3.1662 \(\int (A+B x) (d+e x)^2 (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=118 \[ \frac{e (a+b x)^5 (-3 a B e+A b e+2 b B d)}{5 b^4}+\frac{(a+b x)^4 (b d-a e) (-3 a B e+2 A b e+b B d)}{4 b^4}+\frac{(a+b x)^3 (A b-a B) (b d-a e)^2}{3 b^4}+\frac{B e^2 (a+b x)^6}{6 b^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.129242, antiderivative size = 118, 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, 77} \[ \frac{e (a+b x)^5 (-3 a B e+A b e+2 b B d)}{5 b^4}+\frac{(a+b x)^4 (b d-a e) (-3 a B e+2 A b e+b B d)}{4 b^4}+\frac{(a+b x)^3 (A b-a B) (b d-a e)^2}{3 b^4}+\frac{B e^2 (a+b x)^6}{6 b^4} \]

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]

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

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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

Mathematica [A]  time = 0.0536563, size = 157, normalized size = 1.33 \[ \frac{1}{3} x^3 \left (A \left (a^2 e^2+4 a b d e+b^2 d^2\right )+2 a B d (a e+b d)\right )+\frac{1}{4} x^4 \left (a^2 B e^2+2 a b e (A e+2 B d)+b^2 d (2 A e+B d)\right )+a^2 A d^2 x+\frac{1}{5} b e x^5 (2 a B e+A b e+2 b B d)+\frac{1}{2} a d x^2 (2 A (a e+b d)+a B d)+\frac{1}{6} b^2 B e^2 x^6 \]

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

________________________________________________________________________________________

Maple [A]  time = 0., size = 169, normalized size = 1.4 \begin{align*}{\frac{B{e}^{2}{b}^{2}{x}^{6}}{6}}+{\frac{ \left ( \left ( A{e}^{2}+2\,Bde \right ){b}^{2}+2\,B{e}^{2}ab \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ){b}^{2}+2\, \left ( A{e}^{2}+2\,Bde \right ) ab+{a}^{2}B{e}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{2}{b}^{2}+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{d}^{2}ab+ \left ( 2\,Ade+B{d}^{2} \right ){a}^{2} \right ){x}^{2}}{2}}+A{d}^{2}{a}^{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*e^2*b^2*x^6+1/5*((A*e^2+2*B*d*e)*b^2+2*B*e^2*a*b)*x^5+1/4*((2*A*d*e+B*d^2)*b^2+2*(A*e^2+2*B*d*e)*a*b+a^2
*B*e^2)*x^4+1/3*(A*d^2*b^2+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*d^2*a*b+(2*A*d*e+B*d^2)*a^2
)*x^2+A*d^2*a^2*x

________________________________________________________________________________________

Maxima [A]  time = 1.19813, size = 227, normalized size = 1.92 \begin{align*} \frac{1}{6} \, B b^{2} e^{2} x^{6} + A a^{2} d^{2} x + \frac{1}{5} \,{\left (2 \, B b^{2} d e +{\left (2 \, B a b + A b^{2}\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{2} d^{2} + 2 \,{\left (2 \, B a b + A b^{2}\right )} d e +{\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (A a^{2} e^{2} +{\left (2 \, B a b + A b^{2}\right )} d^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a^{2} d e +{\left (B a^{2} + 2 \, A a b\right )} d^{2}\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*b^2*e^2*x^6 + A*a^2*d^2*x + 1/5*(2*B*b^2*d*e + (2*B*a*b + A*b^2)*e^2)*x^5 + 1/4*(B*b^2*d^2 + 2*(2*B*a*b
+ A*b^2)*d*e + (B*a^2 + 2*A*a*b)*e^2)*x^4 + 1/3*(A*a^2*e^2 + (2*B*a*b + A*b^2)*d^2 + 2*(B*a^2 + 2*A*a*b)*d*e)*
x^3 + 1/2*(2*A*a^2*d*e + (B*a^2 + 2*A*a*b)*d^2)*x^2

________________________________________________________________________________________

Fricas [A]  time = 1.32969, size = 460, normalized size = 3.9 \begin{align*} \frac{1}{6} x^{6} e^{2} b^{2} B + \frac{2}{5} x^{5} e d b^{2} B + \frac{2}{5} x^{5} e^{2} b a B + \frac{1}{5} x^{5} e^{2} b^{2} A + \frac{1}{4} x^{4} d^{2} b^{2} B + x^{4} e d b a B + \frac{1}{4} x^{4} e^{2} a^{2} B + \frac{1}{2} x^{4} e d b^{2} A + \frac{1}{2} x^{4} e^{2} b a A + \frac{2}{3} x^{3} d^{2} b a B + \frac{2}{3} x^{3} e d a^{2} B + \frac{1}{3} x^{3} d^{2} b^{2} A + \frac{4}{3} x^{3} e d b a A + \frac{1}{3} x^{3} e^{2} a^{2} A + \frac{1}{2} x^{2} d^{2} a^{2} B + x^{2} d^{2} b a A + x^{2} e d a^{2} A + x d^{2} a^{2} A \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^2*B + 2/5*x^5*e*d*b^2*B + 2/5*x^5*e^2*b*a*B + 1/5*x^5*e^2*b^2*A + 1/4*x^4*d^2*b^2*B + x^4*e*d*b*
a*B + 1/4*x^4*e^2*a^2*B + 1/2*x^4*e*d*b^2*A + 1/2*x^4*e^2*b*a*A + 2/3*x^3*d^2*b*a*B + 2/3*x^3*e*d*a^2*B + 1/3*
x^3*d^2*b^2*A + 4/3*x^3*e*d*b*a*A + 1/3*x^3*e^2*a^2*A + 1/2*x^2*d^2*a^2*B + x^2*d^2*b*a*A + x^2*e*d*a^2*A + x*
d^2*a^2*A

________________________________________________________________________________________

Sympy [A]  time = 0.090615, size = 202, normalized size = 1.71 \begin{align*} A a^{2} d^{2} x + \frac{B b^{2} e^{2} x^{6}}{6} + x^{5} \left (\frac{A b^{2} e^{2}}{5} + \frac{2 B a b e^{2}}{5} + \frac{2 B b^{2} d e}{5}\right ) + x^{4} \left (\frac{A a b e^{2}}{2} + \frac{A b^{2} d e}{2} + \frac{B a^{2} e^{2}}{4} + B a b d e + \frac{B b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac{A a^{2} e^{2}}{3} + \frac{4 A a b d e}{3} + \frac{A b^{2} d^{2}}{3} + \frac{2 B a^{2} d e}{3} + \frac{2 B a b d^{2}}{3}\right ) + x^{2} \left (A a^{2} d e + A a b d^{2} + \frac{B a^{2} 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*a**2*d**2*x + B*b**2*e**2*x**6/6 + x**5*(A*b**2*e**2/5 + 2*B*a*b*e**2/5 + 2*B*b**2*d*e/5) + x**4*(A*a*b*e**2
/2 + A*b**2*d*e/2 + B*a**2*e**2/4 + B*a*b*d*e + B*b**2*d**2/4) + x**3*(A*a**2*e**2/3 + 4*A*a*b*d*e/3 + A*b**2*
d**2/3 + 2*B*a**2*d*e/3 + 2*B*a*b*d**2/3) + x**2*(A*a**2*d*e + A*a*b*d**2 + B*a**2*d**2/2)

________________________________________________________________________________________

Giac [A]  time = 1.15587, size = 269, normalized size = 2.28 \begin{align*} \frac{1}{6} \, B b^{2} x^{6} e^{2} + \frac{2}{5} \, B b^{2} d x^{5} e + \frac{1}{4} \, B b^{2} d^{2} x^{4} + \frac{2}{5} \, B a b x^{5} e^{2} + \frac{1}{5} \, A b^{2} x^{5} e^{2} + B a b d x^{4} e + \frac{1}{2} \, A b^{2} d x^{4} e + \frac{2}{3} \, B a b d^{2} x^{3} + \frac{1}{3} \, A b^{2} d^{2} x^{3} + \frac{1}{4} \, B a^{2} x^{4} e^{2} + \frac{1}{2} \, A a b x^{4} e^{2} + \frac{2}{3} \, B a^{2} d x^{3} e + \frac{4}{3} \, A a b d x^{3} e + \frac{1}{2} \, B a^{2} d^{2} x^{2} + A a b d^{2} x^{2} + \frac{1}{3} \, A a^{2} x^{3} e^{2} + A a^{2} d x^{2} e + A a^{2} 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*b^2*x^6*e^2 + 2/5*B*b^2*d*x^5*e + 1/4*B*b^2*d^2*x^4 + 2/5*B*a*b*x^5*e^2 + 1/5*A*b^2*x^5*e^2 + B*a*b*d*x^
4*e + 1/2*A*b^2*d*x^4*e + 2/3*B*a*b*d^2*x^3 + 1/3*A*b^2*d^2*x^3 + 1/4*B*a^2*x^4*e^2 + 1/2*A*a*b*x^4*e^2 + 2/3*
B*a^2*d*x^3*e + 4/3*A*a*b*d*x^3*e + 1/2*B*a^2*d^2*x^2 + A*a*b*d^2*x^2 + 1/3*A*a^2*x^3*e^2 + A*a^2*d*x^2*e + A*
a^2*d^2*x