3.137 \(\int (b+2 c x) \left (b x+c x^2\right )^p \, dx\)

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

[Out]

(b*x + c*x^2)^(1 + p)/(1 + p)

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Rubi [A]  time = 0.0102823, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{\left (b x+c x^2\right )^{p+1}}{p+1} \]

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)

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Rubi in Sympy [A]  time = 3.51661, size = 14, normalized size = 0.74 \[ \frac{\left (b x + c x^{2}\right )^{p + 1}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)*(c*x**2+b*x)**p,x)

[Out]

(b*x + c*x**2)**(p + 1)/(p + 1)

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Mathematica [A]  time = 0.03231, 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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Maple [A]  time = 0.006, size = 24, normalized size = 1.3 \[{\frac{x \left ( cx+b \right ) \left ( c{x}^{2}+bx \right ) ^{p}}{1+p}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(c*x^2+b*x)^p,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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]

Exception raised: ValueError

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Fricas [A]  time = 0.298969, size = 35, normalized size = 1.84 \[ \frac{{\left (c x^{2} + b x\right )}{\left (c x^{2} + b x\right )}^{p}}{p + 1} \]

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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Sympy [A]  time = 1.72258, size = 46, normalized size = 2.42 \[ \begin{cases} \frac{b x \left (b x + c x^{2}\right )^{p}}{p + 1} + \frac{c x^{2} \left (b x + c x^{2}\right )^{p}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (x \right )} + \log{\left (\frac{b}{c} + x \right )} & \text{otherwise} \end{cases} \]

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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GIAC/XCAS [A]  time = 0.264448, size = 55, normalized size = 2.89 \[ \frac{c x^{2} e^{\left (p{\rm ln}\left (c x^{2} + b x\right )\right )} + b x e^{\left (p{\rm ln}\left (c x^{2} + b x\right )\right )}}{p + 1} \]

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