3.7.3 \(\int x^m (a+b x^4) \, dx\) [603]

Optimal. Leaf size=25 \[ \frac {a x^{1+m}}{1+m}+\frac {b x^{5+m}}{5+m} \]

[Out]

a*x^(1+m)/(1+m)+b*x^(5+m)/(5+m)

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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \begin {gather*} \frac {a x^{m+1}}{m+1}+\frac {b x^{m+5}}{m+5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^4),x]

[Out]

(a*x^(1 + m))/(1 + m) + (b*x^(5 + m))/(5 + m)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {align*} \int x^m \left (a+b x^4\right ) \, dx &=\int \left (a x^m+b x^{4+m}\right ) \, dx\\ &=\frac {a x^{1+m}}{1+m}+\frac {b x^{5+m}}{5+m}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 25, normalized size = 1.00 \begin {gather*} \frac {a x^{1+m}}{1+m}+\frac {b x^{5+m}}{5+m} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^4),x]

[Out]

(a*x^(1 + m))/(1 + m) + (b*x^(5 + m))/(5 + m)

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Maple [A]
time = 0.02, size = 30, normalized size = 1.20

method	result	size
norman	\(\frac {a x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b \,x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}\)	\(30\)
risch	\(\frac {x \left (b m \,x^{4}+b \,x^{4}+a m +5 a \right ) x^{m}}{\left (5+m \right ) \left (1+m \right )}\)	\(34\)
gosper	\(\frac {x^{1+m} \left (b m \,x^{4}+b \,x^{4}+a m +5 a \right )}{\left (5+m \right ) \left (1+m \right )}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x^4+a),x,method=_RETURNVERBOSE)

[Out]

a/(1+m)*x*exp(m*ln(x))+b/(5+m)*x^5*exp(m*ln(x))

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Maxima [A]
time = 0.29, size = 25, normalized size = 1.00 \begin {gather*} \frac {b x^{m + 5}}{m + 5} + \frac {a x^{m + 1}}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^4+a),x, algorithm="maxima")

[Out]

b*x^(m + 5)/(m + 5) + a*x^(m + 1)/(m + 1)

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Fricas [A]
time = 0.38, size = 33, normalized size = 1.32 \begin {gather*} \frac {{\left ({\left (b m + b\right )} x^{5} + {\left (a m + 5 \, a\right )} x\right )} x^{m}}{m^{2} + 6 \, m + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^4+a),x, algorithm="fricas")

[Out]

((b*m + b)*x^5 + (a*m + 5*a)*x)*x^m/(m^2 + 6*m + 5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (19) = 38\).
time = 0.18, size = 94, normalized size = 3.76 \begin {gather*} \begin {cases} - \frac {a}{4 x^{4}} + b \log {\left (x \right )} & \text {for}\: m = -5 \\a \log {\left (x \right )} + \frac {b x^{4}}{4} & \text {for}\: m = -1 \\\frac {a m x x^{m}}{m^{2} + 6 m + 5} + \frac {5 a x x^{m}}{m^{2} + 6 m + 5} + \frac {b m x^{5} x^{m}}{m^{2} + 6 m + 5} + \frac {b x^{5} x^{m}}{m^{2} + 6 m + 5} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x**4+a),x)

[Out]

Piecewise((-a/(4*x**4) + b*log(x), Eq(m, -5)), (a*log(x) + b*x**4/4, Eq(m, -1)), (a*m*x*x**m/(m**2 + 6*m + 5)
+ 5*a*x*x**m/(m**2 + 6*m + 5) + b*m*x**5*x**m/(m**2 + 6*m + 5) + b*x**5*x**m/(m**2 + 6*m + 5), True))

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Giac [A]
time = 1.53, size = 43, normalized size = 1.72 \begin {gather*} \frac {b m x^{5} x^{m} + b x^{5} x^{m} + a m x x^{m} + 5 \, a x x^{m}}{m^{2} + 6 \, m + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^4+a),x, algorithm="giac")

[Out]

(b*m*x^5*x^m + b*x^5*x^m + a*m*x*x^m + 5*a*x*x^m)/(m^2 + 6*m + 5)

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Mupad [B]
time = 1.24, size = 34, normalized size = 1.36 \begin {gather*} \frac {x^{m+1}\,\left (5\,a+a\,m+b\,x^4+b\,m\,x^4\right )}{m^2+6\,m+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a + b*x^4),x)

[Out]

(x^(m + 1)*(5*a + a*m + b*x^4 + b*m*x^4))/(6*m + m^2 + 5)

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