3.127 \(\int \frac{x^2 \left (b+2 c x^3\right )}{\left (b x^3+c x^6\right )^8} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{21 \left (b x^3+c x^6\right )^7} \]

[Out]

-1/(21*(b*x^3 + c*x^6)^7)

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Rubi [A]  time = 0.0114013, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ -\frac{1}{21 \left (b x^3+c x^6\right )^7} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(b + 2*c*x^3))/(b*x^3 + c*x^6)^8,x]

[Out]

-1/(21*(b*x^3 + c*x^6)^7)

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Rubi in Sympy [A]  time = 13.1662, size = 15, normalized size = 0.88 \[ - \frac{1}{21 x^{21} \left (b + c x^{3}\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(2*c*x**3+b)/(c*x**6+b*x**3)**8,x)

[Out]

-1/(21*x**21*(b + c*x**3)**7)

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Mathematica [A]  time = 0.0589748, size = 16, normalized size = 0.94 \[ -\frac{1}{21 x^{21} \left (b+c x^3\right )^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(b + 2*c*x^3))/(b*x^3 + c*x^6)^8,x]

[Out]

-1/(21*x^21*(b + c*x^3)^7)

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Maple [B]  time = 0.021, size = 197, normalized size = 11.6 \[ -{\frac{{c}^{8}}{3\,{b}^{13}} \left ( -{\frac{{b}^{6}}{7\,c \left ( c{x}^{3}+b \right ) ^{7}}}-66\,{\frac{b}{c \left ( c{x}^{3}+b \right ) ^{2}}}-{\frac{{b}^{5}}{c \left ( c{x}^{3}+b \right ) ^{6}}}-30\,{\frac{{b}^{2}}{c \left ( c{x}^{3}+b \right ) ^{3}}}-12\,{\frac{{b}^{3}}{c \left ( c{x}^{3}+b \right ) ^{4}}}-4\,{\frac{{b}^{4}}{c \left ( c{x}^{3}+b \right ) ^{5}}}-132\,{\frac{1}{c \left ( c{x}^{3}+b \right ) }} \right ) }-{\frac{1}{21\,{b}^{7}{x}^{21}}}-44\,{\frac{{c}^{6}}{{b}^{13}{x}^{3}}}+22\,{\frac{{c}^{5}}{{b}^{12}{x}^{6}}}-10\,{\frac{{c}^{4}}{{b}^{11}{x}^{9}}}+4\,{\frac{{c}^{3}}{{b}^{10}{x}^{12}}}-{\frac{4\,{c}^{2}}{3\,{b}^{9}{x}^{15}}}+{\frac{c}{3\,{b}^{8}{x}^{18}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(2*c*x^3+b)/(c*x^6+b*x^3)^8,x)

[Out]

-1/3*c^8/b^13*(-1/7*b^6/c/(c*x^3+b)^7-66*b/c/(c*x^3+b)^2-b^5/c/(c*x^3+b)^6-30*b^
2/c/(c*x^3+b)^3-12*b^3/c/(c*x^3+b)^4-4*b^4/c/(c*x^3+b)^5-132/c/(c*x^3+b))-1/21/b
^7/x^21-44/b^13*c^6/x^3+22/b^12*c^5/x^6-10/b^11*c^4/x^9+4/b^10*c^3/x^12-4/3/b^9*
c^2/x^15+1/3/b^8*c/x^18

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Maxima [A]  time = 0.767351, size = 109, normalized size = 6.41 \[ -\frac{1}{21 \,{\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 21 \, b^{2} c^{5} x^{36} + 35 \, b^{3} c^{4} x^{33} + 35 \, b^{4} c^{3} x^{30} + 21 \, b^{5} c^{2} x^{27} + 7 \, b^{6} c x^{24} + b^{7} x^{21}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^3 + b)*x^2/(c*x^6 + b*x^3)^8,x, algorithm="maxima")

[Out]

-1/21/(c^7*x^42 + 7*b*c^6*x^39 + 21*b^2*c^5*x^36 + 35*b^3*c^4*x^33 + 35*b^4*c^3*
x^30 + 21*b^5*c^2*x^27 + 7*b^6*c*x^24 + b^7*x^21)

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Fricas [A]  time = 0.274183, size = 109, normalized size = 6.41 \[ -\frac{1}{21 \,{\left (c^{7} x^{42} + 7 \, b c^{6} x^{39} + 21 \, b^{2} c^{5} x^{36} + 35 \, b^{3} c^{4} x^{33} + 35 \, b^{4} c^{3} x^{30} + 21 \, b^{5} c^{2} x^{27} + 7 \, b^{6} c x^{24} + b^{7} x^{21}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^3 + b)*x^2/(c*x^6 + b*x^3)^8,x, algorithm="fricas")

[Out]

-1/21/(c^7*x^42 + 7*b*c^6*x^39 + 21*b^2*c^5*x^36 + 35*b^3*c^4*x^33 + 35*b^4*c^3*
x^30 + 21*b^5*c^2*x^27 + 7*b^6*c*x^24 + b^7*x^21)

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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(x**2*(2*c*x**3+b)/(c*x**6+b*x**3)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.268839, size = 20, normalized size = 1.18 \[ -\frac{1}{21 \,{\left (c x^{6} + b x^{3}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^3 + b)*x^2/(c*x^6 + b*x^3)^8,x, algorithm="giac")

[Out]

-1/21/(c*x^6 + b*x^3)^7