3.1910 \(\int (a+b x) (d+e x) (a^2+2 a b x+b^2 x^2)^2 \, dx\)

Optimal. Leaf size=38 \[ \frac{(a+b x)^6 (b d-a e)}{6 b^2}+\frac{e (a+b x)^7}{7 b^2} \]

[Out]

((b*d - a*e)*(a + b*x)^6)/(6*b^2) + (e*(a + b*x)^7)/(7*b^2)

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Rubi [A]  time = 0.0167707, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 43} \[ \frac{(a+b x)^6 (b d-a e)}{6 b^2}+\frac{e (a+b x)^7}{7 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - a*e)*(a + b*x)^6)/(6*b^2) + (e*(a + b*x)^7)/(7*b^2)

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^5 (d+e x) \, dx\\ &=\int \left (\frac{(b d-a e) (a+b x)^5}{b}+\frac{e (a+b x)^6}{b}\right ) \, dx\\ &=\frac{(b d-a e) (a+b x)^6}{6 b^2}+\frac{e (a+b x)^7}{7 b^2}\\ \end{align*}

Mathematica [B]  time = 0.016037, size = 109, normalized size = 2.87 \[ \frac{5}{2} a^2 b^2 x^4 (a e+b d)+\frac{5}{3} a^3 b x^3 (a e+2 b d)+\frac{1}{2} a^4 x^2 (a e+5 b d)+a^5 d x+\frac{1}{6} b^4 x^6 (5 a e+b d)+a b^3 x^5 (2 a e+b d)+\frac{1}{7} b^5 e x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0., size = 172, normalized size = 4.5 \begin{align*}{\frac{{b}^{5}e{x}^{7}}{7}}+{\frac{ \left ( \left ( ae+bd \right ){b}^{4}+4\,{b}^{4}ea \right ){x}^{6}}{6}}+{\frac{ \left ( ad{b}^{4}+4\, \left ( ae+bd \right ) a{b}^{3}+6\,{b}^{3}e{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,{a}^{2}d{b}^{3}+6\, \left ( ae+bd \right ){a}^{2}{b}^{2}+4\,{b}^{2}e{a}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{a}^{3}d{b}^{2}+4\, \left ( ae+bd \right ){a}^{3}b+be{a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{a}^{4}db+ \left ( ae+bd \right ){a}^{4} \right ){x}^{2}}{2}}+{a}^{5}dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^5*e*x^7+1/6*((a*e+b*d)*b^4+4*b^4*e*a)*x^6+1/5*(a*d*b^4+4*(a*e+b*d)*a*b^3+6*b^3*e*a^2)*x^5+1/4*(4*a^2*d*b
^3+6*(a*e+b*d)*a^2*b^2+4*b^2*e*a^3)*x^4+1/3*(6*a^3*d*b^2+4*(a*e+b*d)*a^3*b+b*e*a^4)*x^3+1/2*(4*a^4*d*b+(a*e+b*
d)*a^4)*x^2+a^5*d*x

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Maxima [B]  time = 1.02191, size = 155, normalized size = 4.08 \begin{align*} \frac{1}{7} \, b^{5} e x^{7} + a^{5} d x + \frac{1}{6} \,{\left (b^{5} d + 5 \, a b^{4} e\right )} x^{6} +{\left (a b^{4} d + 2 \, a^{2} b^{3} e\right )} x^{5} + \frac{5}{2} \,{\left (a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{4} + \frac{5}{3} \,{\left (2 \, a^{3} b^{2} d + a^{4} b e\right )} x^{3} + \frac{1}{2} \,{\left (5 \, a^{4} b d + a^{5} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^5*e*x^7 + a^5*d*x + 1/6*(b^5*d + 5*a*b^4*e)*x^6 + (a*b^4*d + 2*a^2*b^3*e)*x^5 + 5/2*(a^2*b^3*d + a^3*b^2
*e)*x^4 + 5/3*(2*a^3*b^2*d + a^4*b*e)*x^3 + 1/2*(5*a^4*b*d + a^5*e)*x^2

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Fricas [B]  time = 1.31888, size = 275, normalized size = 7.24 \begin{align*} \frac{1}{7} x^{7} e b^{5} + \frac{1}{6} x^{6} d b^{5} + \frac{5}{6} x^{6} e b^{4} a + x^{5} d b^{4} a + 2 x^{5} e b^{3} a^{2} + \frac{5}{2} x^{4} d b^{3} a^{2} + \frac{5}{2} x^{4} e b^{2} a^{3} + \frac{10}{3} x^{3} d b^{2} a^{3} + \frac{5}{3} x^{3} e b a^{4} + \frac{5}{2} x^{2} d b a^{4} + \frac{1}{2} x^{2} e a^{5} + x d a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*e*b^5 + 1/6*x^6*d*b^5 + 5/6*x^6*e*b^4*a + x^5*d*b^4*a + 2*x^5*e*b^3*a^2 + 5/2*x^4*d*b^3*a^2 + 5/2*x^4*
e*b^2*a^3 + 10/3*x^3*d*b^2*a^3 + 5/3*x^3*e*b*a^4 + 5/2*x^2*d*b*a^4 + 1/2*x^2*e*a^5 + x*d*a^5

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Sympy [B]  time = 0.082625, size = 129, normalized size = 3.39 \begin{align*} a^{5} d x + \frac{b^{5} e x^{7}}{7} + x^{6} \left (\frac{5 a b^{4} e}{6} + \frac{b^{5} d}{6}\right ) + x^{5} \left (2 a^{2} b^{3} e + a b^{4} d\right ) + x^{4} \left (\frac{5 a^{3} b^{2} e}{2} + \frac{5 a^{2} b^{3} d}{2}\right ) + x^{3} \left (\frac{5 a^{4} b e}{3} + \frac{10 a^{3} b^{2} d}{3}\right ) + x^{2} \left (\frac{a^{5} e}{2} + \frac{5 a^{4} b d}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*d*x + b**5*e*x**7/7 + x**6*(5*a*b**4*e/6 + b**5*d/6) + x**5*(2*a**2*b**3*e + a*b**4*d) + x**4*(5*a**3*b**
2*e/2 + 5*a**2*b**3*d/2) + x**3*(5*a**4*b*e/3 + 10*a**3*b**2*d/3) + x**2*(a**5*e/2 + 5*a**4*b*d/2)

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Giac [B]  time = 1.12944, size = 171, normalized size = 4.5 \begin{align*} \frac{1}{7} \, b^{5} x^{7} e + \frac{1}{6} \, b^{5} d x^{6} + \frac{5}{6} \, a b^{4} x^{6} e + a b^{4} d x^{5} + 2 \, a^{2} b^{3} x^{5} e + \frac{5}{2} \, a^{2} b^{3} d x^{4} + \frac{5}{2} \, a^{3} b^{2} x^{4} e + \frac{10}{3} \, a^{3} b^{2} d x^{3} + \frac{5}{3} \, a^{4} b x^{3} e + \frac{5}{2} \, a^{4} b d x^{2} + \frac{1}{2} \, a^{5} x^{2} e + a^{5} d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^5*x^7*e + 1/6*b^5*d*x^6 + 5/6*a*b^4*x^6*e + a*b^4*d*x^5 + 2*a^2*b^3*x^5*e + 5/2*a^2*b^3*d*x^4 + 5/2*a^3*
b^2*x^4*e + 10/3*a^3*b^2*d*x^3 + 5/3*a^4*b*x^3*e + 5/2*a^4*b*d*x^2 + 1/2*a^5*x^2*e + a^5*d*x