3.20.11 \(\int (a+b x) (a^2+2 a b x+b^2 x^2)^2 \, dx\) [1911]

Optimal. Leaf size=14 \[ \frac {(a+b x)^6}{6 b} \]

[Out]

1/6*(b*x+a)^6/b

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 32} \begin {gather*} \frac {(a+b x)^6}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(a + b*x)^6/(6*b)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 \, dx\\ &=\frac {(a+b x)^6}{6 b}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {(a+b x)^6}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(a + b*x)^6/(6*b)

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Maple [A]
time = 0.86, size = 24, normalized size = 1.71

method	result	size
default	\(\frac {\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}{6 b}\)	\(24\)
norman	\(\frac {1}{6} b^{5} x^{6}+a \,b^{4} x^{5}+\frac {5}{2} a^{2} b^{3} x^{4}+\frac {10}{3} a^{3} b^{2} x^{3}+\frac {5}{2} a^{4} b \,x^{2}+a^{5} x\)	\(54\)
gosper	\(\frac {x \left (b^{5} x^{5}+6 a \,b^{4} x^{4}+15 a^{2} b^{3} x^{3}+20 a^{3} x^{2} b^{2}+15 a^{4} b x +6 a^{5}\right )}{6}\)	\(55\)
risch	\(\frac {b^{5} x^{6}}{6}+a \,b^{4} x^{5}+\frac {5 a^{2} b^{3} x^{4}}{2}+\frac {10 a^{3} b^{2} x^{3}}{3}+\frac {5 a^{4} b \,x^{2}}{2}+a^{5} x +\frac {a^{6}}{6 b}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/6*(b^2*x^2+2*a*b*x+a^2)^3/b

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Maxima [A]
time = 0.27, size = 23, normalized size = 1.64 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{3}}{6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")

[Out]

1/6*(b^2*x^2 + 2*a*b*x + a^2)^3/b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (12) = 24\).
time = 1.85, size = 53, normalized size = 3.79 \begin {gather*} \frac {1}{6} \, b^{5} x^{6} + a b^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} x^{3} + \frac {5}{2} \, a^{4} b x^{2} + a^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")

[Out]

1/6*b^5*x^6 + a*b^4*x^5 + 5/2*a^2*b^3*x^4 + 10/3*a^3*b^2*x^3 + 5/2*a^4*b*x^2 + a^5*x

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (8) = 16\).
time = 0.01, size = 60, normalized size = 4.29 \begin {gather*} a^{5} x + \frac {5 a^{4} b x^{2}}{2} + \frac {10 a^{3} b^{2} x^{3}}{3} + \frac {5 a^{2} b^{3} x^{4}}{2} + a b^{4} x^{5} + \frac {b^{5} x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

a**5*x + 5*a**4*b*x**2/2 + 10*a**3*b**2*x**3/3 + 5*a**2*b**3*x**4/2 + a*b**4*x**5 + b**5*x**6/6

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (12) = 24\).
time = 1.78, size = 51, normalized size = 3.64 \begin {gather*} \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} a^{4} + \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )}^{2} a^{2} b + \frac {1}{6} \, {\left (b x^{2} + 2 \, a x\right )}^{3} b^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")

[Out]

1/2*(b*x^2 + 2*a*x)*a^4 + 1/2*(b*x^2 + 2*a*x)^2*a^2*b + 1/6*(b*x^2 + 2*a*x)^3*b^2

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Mupad [B]
time = 0.03, size = 53, normalized size = 3.79 \begin {gather*} a^5\,x+\frac {5\,a^4\,b\,x^2}{2}+\frac {10\,a^3\,b^2\,x^3}{3}+\frac {5\,a^2\,b^3\,x^4}{2}+a\,b^4\,x^5+\frac {b^5\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)

[Out]

a^5*x + (b^5*x^6)/6 + (5*a^4*b*x^2)/2 + a*b^4*x^5 + (10*a^3*b^2*x^3)/3 + (5*a^2*b^3*x^4)/2

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