\(\int (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3)^2 \, dx\) [4]

Optimal result

Integrand size = 29, antiderivative size = 14 \[ \int \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^2 \, dx=\frac {(a+b x)^7}{7 b} \]

[Out]

1/7*(b*x+a)^7/b

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2084, 32} \[ \int \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^2 \, dx=\frac {(a+b x)^7}{7 b} \]

[In]

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

[Out]

(a + b*x)^7/(7*b)

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]

Rule 2084

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[u^p, x] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x]
&& IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^6 \, dx \\ & = \frac {(a+b x)^7}{7 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^2 \, dx=\frac {(a+b x)^7}{7 b} \]

[In]

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

[Out]

(a + b*x)^7/(7*b)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(12)=24\).

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 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	\(\frac {1}{7} b^{6} x^{7}+a \,b^{5} x^{6}+3 a^{2} b^{4} x^{5}+5 a^{3} b^{3} x^{4}+5 a^{4} b^{2} x^{3}+3 a^{5} b \,x^{2}+a^{6} x\)	\(65\)
parallelrisch	\(\frac {1}{7} b^{6} x^{7}+a \,b^{5} x^{6}+3 a^{2} b^{4} x^{5}+5 a^{3} b^{3} x^{4}+5 a^{4} b^{2} x^{3}+3 a^{5} b \,x^{2}+a^{6} x\)	\(65\)
gosper	\(\frac {x \left (b^{6} x^{6}+7 a \,b^{5} x^{5}+21 a^{2} b^{4} x^{4}+35 a^{3} b^{3} x^{3}+35 a^{4} b^{2} x^{2}+21 a^{5} b x +7 a^{6}\right )}{7}\)	\(66\)

[In]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (12) = 24\).

Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 4.57 \[ \int \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^2 \, dx=\frac {1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{5} + 5 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{3} + 3 \, a^{5} b x^{2} + a^{6} x \]

[In]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (8) = 16\).

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 4.71 \[ \int \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^2 \, dx=a^{6} x + 3 a^{5} b x^{2} + 5 a^{4} b^{2} x^{3} + 5 a^{3} b^{3} x^{4} + 3 a^{2} b^{4} x^{5} + a b^{5} x^{6} + \frac {b^{6} x^{7}}{7} \]

[In]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (12) = 24\).

Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 7.07 \[ \int \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^2 \, dx=\frac {1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + \frac {9}{5} \, a^{2} b^{4} x^{5} + 3 \, a^{4} b^{2} x^{3} + a^{6} x + \frac {1}{2} \, {\left (b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2}\right )} a^{3} + \frac {3}{10} \, {\left (4 \, b^{3} x^{5} + 15 \, a b^{2} x^{4}\right )} a^{2} b \]

[In]

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

[Out]

1/7*b^6*x^7 + a*b^5*x^6 + 9/5*a^2*b^4*x^5 + 3*a^4*b^2*x^3 + a^6*x + 1/2*(b^3*x^4 + 4*a*b^2*x^3 + 6*a^2*b*x^2)*
a^3 + 3/10*(4*b^3*x^5 + 15*a*b^2*x^4)*a^2*b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (12) = 24\).

Time = 0.34 (sec) , antiderivative size = 64, normalized size of antiderivative = 4.57 \[ \int \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^2 \, dx=\frac {1}{7} \, b^{6} x^{7} + a b^{5} x^{6} + 3 \, a^{2} b^{4} x^{5} + 5 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{3} + 3 \, a^{5} b x^{2} + a^{6} x \]

[In]

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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 4.57 \[ \int \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^2 \, dx=a^6\,x+3\,a^5\,b\,x^2+5\,a^4\,b^2\,x^3+5\,a^3\,b^3\,x^4+3\,a^2\,b^4\,x^5+a\,b^5\,x^6+\frac {b^6\,x^7}{7} \]

[In]

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