3.911 \(\int x^{-1+3 (1+p)} \left (b+c x^3\right )^p \left (b+2 c x^3\right ) \, dx\)

Optimal. Leaf size=27 \[ \frac{x^{3 (p+1)} \left (b+c x^3\right )^{p+1}}{3 (p+1)} \]

[Out]

(x^(3*(1 + p))*(b + c*x^3)^(1 + p))/(3*(1 + p))

_______________________________________________________________________________________

Rubi [A]  time = 0.0210987, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037 \[ \frac{x^{3 (p+1)} \left (b+c x^3\right )^{p+1}}{3 (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 3*(1 + p))*(b + c*x^3)^p*(b + 2*c*x^3),x]

[Out]

(x^(3*(1 + p))*(b + c*x^3)^(1 + p))/(3*(1 + p))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.07563, size = 20, normalized size = 0.74 \[ \frac{x^{3 p + 3} \left (b + c x^{3}\right )^{p + 1}}{3 \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(2+3*p)*(c*x**3+b)**p*(2*c*x**3+b),x)

[Out]

x**(3*p + 3)*(b + c*x**3)**(p + 1)/(3*(p + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.051221, size = 26, normalized size = 0.96 \[ \frac{x^{3 p+3} \left (b+c x^3\right )^{p+1}}{3 p+3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 3*(1 + p))*(b + c*x^3)^p*(b + 2*c*x^3),x]

[Out]

(x^(3 + 3*p)*(b + c*x^3)^(1 + p))/(3 + 3*p)

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 26, normalized size = 1. \[{\frac{{x}^{3+3\,p} \left ( c{x}^{3}+b \right ) ^{1+p}}{3+3\,p}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(2+3*p)*(c*x^3+b)^p*(2*c*x^3+b),x)

[Out]

1/3*x^(3+3*p)*(c*x^3+b)^(1+p)/(1+p)

_______________________________________________________________________________________

Maxima [A]  time = 1.66652, size = 47, normalized size = 1.74 \[ \frac{{\left (c x^{6} + b x^{3}\right )} e^{\left (p \log \left (c x^{3} + b\right ) + 3 \, p \log \left (x\right )\right )}}{3 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(c*x^6 + b*x^3)*e^(p*log(c*x^3 + b) + 3*p*log(x))/(p + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.236144, size = 43, normalized size = 1.59 \[ \frac{{\left (c x^{4} + b x\right )}{\left (c x^{3} + b\right )}^{p} x^{3 \, p + 2}}{3 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(c*x^4 + b*x)*(c*x^3 + b)^p*x^(3*p + 2)/(p + 1)

_______________________________________________________________________________________

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+3*p)*(c*x**3+b)**p*(2*c*x**3+b),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217729, size = 78, normalized size = 2.89 \[ \frac{c x^{4} e^{\left (p{\rm ln}\left (c x^{3} + b\right ) + 3 \, p{\rm ln}\left (x\right ) + 2 \,{\rm ln}\left (x\right )\right )} + b x e^{\left (p{\rm ln}\left (c x^{3} + b\right ) + 3 \, p{\rm ln}\left (x\right ) + 2 \,{\rm ln}\left (x\right )\right )}}{3 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(c*x^4*e^(p*ln(c*x^3 + b) + 3*p*ln(x) + 2*ln(x)) + b*x*e^(p*ln(c*x^3 + b) +
3*p*ln(x) + 2*ln(x)))/(p + 1)