3.377 \(\int (a+b x^3) (c+d x+e x^2+f x^3+g x^4+h x^5) \, dx\)

Optimal. Leaf size=92 \[ \frac{1}{4} x^4 (a f+b c)+\frac{1}{5} x^5 (a g+b d)+\frac{1}{6} x^6 (a h+b e)+a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{7} b f x^7+\frac{1}{8} b g x^8+\frac{1}{9} b h x^9 \]

[Out]

a*c*x + (a*d*x^2)/2 + (a*e*x^3)/3 + ((b*c + a*f)*x^4)/4 + ((b*d + a*g)*x^5)/5 + ((b*e + a*h)*x^6)/6 + (b*f*x^7
)/7 + (b*g*x^8)/8 + (b*h*x^9)/9

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Rubi [A]  time = 0.0734568, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {1850} \[ \frac{1}{4} x^4 (a f+b c)+\frac{1}{5} x^5 (a g+b d)+\frac{1}{6} x^6 (a h+b e)+a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{7} b f x^7+\frac{1}{8} b g x^8+\frac{1}{9} b h x^9 \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]

[Out]

a*c*x + (a*d*x^2)/2 + (a*e*x^3)/3 + ((b*c + a*f)*x^4)/4 + ((b*d + a*g)*x^5)/5 + ((b*e + a*h)*x^6)/6 + (b*f*x^7
)/7 + (b*g*x^8)/8 + (b*h*x^9)/9

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \left (a+b x^3\right ) \left (c+d x+e x^2+f x^3+g x^4+h x^5\right ) \, dx &=\int \left (a c+a d x+a e x^2+(b c+a f) x^3+(b d+a g) x^4+(b e+a h) x^5+b f x^6+b g x^7+b h x^8\right ) \, dx\\ &=a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{4} (b c+a f) x^4+\frac{1}{5} (b d+a g) x^5+\frac{1}{6} (b e+a h) x^6+\frac{1}{7} b f x^7+\frac{1}{8} b g x^8+\frac{1}{9} b h x^9\\ \end{align*}

Mathematica [A]  time = 0.0132528, size = 92, normalized size = 1. \[ \frac{1}{4} x^4 (a f+b c)+\frac{1}{5} x^5 (a g+b d)+\frac{1}{6} x^6 (a h+b e)+a c x+\frac{1}{2} a d x^2+\frac{1}{3} a e x^3+\frac{1}{7} b f x^7+\frac{1}{8} b g x^8+\frac{1}{9} b h x^9 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5),x]

[Out]

a*c*x + (a*d*x^2)/2 + (a*e*x^3)/3 + ((b*c + a*f)*x^4)/4 + ((b*d + a*g)*x^5)/5 + ((b*e + a*h)*x^6)/6 + (b*f*x^7
)/7 + (b*g*x^8)/8 + (b*h*x^9)/9

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Maple [A]  time = 0., size = 77, normalized size = 0.8 \begin{align*} acx+{\frac{ad{x}^{2}}{2}}+{\frac{ae{x}^{3}}{3}}+{\frac{ \left ( af+bc \right ){x}^{4}}{4}}+{\frac{ \left ( ag+bd \right ){x}^{5}}{5}}+{\frac{ \left ( ah+be \right ){x}^{6}}{6}}+{\frac{bf{x}^{7}}{7}}+{\frac{bg{x}^{8}}{8}}+{\frac{bh{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x)

[Out]

a*c*x+1/2*a*d*x^2+1/3*a*e*x^3+1/4*(a*f+b*c)*x^4+1/5*(a*g+b*d)*x^5+1/6*(a*h+b*e)*x^6+1/7*b*f*x^7+1/8*b*g*x^8+1/
9*b*h*x^9

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Maxima [A]  time = 0.933405, size = 103, normalized size = 1.12 \begin{align*} \frac{1}{9} \, b h x^{9} + \frac{1}{8} \, b g x^{8} + \frac{1}{7} \, b f x^{7} + \frac{1}{6} \,{\left (b e + a h\right )} x^{6} + \frac{1}{5} \,{\left (b d + a g\right )} x^{5} + \frac{1}{3} \, a e x^{3} + \frac{1}{4} \,{\left (b c + a f\right )} x^{4} + \frac{1}{2} \, a d x^{2} + a c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="maxima")

[Out]

1/9*b*h*x^9 + 1/8*b*g*x^8 + 1/7*b*f*x^7 + 1/6*(b*e + a*h)*x^6 + 1/5*(b*d + a*g)*x^5 + 1/3*a*e*x^3 + 1/4*(b*c +
a*f)*x^4 + 1/2*a*d*x^2 + a*c*x

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Fricas [A]  time = 0.846603, size = 217, normalized size = 2.36 \begin{align*} \frac{1}{9} x^{9} h b + \frac{1}{8} x^{8} g b + \frac{1}{7} x^{7} f b + \frac{1}{6} x^{6} e b + \frac{1}{6} x^{6} h a + \frac{1}{5} x^{5} d b + \frac{1}{5} x^{5} g a + \frac{1}{4} x^{4} c b + \frac{1}{4} x^{4} f a + \frac{1}{3} x^{3} e a + \frac{1}{2} x^{2} d a + x c a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="fricas")

[Out]

1/9*x^9*h*b + 1/8*x^8*g*b + 1/7*x^7*f*b + 1/6*x^6*e*b + 1/6*x^6*h*a + 1/5*x^5*d*b + 1/5*x^5*g*a + 1/4*x^4*c*b
+ 1/4*x^4*f*a + 1/3*x^3*e*a + 1/2*x^2*d*a + x*c*a

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Sympy [A]  time = 0.069543, size = 87, normalized size = 0.95 \begin{align*} a c x + \frac{a d x^{2}}{2} + \frac{a e x^{3}}{3} + \frac{b f x^{7}}{7} + \frac{b g x^{8}}{8} + \frac{b h x^{9}}{9} + x^{6} \left (\frac{a h}{6} + \frac{b e}{6}\right ) + x^{5} \left (\frac{a g}{5} + \frac{b d}{5}\right ) + x^{4} \left (\frac{a f}{4} + \frac{b c}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c),x)

[Out]

a*c*x + a*d*x**2/2 + a*e*x**3/3 + b*f*x**7/7 + b*g*x**8/8 + b*h*x**9/9 + x**6*(a*h/6 + b*e/6) + x**5*(a*g/5 +
b*d/5) + x**4*(a*f/4 + b*c/4)

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Giac [A]  time = 1.06718, size = 113, normalized size = 1.23 \begin{align*} \frac{1}{9} \, b h x^{9} + \frac{1}{8} \, b g x^{8} + \frac{1}{7} \, b f x^{7} + \frac{1}{6} \, a h x^{6} + \frac{1}{6} \, b x^{6} e + \frac{1}{5} \, b d x^{5} + \frac{1}{5} \, a g x^{5} + \frac{1}{4} \, b c x^{4} + \frac{1}{4} \, a f x^{4} + \frac{1}{3} \, a x^{3} e + \frac{1}{2} \, a d x^{2} + a c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c),x, algorithm="giac")

[Out]

1/9*b*h*x^9 + 1/8*b*g*x^8 + 1/7*b*f*x^7 + 1/6*a*h*x^6 + 1/6*b*x^6*e + 1/5*b*d*x^5 + 1/5*a*g*x^5 + 1/4*b*c*x^4
+ 1/4*a*f*x^4 + 1/3*a*x^3*e + 1/2*a*d*x^2 + a*c*x