3.3.39 \(\int \frac {(a+b x+c x^2+d x^3)^p (-a+b p x+c (1+2 p) x^2+d (2+3 p) x^3)}{x^2} \, dx\) [239]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {1604} \begin {gather*} \frac {\left (a+b x+c x^2+d x^3\right )^{p+1}}{x} \end {gather*}

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

Rule 1604

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*(Rr^(n + 1)/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x,
r])), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps

\begin {align*} \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 &=\frac {\left (a+b x+c x^2+d x^3\right )^{1+p}}{x}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.71, size = 21, normalized size = 0.91 \begin {gather*} \frac {(a+x (b+x (c+d x)))^{1+p}}{x} \end {gather*}

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

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 24, normalized size = 1.04

method	result	size
gosper	\(\frac {\left (d \,x^{3}+c \,x^{2}+b x +a \right )^{1+p}}{x}\)	\(24\)
risch	\(\frac {\left (d \,x^{3}+c \,x^{2}+b x +a \right ) \left (d \,x^{3}+c \,x^{2}+b x +a \right )^{p}}{x}\)	\(37\)
norman	\(\frac {a \,{\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+b x \,{\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+c \,x^{2} {\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+d \,x^{3} {\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}}{x}\)	\(97\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 36, normalized size = 1.57 \begin {gather*} \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} \end {gather*}

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, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.44, size = 36, normalized size = 1.57 \begin {gather*} \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} \end {gather*}

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, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, 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)

________________________________________________________________________________________

Mupad [B]
time = 3.20, size = 23, normalized size = 1.00 \begin {gather*} \frac {{\left (d\,x^3+c\,x^2+b\,x+a\right )}^{p+1}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a + b*x + c*x^2 + d*x^3)^(p + 1)/x

________________________________________________________________________________________