3.217 \(\int \left (b x+c x^2\right ) \left (1+\left (\frac{b x^2}{2}+\frac{c x^3}{3}\right )^n\right ) \, dx\)

Optimal. Leaf size=44 \[ \frac{\left (\frac{b x^2}{2}+\frac{c x^3}{3}\right )^{n+1}}{n+1}+\frac{b x^2}{2}+\frac{c x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0206251, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032 \[ \frac{\left (\frac{b x^2}{2}+\frac{c x^3}{3}\right )^{n+1}}{n+1}+\frac{b x^2}{2}+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)*(1 + ((b*x^2)/2 + (c*x^3)/3)^n),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.32383, size = 32, normalized size = 0.73 \[ \frac{b x^{2}}{2} + \frac{c x^{3}}{3} + \frac{\left (\frac{b x^{2}}{2} + \frac{c x^{3}}{3}\right )^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)*(1+(1/2*b*x**2+1/3*c*x**3)**n),x)

[Out]

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

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Mathematica [A]  time = 0.0371577, size = 42, normalized size = 0.95 \[ \frac{x^2 (3 b+2 c x) \left (\left (\frac{b x^2}{2}+\frac{c x^3}{3}\right )^n+n+1\right )}{6 (n+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)*(1 + ((b*x^2)/2 + (c*x^3)/3)^n),x]

[Out]

(x^2*(3*b + 2*c*x)*(1 + n + ((b*x^2)/2 + (c*x^3)/3)^n))/(6*(1 + n))

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Maple [A]  time = 0.004, size = 37, normalized size = 0.8 \[{\frac{b{x}^{2}}{2}}+{\frac{c{x}^{3}}{3}}+{\frac{1}{1+n} \left ({\frac{b{x}^{2}}{2}}+{\frac{c{x}^{3}}{3}} \right ) ^{1+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)*(1+(1/2*b*x^2+1/3*c*x^3)^n),x)

[Out]

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

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Maxima [A]  time = 0.992911, size = 96, normalized size = 2.18 \[ \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + \frac{{\left (2 \, c x^{3} + 3 \, b x^{2}\right )} e^{\left (n \log \left (2 \, c x + 3 \, b\right ) + 2 \, n \log \left (x\right )\right )}}{3^{n + 1} 2^{n + 1} n + 3^{n + 1} 2^{n + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*((1/3*c*x^3 + 1/2*b*x^2)^n + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.279799, size = 77, normalized size = 1.75 \[ \frac{2 \,{\left (c n + c\right )} x^{3} + 3 \,{\left (b n + b\right )} x^{2} +{\left (2 \, c x^{3} + 3 \, b x^{2}\right )}{\left (\frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2}\right )}^{n}}{6 \,{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*((1/3*c*x^3 + 1/2*b*x^2)^n + 1),x, algorithm="fricas")

[Out]

1/6*(2*(c*n + c)*x^3 + 3*(b*n + b)*x^2 + (2*c*x^3 + 3*b*x^2)*(1/3*c*x^3 + 1/2*b*
x^2)^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((c*x**2+b*x)*(1+(1/2*b*x**2+1/3*c*x**3)**n),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.260366, size = 49, normalized size = 1.11 \[ \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + \frac{{\left (\frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2}\right )}^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*((1/3*c*x^3 + 1/2*b*x^2)^n + 1),x, algorithm="giac")

[Out]

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