3.172 \(\int (b+2 c x) (b x+c x^2)^p \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {629} \[ \frac {\left (b x+c x^2\right )^{p+1}}{p+1} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(b*x + c*x^2)^p,x]

[Out]

(b*x + c*x^2)^(1 + p)/(1 + p)

Rule 629

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

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.89 \[ \frac {(x (b+c x))^{p+1}}{p+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b + c*x))^(1 + p)/(1 + p)

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fricas [A]  time = 0.78, size = 26, normalized size = 1.37 \[ \frac {{\left (c x^{2} + b x\right )} {\left (c x^{2} + b x\right )}^{p}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x^2 + b*x)*(c*x^2 + b*x)^p/(p + 1)

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giac [A]  time = 0.37, size = 19, normalized size = 1.00 \[ \frac {{\left (c x^{2} + b x\right )}^{p + 1}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x^2 + b*x)^(p + 1)/(p + 1)

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maple [A]  time = 0.00, size = 24, normalized size = 1.26 \[ \frac {\left (c x +b \right ) x \left (c \,x^{2}+b x \right )^{p}}{p +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.56, size = 19, normalized size = 1.00 \[ \frac {{\left (c x^{2} + b x\right )}^{p + 1}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x^2 + b*x)^(p + 1)/(p + 1)

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mupad [B]  time = 2.10, size = 23, normalized size = 1.21 \[ \frac {x\,{\left (c\,x^2+b\,x\right )}^p\,\left (b+c\,x\right )}{p+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^p*(b + 2*c*x),x)

[Out]

(x*(b*x + c*x^2)^p*(b + c*x))/(p + 1)

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sympy [A]  time = 0.71, size = 46, normalized size = 2.42 \[ \begin {cases} \frac {b x \left (b x + c x^{2}\right )^{p}}{p + 1} + \frac {c x^{2} \left (b x + c x^{2}\right )^{p}}{p + 1} & \text {for}\: p \neq -1 \\\log {\relax (x )} + \log {\left (\frac {b}{c} + x \right )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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