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

Optimal. Leaf size=22 \[ \frac{\left (a+c x^2+d x^3\right )^{n+1}}{n+1} \]

[Out]

(a + c*x^2 + d*x^3)^(1 + n)/(1 + n)

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Rubi [A]  time = 0.0148789, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{\left (a+c x^2+d x^3\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

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

[Out]

(a + c*x^2 + d*x^3)^(1 + n)/(1 + n)

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Rubi in Sympy [A]  time = 7.3025, size = 17, normalized size = 0.77 \[ \frac{\left (a + c x^{2} + d x^{3}\right )^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.034506, size = 21, normalized size = 0.95 \[ \frac{\left (a+x^2 (c+d x)\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

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

[Out]

(a + x^2*(c + d*x))^(1 + n)/(1 + n)

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Maple [A]  time = 0.004, size = 23, normalized size = 1.1 \[{\frac{ \left ( d{x}^{3}+c{x}^{2}+a \right ) ^{1+n}}{1+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x^3+c*x^2+a)^(1+n)/(1+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)*(d*x^3 + c*x^2 + a)^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270943, size = 43, normalized size = 1.95 \[ \frac{{\left (d x^{3} + c x^{2} + a\right )}{\left (d x^{3} + c x^{2} + a\right )}^{n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275085, size = 90, normalized size = 4.09 \[ \frac{d x^{3} e^{\left (n{\rm ln}\left (d x^{3} + c x^{2} + a\right )\right )} + c x^{2} e^{\left (n{\rm ln}\left (d x^{3} + c x^{2} + a\right )\right )} + a e^{\left (n{\rm ln}\left (d x^{3} + c x^{2} + a\right )\right )}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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