3.11 \(\int \left (a+b x^n+c x^{2 n}\right )^p \left (a+b (1+n+n p) x^n+c (1+2 n (1+p)) x^{2 n}\right ) \, dx\)

Optimal. Leaf size=20 \[ x \left (a+b x^n+c x^{2 n}\right )^{p+1} \]

[Out]

x*(a + b*x^n + c*x^(2*n))^(1 + p)

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Rubi [A]  time = 0.0406958, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.022 \[ x \left (a+b x^n+c x^{2 n}\right )^{p+1} \]

Antiderivative was successfully verified.

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

[Out]

x*(a + b*x^n + c*x^(2*n))^(1 + p)

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Rubi in Sympy [A]  time = 23.3869, size = 17, normalized size = 0.85 \[ x \left (a + b x^{n} + c x^{2 n}\right )^{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a + b*x**n + c*x**(2*n))**(p + 1)

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Mathematica [A]  time = 0.134699, size = 19, normalized size = 0.95 \[ x \left (a+x^n \left (b+c x^n\right )\right )^{p+1} \]

Antiderivative was successfully verified.

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

[Out]

x*(a + x^n*(b + c*x^n))^(1 + p)

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Maple [A]  time = 0.078, size = 33, normalized size = 1.7 \[ x \left ( a+b{x}^{n}+c \left ({x}^{n} \right ) ^{2} \right ) \left ( a+b{x}^{n}+c \left ({x}^{n} \right ) ^{2} \right ) ^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a+b*x^n+c*(x^n)^2)*(a+b*x^n+c*(x^n)^2)^p

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Maxima [A]  time = 0.942247, size = 47, normalized size = 2.35 \[{\left (c x x^{2 \, n} + b x x^{n} + a x\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x*x^(2*n) + b*x*x^n + a*x)*(c*x^(2*n) + b*x^n + a)^p

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Fricas [A]  time = 0.302557, size = 47, normalized size = 2.35 \[{\left (c x x^{2 \, n} + b x x^{n} + a x\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x*x^(2*n) + b*x*x^n + a*x)*(c*x^(2*n) + b*x^n + a)^p

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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((a+b*x**n+c*x**(2*n))**p*(a+b*(n*p+n+1)*x**n+c*(1+2*n*(1+p))*x**(2*n)),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287173, size = 113, normalized size = 5.65 \[ c x e^{\left (p{\rm ln}\left (c e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + 2 \, n{\rm ln}\left (x\right )\right )} + b x e^{\left (p{\rm ln}\left (c e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + n{\rm ln}\left (x\right )\right )} + a x e^{\left (p{\rm ln}\left (c e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x*e^(p*ln(c*e^(2*n*ln(x)) + b*e^(n*ln(x)) + a) + 2*n*ln(x)) + b*x*e^(p*ln(c*e^
(2*n*ln(x)) + b*e^(n*ln(x)) + a) + n*ln(x)) + a*x*e^(p*ln(c*e^(2*n*ln(x)) + b*e^
(n*ln(x)) + a))