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

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))

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Rubi [A]  time = 0.0186476, antiderivative size = 27, 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 \[ \frac{x^{p+1} \left (b x+c x^3\right )^{p+1}}{2 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.5, size = 20, normalized size = 0.74 \[ \frac{x^{p + 1} \left (b x + c x^{3}\right )^{p + 1}}{2 \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0508187, 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[x^(1 + p)*(b + 2*c*x^2)*(b*x + c*x^3)^p,x]

[Out]

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

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Maple [A]  time = 0.006, size = 31, normalized size = 1.2 \[{\frac{{x}^{2+p} \left ( c{x}^{2}+b \right ) \left ( c{x}^{3}+bx \right ) ^{p}}{2+2\,p}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.929879, 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^2 + b)*(c*x^3 + b*x)^p*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)

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Fricas [A]  time = 0.275371, size = 43, normalized size = 1.59 \[ \frac{{\left (c x^{3} + b x\right )}{\left (c x^{3} + b x\right )}^{p} x^{p + 1}}{2 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.268733, size = 73, normalized size = 2.7 \[ \frac{c x^{3} e^{\left (p{\rm ln}\left (c x^{2} + b\right ) + 2 \, p{\rm ln}\left (x\right ) +{\rm ln}\left (x\right )\right )} + b x e^{\left (p{\rm ln}\left (c x^{2} + b\right ) + 2 \, p{\rm ln}\left (x\right ) +{\rm ln}\left (x\right )\right )}}{2 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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