3.234 \(\int x^m \left (a+b x+c x^2+d x^3\right )^p (a (1+m)+x (b (2+m+p)+x (c (3+m+2 p)+d (4+m+3 p) x))) \, dx\)

Optimal. Leaf size=25 \[ x^{m+1} \left (a+b x+c x^2+d x^3\right )^{p+1} \]

[Out]

x^(1 + m)*(a + b*x + c*x^2 + d*x^3)^(1 + p)

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Rubi [A]  time = 0.0304605, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 56, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.018 \[ x^{m+1} \left (a+b x+c x^2+d x^3\right )^{p+1} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x + c*x^2 + d*x^3)^p*(a*(1 + m) + x*(b*(2 + m + p) + x*(c*(3 + m + 2*p) + d*(4 + m + 3*p)*x))),x]

[Out]

x^(1 + m)*(a + b*x + c*x^2 + d*x^3)^(1 + p)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(d*x**3+c*x**2+b*x+a)**p*(a*(1+m)+x*(b*(2+m+p)+x*(c*(3+m+2*p)+d*(4+m+3*p)*x))),x)

[Out]

Timed out

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Mathematica [A]  time = 0.137723, size = 23, normalized size = 0.92 \[ x^{m+1} (a+x (b+x (c+d x)))^{p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x + c*x^2 + d*x^3)^p*(a*(1 + m) + x*(b*(2 + m + p) + x*(c*(3 + m + 2*p) + d*(4 + m + 3*p)*x))),x]

[Out]

x^(1 + m)*(a + x*(b + x*(c + d*x)))^(1 + p)

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Maple [A]  time = 0.011, size = 26, normalized size = 1. \[{x}^{1+m} \left ( d{x}^{3}+c{x}^{2}+bx+a \right ) ^{1+p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(d*x^3+c*x^2+b*x+a)^p*(a*(1+m)+x*(b*(2+m+p)+x*(c*(3+m+2*p)+d*(4+m+3*p)*x))),x)

[Out]

x^(1+m)*(d*x^3+c*x^2+b*x+a)^(1+p)

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Maxima [A]  time = 0.993762, size = 59, normalized size = 2.36 \[{\left (d x^{4} + c x^{3} + b x^{2} + a x\right )} e^{\left (p \log \left (d x^{3} + c x^{2} + b x + a\right ) + m \log \left (x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*(m + 1) + (b*(m + p + 2) + (d*(m + 3*p + 4)*x + c*(m + 2*p + 3))*x)*x)*(d*x^3 + c*x^2 + b*x + a)^p*x^m,x, algorithm="maxima")

[Out]

(d*x^4 + c*x^3 + b*x^2 + a*x)*e^(p*log(d*x^3 + c*x^2 + b*x + a) + m*log(x))

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Fricas [A]  time = 0.506426, size = 54, normalized size = 2.16 \[{\left (d x^{4} + c x^{3} + b x^{2} + a x\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} x^{m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*(m + 1) + (b*(m + p + 2) + (d*(m + 3*p + 4)*x + c*(m + 2*p + 3))*x)*x)*(d*x^3 + c*x^2 + b*x + a)^p*x^m,x, algorithm="fricas")

[Out]

(d*x^4 + c*x^3 + b*x^2 + a*x)*(d*x^3 + c*x^2 + b*x + a)^p*x^m

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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**m*(d*x**3+c*x**2+b*x+a)**p*(a*(1+m)+x*(b*(2+m+p)+x*(c*(3+m+2*p)+d*(4+m+3*p)*x))),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.414685, size = 155, normalized size = 6.2 \[ d x^{4} e^{\left (p{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right ) + m{\rm ln}\left (x\right )\right )} + c x^{3} e^{\left (p{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right ) + m{\rm ln}\left (x\right )\right )} + b x^{2} e^{\left (p{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right ) + m{\rm ln}\left (x\right )\right )} + a x e^{\left (p{\rm ln}\left (d x^{3} + c x^{2} + b x + a\right ) + m{\rm ln}\left (x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*(m + 1) + (b*(m + p + 2) + (d*(m + 3*p + 4)*x + c*(m + 2*p + 3))*x)*x)*(d*x^3 + c*x^2 + b*x + a)^p*x^m,x, algorithm="giac")

[Out]

d*x^4*e^(p*ln(d*x^3 + c*x^2 + b*x + a) + m*ln(x)) + c*x^3*e^(p*ln(d*x^3 + c*x^2
+ b*x + a) + m*ln(x)) + b*x^2*e^(p*ln(d*x^3 + c*x^2 + b*x + a) + m*ln(x)) + a*x*
e^(p*ln(d*x^3 + c*x^2 + b*x + a) + m*ln(x))