\(\int x^3 (a+b x) \, dx\) [43]

Optimal result

Integrand size = 9, antiderivative size = 17 \[ \int x^3 (a+b x) \, dx=\frac {a x^4}{4}+\frac {b x^5}{5} \]

[Out]

1/4*a*x^4+1/5*b*x^5

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int x^3 (a+b x) \, dx=\frac {a x^4}{4}+\frac {b x^5}{5} \]

[In]

Int[x^3*(a + b*x),x]

[Out]

(a*x^4)/4 + (b*x^5)/5

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (a x^3+b x^4\right ) \, dx \\ & = \frac {a x^4}{4}+\frac {b x^5}{5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int x^3 (a+b x) \, dx=\frac {a x^4}{4}+\frac {b x^5}{5} \]

[In]

Integrate[x^3*(a + b*x),x]

[Out]

(a*x^4)/4 + (b*x^5)/5

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
gosper	\(\frac {1}{4} a \,x^{4}+\frac {1}{5} b \,x^{5}\)	\(14\)
default	\(\frac {1}{4} a \,x^{4}+\frac {1}{5} b \,x^{5}\)	\(14\)
norman	\(\frac {1}{4} a \,x^{4}+\frac {1}{5} b \,x^{5}\)	\(14\)
risch	\(\frac {1}{4} a \,x^{4}+\frac {1}{5} b \,x^{5}\)	\(14\)
parallelrisch	\(\frac {1}{4} a \,x^{4}+\frac {1}{5} b \,x^{5}\)	\(14\)

[In]

int(x^3*(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/4*a*x^4+1/5*b*x^5

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int x^3 (a+b x) \, dx=\frac {1}{5} \, b x^{5} + \frac {1}{4} \, a x^{4} \]

[In]

integrate(x^3*(b*x+a),x, algorithm="fricas")

[Out]

1/5*b*x^5 + 1/4*a*x^4

Sympy [A] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int x^3 (a+b x) \, dx=\frac {a x^{4}}{4} + \frac {b x^{5}}{5} \]

[In]

integrate(x**3*(b*x+a),x)

[Out]

a*x**4/4 + b*x**5/5

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int x^3 (a+b x) \, dx=\frac {1}{5} \, b x^{5} + \frac {1}{4} \, a x^{4} \]

[In]

integrate(x^3*(b*x+a),x, algorithm="maxima")

[Out]

1/5*b*x^5 + 1/4*a*x^4

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int x^3 (a+b x) \, dx=\frac {1}{5} \, b x^{5} + \frac {1}{4} \, a x^{4} \]

[In]

integrate(x^3*(b*x+a),x, algorithm="giac")

[Out]

1/5*b*x^5 + 1/4*a*x^4

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int x^3 (a+b x) \, dx=\frac {x^4\,\left (5\,a+4\,b\,x\right )}{20} \]

[In]

int(x^3*(a + b*x),x)

[Out]

(x^4*(5*a + 4*b*x))/20