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

Optimal. Leaf size=24 \[ \frac{x^{p+1} \left (b x+c x^2\right )^{p+1}}{p+1} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0166814, antiderivative size = 24, 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 \[ \frac{x^{p+1} \left (b x+c x^2\right )^{p+1}}{p+1} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.72861, size = 19, normalized size = 0.79 \[ \frac{x^{p + 1} \left (b x + c x^{2}\right )^{p + 1}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1+p)*(3*c*x+2*b)*(c*x**2+b*x)**p,x)

[Out]

x**(p + 1)*(b*x + c*x**2)**(p + 1)/(p + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0500025, size = 22, normalized size = 0.92 \[ \frac{x^{p+1} (x (b+c x))^{p+1}}{p+1} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 28, normalized size = 1.2 \[{\frac{{x}^{2+p} \left ( cx+b \right ) \left ( c{x}^{2}+bx \right ) ^{p}}{1+p}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.790651, size = 43, normalized size = 1.79 \[ \frac{{\left (c x^{3} + b x^{2}\right )} e^{\left (p \log \left (c x + b\right ) + 2 \, p \log \left (x\right )\right )}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*c*x + 2*b)*(c*x^2 + b*x)^p*x^(p + 1),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.313436, size = 42, normalized size = 1.75 \[ \frac{{\left (c x^{2} + b x\right )}{\left (c x^{2} + b x\right )}^{p} x^{p + 1}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*c*x + 2*b)*(c*x^2 + b*x)^p*x^(p + 1),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.274724, size = 66, normalized size = 2.75 \[ \frac{c x^{2} e^{\left (p{\rm ln}\left (c x + b\right ) + 2 \, p{\rm ln}\left (x\right ) +{\rm ln}\left (x\right )\right )} + b x e^{\left (p{\rm ln}\left (c x + b\right ) + 2 \, p{\rm ln}\left (x\right ) +{\rm ln}\left (x\right )\right )}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*c*x + 2*b)*(c*x^2 + b*x)^p*x^(p + 1),x, algorithm="giac")

[Out]

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