3.909 \(\int x^p (b+c x)^p (b+2 c x) \, dx\)

Optimal. Leaf size=20 \[ \frac{x^{p+1} (b+c x)^{p+1}}{p+1} \]

[Out]

(x^(1 + p)*(b + c*x)^(1 + p))/(1 + p)

_______________________________________________________________________________________

Rubi [A]  time = 0.0190531, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{x^{p+1} (b+c x)^{p+1}}{p+1} \]

Antiderivative was successfully verified.

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

[Out]

(x^(1 + p)*(b + c*x)^(1 + p))/(1 + p)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.73899, size = 15, normalized size = 0.75 \[ \frac{x^{p + 1} \left (b + c x\right )^{p + 1}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(p + 1)*(b + c*x)**(p + 1)/(p + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0393934, size = 20, normalized size = 1. \[ \frac{x^{p+1} (b+c x)^{p+1}}{p+1} \]

Antiderivative was successfully verified.

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

[Out]

(x^(1 + p)*(b + c*x)^(1 + p))/(1 + p)

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 21, normalized size = 1.1 \[{\frac{{x}^{1+p} \left ( cx+b \right ) ^{1+p}}{1+p}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^p*(c*x+b)^p*(2*c*x+b),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.54331, size = 39, normalized size = 1.95 \[ \frac{{\left (c x^{2} + b x\right )} e^{\left (p \log \left (c x + b\right ) + p \log \left (x\right )\right )}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x^2 + b*x)*e^(p*log(c*x + b) + p*log(x))/(p + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.234496, size = 34, normalized size = 1.7 \[ \frac{{\left (c x^{2} + b x\right )}{\left (c x + b\right )}^{p} x^{p}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x^2 + b*x)*(c*x + b)^p*x^p/(p + 1)

_______________________________________________________________________________________

Sympy [A]  time = 5.20877, size = 46, normalized size = 2.3 \[ \begin{cases} \frac{b x x^{p} \left (b + c x\right )^{p}}{p + 1} + \frac{c x^{2} x^{p} \left (b + c x\right )^{p}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (x \right )} + \log{\left (\frac{b}{c} + x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((b*x*x**p*(b + c*x)**p/(p + 1) + c*x**2*x**p*(b + c*x)**p/(p + 1), Ne(
p, -1)), (log(x) + log(b/c + x), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212511, size = 58, normalized size = 2.9 \[ \frac{c x^{2} e^{\left (p{\rm ln}\left (c x + b\right ) + p{\rm ln}\left (x\right )\right )} + b x e^{\left (p{\rm ln}\left (c x + b\right ) + p{\rm ln}\left (x\right )\right )}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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