3.1898 \(\int (a+b x) (d+e x)^2 (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=65 \[ \frac{2 e (a+b x)^5 (b d-a e)}{5 b^3}+\frac{(a+b x)^4 (b d-a e)^2}{4 b^3}+\frac{e^2 (a+b x)^6}{6 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0666271, antiderivative size = 65, 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} \[ \frac{2 e (a+b x)^5 (b d-a e)}{5 b^3}+\frac{(a+b x)^4 (b d-a e)^2}{4 b^3}+\frac{e^2 (a+b x)^6}{6 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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)^2 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 (d+e x)^2 \, dx\\ &=\int \left (\frac{(b d-a e)^2 (a+b x)^3}{b^2}+\frac{2 e (b d-a e) (a+b x)^4}{b^2}+\frac{e^2 (a+b x)^5}{b^2}\right ) \, dx\\ &=\frac{(b d-a e)^2 (a+b x)^4}{4 b^3}+\frac{2 e (b d-a e) (a+b x)^5}{5 b^3}+\frac{e^2 (a+b x)^6}{6 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0197151, size = 122, normalized size = 1.88 \[ \frac{1}{4} b x^4 \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )+\frac{1}{3} a x^3 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )+\frac{1}{2} a^2 d x^2 (2 a e+3 b d)+a^3 d^2 x+\frac{1}{5} b^2 e x^5 (3 a e+2 b d)+\frac{1}{6} b^3 e^2 x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.001, size = 169, normalized size = 2.6 \begin{align*}{\frac{{b}^{3}{e}^{2}{x}^{6}}{6}}+{\frac{ \left ( \left ( a{e}^{2}+2\,bde \right ){b}^{2}+2\,{b}^{2}{e}^{2}a \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 2\,ade+b{d}^{2} \right ){b}^{2}+2\, \left ( a{e}^{2}+2\,bde \right ) ab+b{e}^{2}{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{2}{d}^{2}a+2\, \left ( 2\,ade+b{d}^{2} \right ) ab+ \left ( a{e}^{2}+2\,bde \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}{d}^{2}b+ \left ( 2\,ade+b{d}^{2} \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{3}{d}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.95951, size = 167, normalized size = 2.57 \begin{align*} \frac{1}{6} \, b^{3} e^{2} x^{6} + a^{3} d^{2} x + \frac{1}{5} \,{\left (2 \, b^{3} d e + 3 \, a b^{2} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} d^{2} + 6 \, a b^{2} d e + 3 \, a^{2} b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a b^{2} d^{2} + 6 \, a^{2} b d e + a^{3} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d^{2} + 2 \, a^{3} d e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.26756, size = 285, normalized size = 4.38 \begin{align*} \frac{1}{6} x^{6} e^{2} b^{3} + \frac{2}{5} x^{5} e d b^{3} + \frac{3}{5} x^{5} e^{2} b^{2} a + \frac{1}{4} x^{4} d^{2} b^{3} + \frac{3}{2} x^{4} e d b^{2} a + \frac{3}{4} x^{4} e^{2} b a^{2} + x^{3} d^{2} b^{2} a + 2 x^{3} e d b a^{2} + \frac{1}{3} x^{3} e^{2} a^{3} + \frac{3}{2} x^{2} d^{2} b a^{2} + x^{2} e d a^{3} + x d^{2} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.080874, size = 133, normalized size = 2.05 \begin{align*} a^{3} d^{2} x + \frac{b^{3} e^{2} x^{6}}{6} + x^{5} \left (\frac{3 a b^{2} e^{2}}{5} + \frac{2 b^{3} d e}{5}\right ) + x^{4} \left (\frac{3 a^{2} b e^{2}}{4} + \frac{3 a b^{2} d e}{2} + \frac{b^{3} d^{2}}{4}\right ) + x^{3} \left (\frac{a^{3} e^{2}}{3} + 2 a^{2} b d e + a b^{2} d^{2}\right ) + x^{2} \left (a^{3} d e + \frac{3 a^{2} b d^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.12962, size = 176, normalized size = 2.71 \begin{align*} \frac{1}{6} \, b^{3} x^{6} e^{2} + \frac{2}{5} \, b^{3} d x^{5} e + \frac{1}{4} \, b^{3} d^{2} x^{4} + \frac{3}{5} \, a b^{2} x^{5} e^{2} + \frac{3}{2} \, a b^{2} d x^{4} e + a b^{2} d^{2} x^{3} + \frac{3}{4} \, a^{2} b x^{4} e^{2} + 2 \, a^{2} b d x^{3} e + \frac{3}{2} \, a^{2} b d^{2} x^{2} + \frac{1}{3} \, a^{3} x^{3} e^{2} + a^{3} d x^{2} e + a^{3} d^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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