3.133 \(\int (b+2 c x) \left (-a+b x+c x^2\right )^p \, dx\)

Optimal. Leaf size=22 \[ \frac{\left (-a+b x+c x^2\right )^{p+1}}{p+1} \]

[Out]

(-a + b*x + c*x^2)^(1 + p)/(1 + p)

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Rubi [A]  time = 0.0108548, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{\left (-a+b x+c x^2\right )^{p+1}}{p+1} \]

Antiderivative was successfully verified.

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

[Out]

(-a + b*x + c*x^2)^(1 + p)/(1 + p)

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Rubi in Sympy [A]  time = 4.38538, size = 15, normalized size = 0.68 \[ \frac{\left (- a + b x + c x^{2}\right )^{p + 1}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-a + b*x + c*x**2)**(p + 1)/(p + 1)

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Mathematica [A]  time = 0.0310867, size = 21, normalized size = 0.95 \[ \frac{(x (b+c x)-a)^{p+1}}{p+1} \]

Antiderivative was successfully verified.

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

[Out]

(-a + x*(b + c*x))^(1 + p)/(1 + p)

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Maple [A]  time = 0.004, size = 23, normalized size = 1.1 \[{\frac{ \left ( c{x}^{2}+bx-a \right ) ^{1+p}}{1+p}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x^2+b*x-a)^(1+p)/(1+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 - a)^p,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281159, size = 43, normalized size = 1.95 \[ \frac{{\left (c x^{2} + b x - a\right )}{\left (c x^{2} + b x - a\right )}^{p}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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((2*c*x+b)*(c*x**2+b*x-a)**p,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.266364, size = 89, normalized size = 4.05 \[ \frac{c x^{2} e^{\left (p{\rm ln}\left (c x^{2} + b x - a\right )\right )} + b x e^{\left (p{\rm ln}\left (c x^{2} + b x - a\right )\right )} - a e^{\left (p{\rm ln}\left (c x^{2} + b x - a\right )\right )}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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