3.181 \(\int \left (b+2 c x+3 d x^2\right ) \left (b x+c x^2+d x^3\right )^n \, dx\)

Optimal. Leaf size=24 \[ \frac{\left (b x+c x^2+d x^3\right )^{n+1}}{n+1} \]

[Out]

(b*x + c*x^2 + d*x^3)^(1 + n)/(1 + n)

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Rubi [A]  time = 0.0160103, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034 \[ \frac{\left (b x+c x^2+d x^3\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]  Int[(b + 2*c*x + 3*d*x^2)*(b*x + c*x^2 + d*x^3)^n,x]

[Out]

(b*x + c*x^2 + d*x^3)^(1 + n)/(1 + n)

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Rubi in Sympy [A]  time = 7.36209, size = 19, normalized size = 0.79 \[ \frac{\left (b x + c x^{2} + d x^{3}\right )^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*d*x**2+2*c*x+b)*(d*x**3+c*x**2+b*x)**n,x)

[Out]

(b*x + c*x**2 + d*x**3)**(n + 1)/(n + 1)

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Mathematica [A]  time = 0.051956, size = 21, normalized size = 0.88 \[ \frac{(x (b+x (c+d x)))^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(b + 2*c*x + 3*d*x^2)*(b*x + c*x^2 + d*x^3)^n,x]

[Out]

(x*(b + x*(c + d*x)))^(1 + n)/(1 + n)

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Maple [A]  time = 0.006, size = 34, normalized size = 1.4 \[{\frac{x \left ( d{x}^{2}+cx+b \right ) \left ( d{x}^{3}+c{x}^{2}+bx \right ) ^{n}}{1+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*d*x^2+2*c*x+b)*(d*x^3+c*x^2+b*x)^n,x)

[Out]

x*(d*x^2+c*x+b)/(1+n)*(d*x^3+c*x^2+b*x)^n

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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((3*d*x^2 + 2*c*x + b)*(d*x^3 + c*x^2 + b*x)^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.269827, size = 49, normalized size = 2.04 \[ \frac{{\left (d x^{3} + c x^{2} + b x\right )}{\left (d x^{3} + c x^{2} + b x\right )}^{n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*d*x^2 + 2*c*x + b)*(d*x^3 + c*x^2 + b*x)^n,x, algorithm="fricas")

[Out]

(d*x^3 + c*x^2 + b*x)*(d*x^3 + c*x^2 + b*x)^n/(n + 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((3*d*x**2+2*c*x+b)*(d*x**3+c*x**2+b*x)**n,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.270205, size = 100, normalized size = 4.17 \[ \frac{d x^{3} e^{\left (n{\rm ln}\left (d x^{3} + c x^{2} + b x\right )\right )} + c x^{2} e^{\left (n{\rm ln}\left (d x^{3} + c x^{2} + b x\right )\right )} + b x e^{\left (n{\rm ln}\left (d x^{3} + c x^{2} + b x\right )\right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*d*x^2 + 2*c*x + b)*(d*x^3 + c*x^2 + b*x)^n,x, algorithm="giac")

[Out]

(d*x^3*e^(n*ln(d*x^3 + c*x^2 + b*x)) + c*x^2*e^(n*ln(d*x^3 + c*x^2 + b*x)) + b*x
*e^(n*ln(d*x^3 + c*x^2 + b*x)))/(n + 1)