3.5.77 \(\int (c+d x+e x^2+f x^3) (a+b x^4) \, dx\) [477]

Optimal. Leaf size=68 \[ a c x+\frac {1}{2} a d x^2+\frac {1}{3} a e x^3+\frac {1}{4} a f x^4+\frac {1}{5} b c x^5+\frac {1}{6} b d x^6+\frac {1}{7} b e x^7+\frac {1}{8} b f x^8 \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1864} \begin {gather*} a c x+\frac {1}{2} a d x^2+\frac {1}{3} a e x^3+\frac {1}{4} a f x^4+\frac {1}{5} b c x^5+\frac {1}{6} b d x^6+\frac {1}{7} b e x^7+\frac {1}{8} b f x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1864

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

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Mathematica [A]
time = 0.00, size = 68, normalized size = 1.00 \begin {gather*} a c x+\frac {1}{2} a d x^2+\frac {1}{3} a e x^3+\frac {1}{4} a f x^4+\frac {1}{5} b c x^5+\frac {1}{6} b d x^6+\frac {1}{7} b e x^7+\frac {1}{8} b f x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.12, size = 55, normalized size = 0.81

method	result	size
gosper	\(a c x +\frac {1}{2} a d \,x^{2}+\frac {1}{3} a e \,x^{3}+\frac {1}{4} a f \,x^{4}+\frac {1}{5} b c \,x^{5}+\frac {1}{6} b d \,x^{6}+\frac {1}{7} b e \,x^{7}+\frac {1}{8} b f \,x^{8}\)	\(55\)
default	\(a c x +\frac {1}{2} a d \,x^{2}+\frac {1}{3} a e \,x^{3}+\frac {1}{4} a f \,x^{4}+\frac {1}{5} b c \,x^{5}+\frac {1}{6} b d \,x^{6}+\frac {1}{7} b e \,x^{7}+\frac {1}{8} b f \,x^{8}\)	\(55\)
norman	\(a c x +\frac {1}{2} a d \,x^{2}+\frac {1}{3} a e \,x^{3}+\frac {1}{4} a f \,x^{4}+\frac {1}{5} b c \,x^{5}+\frac {1}{6} b d \,x^{6}+\frac {1}{7} b e \,x^{7}+\frac {1}{8} b f \,x^{8}\)	\(55\)
risch	\(a c x +\frac {1}{2} a d \,x^{2}+\frac {1}{3} a e \,x^{3}+\frac {1}{4} a f \,x^{4}+\frac {1}{5} b c \,x^{5}+\frac {1}{6} b d \,x^{6}+\frac {1}{7} b e \,x^{7}+\frac {1}{8} b f \,x^{8}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^3+e*x^2+d*x+c)*(b*x^4+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 56, normalized size = 0.82 \begin {gather*} \frac {1}{8} \, b f x^{8} + \frac {1}{7} \, b x^{7} e + \frac {1}{6} \, b d x^{6} + \frac {1}{5} \, b c x^{5} + \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 54, normalized size = 0.79 \begin {gather*} \frac {1}{8} \, b f x^{8} + \frac {1}{7} \, b e x^{7} + \frac {1}{6} \, b d x^{6} + \frac {1}{5} \, b c x^{5} + \frac {1}{4} \, a f x^{4} + \frac {1}{3} \, a e x^{3} + \frac {1}{2} \, a d x^{2} + a c x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.01, size = 63, normalized size = 0.93 \begin {gather*} a c x + \frac {a d x^{2}}{2} + \frac {a e x^{3}}{3} + \frac {a f x^{4}}{4} + \frac {b c x^{5}}{5} + \frac {b d x^{6}}{6} + \frac {b e x^{7}}{7} + \frac {b f x^{8}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x + a*d*x**2/2 + a*e*x**3/3 + a*f*x**4/4 + b*c*x**5/5 + b*d*x**6/6 + b*e*x**7/7 + b*f*x**8/8

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Giac [A]
time = 0.59, size = 56, normalized size = 0.82 \begin {gather*} \frac {1}{8} \, b f x^{8} + \frac {1}{7} \, b x^{7} e + \frac {1}{6} \, b d x^{6} + \frac {1}{5} \, b c x^{5} + \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.04, size = 54, normalized size = 0.79 \begin {gather*} \frac {b\,f\,x^8}{8}+\frac {b\,e\,x^7}{7}+\frac {b\,d\,x^6}{6}+\frac {b\,c\,x^5}{5}+\frac {a\,f\,x^4}{4}+\frac {a\,e\,x^3}{3}+\frac {a\,d\,x^2}{2}+a\,c\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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