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

Optimal. Leaf size=27 \[ \frac{x^{p+1} \left (b x+c x^3\right )^{p+1}}{2 (p+1)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [C]  time = 0.221907, antiderivative size = 116, normalized size of antiderivative = 4.3, number of steps used = 7, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079 \[ \frac{b x^{p+2} \left (b x+c x^3\right )^p \left (\frac{c x^2}{b}+1\right )^{-p} \, _2F_1\left (-p,p+1;p+2;-\frac{c x^2}{b}\right )}{2 (p+1)}+\frac{c x^{p+4} \left (b x+c x^3\right )^p \left (\frac{c x^2}{b}+1\right )^{-p} \, _2F_1\left (-p,p+2;p+3;-\frac{c x^2}{b}\right )}{p+2} \]

Antiderivative was successfully verified.

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

[Out]

(b*x^(2 + p)*(b*x + c*x^3)^p*Hypergeometric2F1[-p, 1 + p, 2 + p, -((c*x^2)/b)])/
(2*(1 + p)*(1 + (c*x^2)/b)^p) + (c*x^(4 + p)*(b*x + c*x^3)^p*Hypergeometric2F1[-
p, 2 + p, 3 + p, -((c*x^2)/b)])/((2 + p)*(1 + (c*x^2)/b)^p)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.2655, size = 102, normalized size = 3.78 \[ \frac{b x^{- p} x^{2 p + 2} \left (1 + \frac{c x^{2}}{b}\right )^{- p} \left (b x + c x^{3}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 1 \\ p + 2 \end{matrix}\middle |{- \frac{c x^{2}}{b}} \right )}}{2 \left (p + 1\right )} + \frac{c x^{- p} x^{2 p + 4} \left (1 + \frac{c x^{2}}{b}\right )^{- p} \left (b x + c x^{3}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 2 \\ p + 3 \end{matrix}\middle |{- \frac{c x^{2}}{b}} \right )}}{p + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x**(-p)*x**(2*p + 2)*(1 + c*x**2/b)**(-p)*(b*x + c*x**3)**p*hyper((-p, p + 1),
(p + 2,), -c*x**2/b)/(2*(p + 1)) + c*x**(-p)*x**(2*p + 4)*(1 + c*x**2/b)**(-p)*
(b*x + c*x**3)**p*hyper((-p, p + 2), (p + 3,), -c*x**2/b)/(p + 2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0291373, size = 27, normalized size = 1. \[ \frac{x^{p+1} \left (x \left (b+c x^2\right )\right )^{p+1}}{2 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.173, size = 142, normalized size = 5.3 \[{\frac{x \left ( c{x}^{2}+b \right ){x}^{1+p}}{2+2\,p}{{\rm e}^{-{\frac{p \left ( i \left ({\it csgn} \left ( ix \left ( c{x}^{2}+b \right ) \right ) \right ) ^{3}\pi -i \left ({\it csgn} \left ( ix \left ( c{x}^{2}+b \right ) \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i \left ({\it csgn} \left ( ix \left ( c{x}^{2}+b \right ) \right ) \right ) ^{2}{\it csgn} \left ( i \left ( c{x}^{2}+b \right ) \right ) \pi +i{\it csgn} \left ( ix \left ( c{x}^{2}+b \right ) \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( i \left ( c{x}^{2}+b \right ) \right ) \pi -2\,\ln \left ( c{x}^{2}+b \right ) -2\,\ln \left ( x \right ) \right ) }{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(c*x^2+b)*x*x^(1+p)/(1+p)*exp(-1/2*p*(I*csgn(I*x*(c*x^2+b))^3*Pi-I*csgn(I*x*
(c*x^2+b))^2*csgn(I*x)*Pi-I*csgn(I*x*(c*x^2+b))^2*csgn(I*(c*x^2+b))*Pi+I*csgn(I*
x*(c*x^2+b))*csgn(I*x)*csgn(I*(c*x^2+b))*Pi-2*ln(c*x^2+b)-2*ln(x)))

_______________________________________________________________________________________

Maxima [A]  time = 0.920708, size = 47, normalized size = 1.74 \[ \frac{{\left (c x^{4} + b x^{2}\right )} e^{\left (p \log \left (c x^{2} + b\right ) + 2 \, p \log \left (x\right )\right )}}{2 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.273761, size = 45, normalized size = 1.67 \[ \frac{{\left (c x^{2} + b\right )}{\left (c x^{3} + b x\right )}^{p} x^{p + 3}}{2 \,{\left (p + 1\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int 2 \,{\left (c x^{3} + b x\right )}^{p} c x^{p + 3} +{\left (c x^{3} + b x\right )}^{p} b x^{p + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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