3.238 \(\int \frac{\left (a+b x+c x^2+d x^3\right )^p \left (b (1+p) x+c (2+2 p) x^2+d (3+3 p) x^3\right )}{x} \, dx\)

Optimal. Leaf size=19 \[ \left (a+b x+c x^2+d x^3\right )^{p+1} \]

[Out]

(a + b*x + c*x^2 + d*x^3)^(1 + p)

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Rubi [A]  time = 0.0220254, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.021 \[ \left (a+b x+c x^2+d x^3\right )^{p+1} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x + c*x^2 + d*x^3)^(1 + p)

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Rubi in Sympy [A]  time = 17.5511, size = 17, normalized size = 0.89 \[ \left (a + b x + c x^{2} + d x^{3}\right )^{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x + c*x**2 + d*x**3)**(p + 1)

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Mathematica [A]  time = 0.0399336, size = 17, normalized size = 0.89 \[ (a+x (b+x (c+d x)))^{p+1} \]

Antiderivative was successfully verified.

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

[Out]

(a + x*(b + x*(c + d*x)))^(1 + p)

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Maple [A]  time = 0.007, size = 20, normalized size = 1.1 \[ \left ( d{x}^{3}+c{x}^{2}+bx+a \right ) ^{1+p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.923027, size = 45, normalized size = 2.37 \[{\left (d x^{3} + c x^{2} + b x + a\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.289565, size = 45, normalized size = 2.37 \[{\left (d x^{3} + c x^{2} + b x + a\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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