3.15.40 \(\int (b d+2 c d x) (a+b x+c x^2)^p \, dx\) [1440]

Optimal. Leaf size=21 \[ \frac {d \left (a+b x+c x^2\right )^{1+p}}{1+p} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {643} \begin {gather*} \frac {d \left (a+b x+c x^2\right )^{p+1}}{p+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(a + b*x + c*x^2)^(1 + p))/(1 + p)

Rule 643

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*((a + b*x + c*x^2)^(p +
1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 20, normalized size = 0.95 \begin {gather*} \frac {d (a+x (b+c x))^{1+p}}{1+p} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(a + x*(b + c*x))^(1 + p))/(1 + p)

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Maple [A]
time = 0.68, size = 22, normalized size = 1.05

method	result	size
gosper	\(\frac {d \left (c \,x^{2}+b x +a \right )^{1+p}}{1+p}\)	\(22\)
risch	\(\frac {\left (c \,x^{2}+b x +a \right )^{p} \left (c \,x^{2}+b x +a \right ) d}{1+p}\)	\(30\)
norman	\(\frac {a d \,{\mathrm e}^{p \ln \left (c \,x^{2}+b x +a \right )}}{1+p}+\frac {b d x \,{\mathrm e}^{p \ln \left (c \,x^{2}+b x +a \right )}}{1+p}+\frac {c d \,x^{2} {\mathrm e}^{p \ln \left (c \,x^{2}+b x +a \right )}}{1+p}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)*(c*x^2+b*x+a)^p,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 21, normalized size = 1.00 \begin {gather*} \frac {{\left (c x^{2} + b x + a\right )}^{p + 1} d}{p + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)*(c*x^2+b*x+a)^p,x, algorithm="maxima")

[Out]

(c*x^2 + b*x + a)^(p + 1)*d/(p + 1)

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Fricas [A]
time = 3.03, size = 32, normalized size = 1.52 \begin {gather*} \frac {{\left (c d x^{2} + b d x + a d\right )} {\left (c x^{2} + b x + a\right )}^{p}}{p + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)*(c*x^2+b*x+a)^p,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (17) = 34\).
time = 30.86, size = 112, normalized size = 5.33 \begin {gather*} \begin {cases} \frac {a d \left (a + b x + c x^{2}\right )^{p}}{p + 1} + \frac {b d x \left (a + b x + c x^{2}\right )^{p}}{p + 1} + \frac {c d x^{2} \left (a + b x + c x^{2}\right )^{p}}{p + 1} & \text {for}\: p \neq -1 \\d \log {\left (\frac {b}{2 c} + x - \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )} + d \log {\left (\frac {b}{2 c} + x + \frac {\sqrt {- 4 a c + b^{2}}}{2 c} \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)*(c*x**2+b*x+a)**p,x)

[Out]

Piecewise((a*d*(a + b*x + c*x**2)**p/(p + 1) + b*d*x*(a + b*x + c*x**2)**p/(p + 1) + c*d*x**2*(a + b*x + c*x**
2)**p/(p + 1), Ne(p, -1)), (d*log(b/(2*c) + x - sqrt(-4*a*c + b**2)/(2*c)) + d*log(b/(2*c) + x + sqrt(-4*a*c +
b**2)/(2*c)), True))

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Giac [A]
time = 2.36, size = 21, normalized size = 1.00 \begin {gather*} \frac {{\left (c x^{2} + b x + a\right )}^{p + 1} d}{p + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)*(c*x^2+b*x+a)^p,x, algorithm="giac")

[Out]

(c*x^2 + b*x + a)^(p + 1)*d/(p + 1)

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Mupad [B]
time = 0.52, size = 42, normalized size = 2.00 \begin {gather*} {\left (c\,x^2+b\,x+a\right )}^p\,\left (\frac {a\,d}{p+1}+\frac {c\,d\,x^2}{p+1}+\frac {b\,d\,x}{p+1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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