3.1429 \(\int (b d+2 c d x) \left (a+b x+c x^2\right )^p \, dx\)

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

[Out]

(d*(a + b*x + c*x^2)^(1 + p))/(1 + p)

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Rubi [A]  time = 0.0152456, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{d \left (a+b x+c x^2\right )^{p+1}}{p+1} \]

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)

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Rubi in Sympy [A]  time = 5.40311, size = 17, normalized size = 0.81 \[ \frac{d \left (a + b x + c x^{2}\right )^{p + 1}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(a + b*x + c*x**2)**(p + 1)/(p + 1)

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Mathematica [A]  time = 0.0283691, size = 20, normalized size = 0.95 \[ \frac{d (a+x (b+c x))^{p+1}}{p+1} \]

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 = 0.005, size = 22, normalized size = 1.1 \[{\frac{d \left ( c{x}^{2}+bx+a \right ) ^{1+p}}{1+p}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.265021, size = 43, normalized size = 2.05 \[ \frac{{\left (c d x^{2} + b d x + a d\right )}{\left (c x^{2} + b x + a\right )}^{p}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215773, size = 84, normalized size = 4. \[ \frac{c d x^{2} e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} + b d x e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )} + a d e^{\left (p{\rm ln}\left (c x^{2} + b x + a\right )\right )}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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