3.336 \(\int (a x^m+b x^{1+6 m})^5 \, dx\)

Optimal. Leaf size=27 \[ \frac{\left (a+b x^{5 m+1}\right )^6}{6 b (5 m+1)} \]

[Out]

(a + b*x^(1 + 5*m))^6/(6*b*(1 + 5*m))

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Rubi [A]  time = 0.0102364, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1593, 261} \[ \frac{\left (a+b x^{5 m+1}\right )^6}{6 b (5 m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^m + b*x^(1 + 6*m))^5,x]

[Out]

(a + b*x^(1 + 5*m))^6/(6*b*(1 + 5*m))

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \left (a x^m+b x^{1+6 m}\right )^5 \, dx &=\int x^{5 m} \left (a+b x^{1+5 m}\right )^5 \, dx\\ &=\frac{\left (a+b x^{1+5 m}\right )^6}{6 b (1+5 m)}\\ \end{align*}

Mathematica [A]  time = 0.0133221, size = 27, normalized size = 1. \[ \frac{\left (a+b x^{5 m+1}\right )^6}{6 b (5 m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^m + b*x^(1 + 6*m))^5,x]

[Out]

(a + b*x^(1 + 5*m))^6/(6*b*(1 + 5*m))

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Maple [B]  time = 0.018, size = 126, normalized size = 4.7 \begin{align*}{\frac{{b}^{5}{x}^{6} \left ({x}^{m} \right ) ^{30}}{6+30\,m}}+{\frac{a{b}^{4}{x}^{5} \left ({x}^{m} \right ) ^{25}}{1+5\,m}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{4} \left ({x}^{m} \right ) ^{20}}{2+10\,m}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{3} \left ({x}^{m} \right ) ^{15}}{3+15\,m}}+{\frac{5\,{a}^{4}b{x}^{2} \left ({x}^{m} \right ) ^{10}}{2+10\,m}}+{\frac{{a}^{5}x \left ({x}^{m} \right ) ^{5}}{1+5\,m}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^m+b*x^(1+6*m))^5,x)

[Out]

1/6*b^5*x^6/(1+5*m)*(x^m)^30+a*b^4*x^5/(1+5*m)*(x^m)^25+5/2*a^2*b^3*x^4/(1+5*m)*(x^m)^20+10/3*a^3*b^2*x^3/(1+5
*m)*(x^m)^15+5/2*a^4*b*x^2/(1+5*m)*(x^m)^10+a^5/(1+5*m)*x*(x^m)^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^m+b*x^(1+6*m))^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.85108, size = 204, normalized size = 7.56 \begin{align*} \frac{b^{5} x^{6} x^{30 \, m} + 6 \, a b^{4} x^{5} x^{25 \, m} + 15 \, a^{2} b^{3} x^{4} x^{20 \, m} + 20 \, a^{3} b^{2} x^{3} x^{15 \, m} + 15 \, a^{4} b x^{2} x^{10 \, m} + 6 \, a^{5} x x^{5 \, m}}{6 \,{\left (5 \, m + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^m+b*x^(1+6*m))^5,x, algorithm="fricas")

[Out]

1/6*(b^5*x^6*x^(30*m) + 6*a*b^4*x^5*x^(25*m) + 15*a^2*b^3*x^4*x^(20*m) + 20*a^3*b^2*x^3*x^(15*m) + 15*a^4*b*x^
2*x^(10*m) + 6*a^5*x*x^(5*m))/(5*m + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**m+b*x**(1+6*m))**5,x)

[Out]

Timed out

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Giac [B]  time = 1.12316, size = 126, normalized size = 4.67 \begin{align*} \frac{b^{5} x^{6} x^{30 \, m} + 6 \, a b^{4} x^{5} x^{25 \, m} + 15 \, a^{2} b^{3} x^{4} x^{20 \, m} + 20 \, a^{3} b^{2} x^{3} x^{15 \, m} + 15 \, a^{4} b x^{2} x^{10 \, m} + 6 \, a^{5} x x^{5 \, m}}{6 \,{\left (5 \, m + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^m+b*x^(1+6*m))^5,x, algorithm="giac")

[Out]

1/6*(b^5*x^6*x^(30*m) + 6*a*b^4*x^5*x^(25*m) + 15*a^2*b^3*x^4*x^(20*m) + 20*a^3*b^2*x^3*x^(15*m) + 15*a^4*b*x^
2*x^(10*m) + 6*a^5*x*x^(5*m))/(5*m + 1)