3.15.89 \(\int (a^2+2 a b x+b^2 x^2)^3 \, dx\) [1489]

Optimal. Leaf size=14 \[ \frac {(a+b x)^7}{7 b} \]

[Out]

1/7*(b*x+a)^7/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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 32} \begin {gather*} \frac {(a+b x)^7}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(a + b*x)^7/(7*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 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 \, dx\\ &=\frac {(a+b x)^7}{7 b}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {(a+b x)^7}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(a + b*x)^7/(7*b)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(12)=24\).
time = 0.64, size = 65, normalized size = 4.64

method	result	size
default	\(\frac {1}{7} b^{6} x^{7}+a \,b^{5} x^{6}+3 a^{2} b^{4} x^{5}+5 a^{3} b^{3} x^{4}+5 a^{4} b^{2} x^{3}+3 a^{5} b \,x^{2}+a^{6} x\)	\(65\)
norman	\(\frac {1}{7} b^{6} x^{7}+a \,b^{5} x^{6}+3 a^{2} b^{4} x^{5}+5 a^{3} b^{3} x^{4}+5 a^{4} b^{2} x^{3}+3 a^{5} b \,x^{2}+a^{6} x\)	\(65\)
risch	\(\frac {1}{7} b^{6} x^{7}+a \,b^{5} x^{6}+3 a^{2} b^{4} x^{5}+5 a^{3} b^{3} x^{4}+5 a^{4} b^{2} x^{3}+3 a^{5} b \,x^{2}+a^{6} x\)	\(65\)
gosper	\(\frac {x \left (b^{6} x^{6}+7 a \,b^{5} x^{5}+21 a^{2} b^{4} x^{4}+35 a^{3} b^{3} x^{3}+35 a^{4} b^{2} x^{2}+21 a^{5} b x +7 a^{6}\right )}{7}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/7*b^6*x^7+a*b^5*x^6+3*a^2*b^4*x^5+5*a^3*b^3*x^4+5*a^4*b^2*x^3+3*a^5*b*x^2+a^6*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (12) = 24\).
time = 0.28, size = 97, normalized size = 6.93 \begin {gather*} \frac {1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + \frac {12}{5} \, a^{2} b^{4} x^{5} + 2 \, a^{3} b^{3} x^{4} + a^{6} x + {\left (b^{2} x^{3} + 3 \, a b x^{2}\right )} a^{4} + \frac {1}{5} \, {\left (3 \, b^{4} x^{5} + 15 \, a b^{3} x^{4} + 20 \, a^{2} b^{2} x^{3}\right )} a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/7*b^6*x^7 + a*b^5*x^6 + 12/5*a^2*b^4*x^5 + 2*a^3*b^3*x^4 + a^6*x + (b^2*x^3 + 3*a*b*x^2)*a^4 + 1/5*(3*b^4*x^
5 + 15*a*b^3*x^4 + 20*a^2*b^2*x^3)*a^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (12) = 24\).
time = 2.36, size = 64, normalized size = 4.57 \begin {gather*} \frac {1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{5} + 5 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{3} + 3 \, a^{5} b x^{2} + a^{6} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/7*b^6*x^7 + a*b^5*x^6 + 3*a^2*b^4*x^5 + 5*a^3*b^3*x^4 + 5*a^4*b^2*x^3 + 3*a^5*b*x^2 + a^6*x

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (8) = 16\).
time = 0.01, size = 66, normalized size = 4.71 \begin {gather*} a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac {b^{6} x^{7}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**6*x + 3*a**5*b*x**2 + 5*a**4*b**2*x**3 + 5*a**3*b**3*x**4 + 3*a**2*b**4*x**5 + a*b**5*x**6 + b**6*x**7/7

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (12) = 24\).
time = 1.53, size = 64, normalized size = 4.57 \begin {gather*} \frac {1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{5} + 5 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{3} + 3 \, a^{5} b x^{2} + a^{6} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/7*b^6*x^7 + a*b^5*x^6 + 3*a^2*b^4*x^5 + 5*a^3*b^3*x^4 + 5*a^4*b^2*x^3 + 3*a^5*b*x^2 + a^6*x

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Mupad [B]
time = 0.03, size = 64, normalized size = 4.57 \begin {gather*} a^6\,x+3\,a^5\,b\,x^2+5\,a^4\,b^2\,x^3+5\,a^3\,b^3\,x^4+3\,a^2\,b^4\,x^5+a\,b^5\,x^6+\frac {b^6\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

a^6*x + (b^6*x^7)/7 + 3*a^5*b*x^2 + a*b^5*x^6 + 5*a^4*b^2*x^3 + 5*a^3*b^3*x^4 + 3*a^2*b^4*x^5

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