4} x^6 \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+6 \sqrt{2} b^{3/4} x^6 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )-3 \sqrt{2} b^{3/4} x^6 \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-3 \sqrt{2} b^{3/4} x^6 \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+3 \sqrt{2} b^{3/4} x^6 \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+3 \sqrt{2} b^{3/4} x^6 \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{48 a^{7/4} x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(a + b*x^8)),x]

[Out]

(-8*a^(3/4) + 6*Sqrt[2]*b^(3/4)*x^6*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(
1/8)] + 6*Sqrt[2]*b^(3/4)*x^6*ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]
- 6*Sqrt[2]*b^(3/4)*x^6*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]] + 6*Sq
rt[2]*b^(3/4)*x^6*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]] + 3*Sqrt[2]*
b^(3/4)*x^6*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + 3*Sqrt[
2]*b^(3/4)*x^6*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] - 3*Sq
rt[2]*b^(3/4)*x^6*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]] - 3
*Sqrt[2]*b^(3/4)*x^6*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]])
/(48*a^(7/4)*x^6)

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 147, normalized size = 0.7 \[ -{\frac{b\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ({x}^{4}+\sqrt [4]{{\frac{a}{b}}}{x}^{2}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{4}-\sqrt [4]{{\frac{a}{b}}}{x}^{2}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{b\sqrt{2}}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({{x}^{2}\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{b\sqrt{2}}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({{x}^{2}\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{1}{6\,{x}^{6}a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(b*x^8+a),x)

[Out]

-1/16*b/a^2*(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^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(
1/2)/(a/b)^(1/4)*x^2+1)-1/8*b/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)
*x^2-1)-1/6/x^6/a

_______________________________________________________________________________________

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(1/((b*x^8 + a)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.235699, size = 208, normalized size = 1.02 \[ \frac{12 \, a x^{6} \left (-\frac{b^{3}}{a^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} \left (-\frac{b^{3}}{a^{7}}\right )^{\frac{1}{4}}}{b x^{2} + b \sqrt{\frac{b^{2} x^{4} + a^{4} \sqrt{-\frac{b^{3}}{a^{7}}}}{b^{2}}}}\right ) - 3 \, a x^{6} \left (-\frac{b^{3}}{a^{7}}\right )^{\frac{1}{4}} \log \left (b x^{2} + a^{2} \left (-\frac{b^{3}}{a^{7}}\right )^{\frac{1}{4}}\right ) + 3 \, a x^{6} \left (-\frac{b^{3}}{a^{7}}\right )^{\frac{1}{4}} \log \left (b x^{2} - a^{2} \left (-\frac{b^{3}}{a^{7}}\right )^{\frac{1}{4}}\right ) - 4}{24 \, a x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^7),x, algorithm="fricas")

[Out]

1/24*(12*a*x^6*(-b^3/a^7)^(1/4)*arctan(a^2*(-b^3/a^7)^(1/4)/(b*x^2 + b*sqrt((b^2
*x^4 + a^4*sqrt(-b^3/a^7))/b^2))) - 3*a*x^6*(-b^3/a^7)^(1/4)*log(b*x^2 + a^2*(-b
^3/a^7)^(1/4)) + 3*a*x^6*(-b^3/a^7)^(1/4)*log(b*x^2 - a^2*(-b^3/a^7)^(1/4)) - 4)
/(a*x^6)

_______________________________________________________________________________________

Sympy [A]  time = 9.15608, size = 34, normalized size = 0.17 \[ \operatorname{RootSum}{\left (4096 t^{4} a^{7} + b^{3}, \left ( t \mapsto t \log{\left (- \frac{8 t a^{2}}{b} + x^{2} \right )} \right )\right )} - \frac{1}{6 a x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(b*x**8+a),x)

[Out]

RootSum(4096*_t**4*a**7 + b**3, Lambda(_t, _t*log(-8*_t*a**2/b + x**2))) - 1/(6*
a*x**6)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.23065, size = 247, normalized size = 1.22 \[ -\frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x^{2} + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x^{2} - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{4} + \sqrt{2} x^{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2}} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{4} - \sqrt{2} x^{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2}} - \frac{1}{6 \, a x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^8 + a)*x^7),x, algorithm="giac")

[Out]

-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))/a^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))/a^2 - 1/16*sqrt(2)*(a*b^3)^(1/4)*ln(x^4 + sqrt(2)*x^2*(a/
b)^(1/4) + sqrt(a/b))/a^2 + 1/16*sqrt(2)*(a*b^3)^(1/4)*ln(x^4 - sqrt(2)*x^2*(a/b
)^(1/4) + sqrt(a/b))/a^2 - 1/6/(a*x^6)