3.169 \(\int \frac{b+2 c x^2}{x^7 \left (b x+c x^3\right )^8} \, dx\)

Optimal. Leaf size=18 \[ -\frac{1}{14 x^7 \left (b x+c x^3\right )^7} \]

[Out]

-1/(14*x^7*(b*x + c*x^3)^7)

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Rubi [A]  time = 0.0145864, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{1}{14 x^7 \left (b x+c x^3\right )^7} \]

Antiderivative was successfully verified.

[In]  Int[(b + 2*c*x^2)/(x^7*(b*x + c*x^3)^8),x]

[Out]

-1/(14*x^7*(b*x + c*x^3)^7)

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Rubi in Sympy [A]  time = 12.5887, size = 15, normalized size = 0.83 \[ - \frac{1}{14 x^{14} \left (b + c x^{2}\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x**2+b)/x**7/(c*x**3+b*x)**8,x)

[Out]

-1/(14*x**14*(b + c*x**2)**7)

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Mathematica [A]  time = 0.0482352, size = 16, normalized size = 0.89 \[ -\frac{1}{14 x^{14} \left (b+c x^2\right )^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(b + 2*c*x^2)/(x^7*(b*x + c*x^3)^8),x]

[Out]

-1/(14*x^14*(b + c*x^2)^7)

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Maple [B]  time = 0.021, size = 197, normalized size = 10.9 \[ -{\frac{{c}^{8}}{2\,{b}^{13}} \left ( -{\frac{{b}^{6}}{7\,c \left ( c{x}^{2}+b \right ) ^{7}}}-66\,{\frac{b}{c \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{{b}^{5}}{c \left ( c{x}^{2}+b \right ) ^{6}}}-30\,{\frac{{b}^{2}}{c \left ( c{x}^{2}+b \right ) ^{3}}}-12\,{\frac{{b}^{3}}{c \left ( c{x}^{2}+b \right ) ^{4}}}-4\,{\frac{{b}^{4}}{c \left ( c{x}^{2}+b \right ) ^{5}}}-132\,{\frac{1}{ \left ( c{x}^{2}+b \right ) c}} \right ) }-{\frac{1}{14\,{b}^{7}{x}^{14}}}-66\,{\frac{{c}^{6}}{{b}^{13}{x}^{2}}}+33\,{\frac{{c}^{5}}{{b}^{12}{x}^{4}}}-15\,{\frac{{c}^{4}}{{b}^{11}{x}^{6}}}+6\,{\frac{{c}^{3}}{{b}^{10}{x}^{8}}}-2\,{\frac{{c}^{2}}{{b}^{9}{x}^{10}}}+{\frac{c}{2\,{b}^{8}{x}^{12}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x^2+b)/x^7/(c*x^3+b*x)^8,x)

[Out]

-1/2*c^8/b^13*(-1/7*b^6/c/(c*x^2+b)^7-66*b/c/(c*x^2+b)^2-b^5/c/(c*x^2+b)^6-30*b^
2/c/(c*x^2+b)^3-12*b^3/c/(c*x^2+b)^4-4*b^4/c/(c*x^2+b)^5-132/(c*x^2+b)/c)-1/14/b
^7/x^14-66/b^13*c^6/x^2+33/b^12*c^5/x^4-15/b^11*c^4/x^6+6/b^10*c^3/x^8-2/b^9*c^2
/x^10+1/2/b^8*c/x^12

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Maxima [A]  time = 0.832899, size = 109, normalized size = 6.06 \[ -\frac{1}{14 \,{\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 21 \, b^{2} c^{5} x^{24} + 35 \, b^{3} c^{4} x^{22} + 35 \, b^{4} c^{3} x^{20} + 21 \, b^{5} c^{2} x^{18} + 7 \, b^{6} c x^{16} + b^{7} x^{14}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^2 + b)/((c*x^3 + b*x)^8*x^7),x, algorithm="maxima")

[Out]

-1/14/(c^7*x^28 + 7*b*c^6*x^26 + 21*b^2*c^5*x^24 + 35*b^3*c^4*x^22 + 35*b^4*c^3*
x^20 + 21*b^5*c^2*x^18 + 7*b^6*c*x^16 + b^7*x^14)

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Fricas [A]  time = 0.258191, size = 109, normalized size = 6.06 \[ -\frac{1}{14 \,{\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 21 \, b^{2} c^{5} x^{24} + 35 \, b^{3} c^{4} x^{22} + 35 \, b^{4} c^{3} x^{20} + 21 \, b^{5} c^{2} x^{18} + 7 \, b^{6} c x^{16} + b^{7} x^{14}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^2 + b)/((c*x^3 + b*x)^8*x^7),x, algorithm="fricas")

[Out]

-1/14/(c^7*x^28 + 7*b*c^6*x^26 + 21*b^2*c^5*x^24 + 35*b^3*c^4*x^22 + 35*b^4*c^3*
x^20 + 21*b^5*c^2*x^18 + 7*b^6*c*x^16 + b^7*x^14)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x**2+b)/x**7/(c*x**3+b*x)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.264415, size = 20, normalized size = 1.11 \[ -\frac{1}{14 \,{\left (c x^{4} + b x^{2}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^2 + b)/((c*x^3 + b*x)^8*x^7),x, algorithm="giac")

[Out]

-1/14/(c*x^4 + b*x^2)^7