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

Optimal. Leaf size=23 \[ \frac{\left (a+b x+c x^2+d x^3\right )^{p+1}}{x} \]

[Out]

(a + b*x + c*x^2 + d*x^3)^(1 + p)/x

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Rubi [A]  time = 0.0211701, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.02 \[ \frac{\left (a+b x+c x^2+d x^3\right )^{p+1}}{x} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x + c*x^2 + d*x^3)^(1 + p)/x

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

[Out]

Timed out

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Mathematica [A]  time = 0.0715709, size = 21, normalized size = 0.91 \[ \frac{(a+x (b+x (c+d x)))^{p+1}}{x} \]

Antiderivative was successfully verified.

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

[Out]

(a + x*(b + x*(c + d*x)))^(1 + p)/x

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Maple [A]  time = 0.009, size = 24, normalized size = 1. \[{\frac{ \left ( d{x}^{3}+c{x}^{2}+bx+a \right ) ^{1+p}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.901203, size = 49, normalized size = 2.13 \[ \frac{{\left (d x^{3} + c x^{2} + b x + a\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.320247, size = 49, normalized size = 2.13 \[ \frac{{\left (d x^{3} + c x^{2} + b x + a\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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