3.1660 \(\int (A+B x) (d+e x)^4 (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=120 \[ -\frac{b (d+e x)^7 (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac{(d+e x)^6 (b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4}-\frac{(d+e x)^5 (b d-a e)^2 (B d-A e)}{5 e^4}+\frac{b^2 B (d+e x)^8}{8 e^4} \]

[Out]

-((b*d - a*e)^2*(B*d - A*e)*(d + e*x)^5)/(5*e^4) + ((b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^6)/(6*e^
4) - (b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^7)/(7*e^4) + (b^2*B*(d + e*x)^8)/(8*e^4)

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Rubi [A]  time = 0.213899, antiderivative size = 120, 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} \[ -\frac{b (d+e x)^7 (-2 a B e-A b e+3 b B d)}{7 e^4}+\frac{(d+e x)^6 (b d-a e) (-a B e-2 A b e+3 b B d)}{6 e^4}-\frac{(d+e x)^5 (b d-a e)^2 (B d-A e)}{5 e^4}+\frac{b^2 B (d+e x)^8}{8 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

-((b*d - a*e)^2*(B*d - A*e)*(d + e*x)^5)/(5*e^4) + ((b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^6)/(6*e^
4) - (b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^7)/(7*e^4) + (b^2*B*(d + e*x)^8)/(8*e^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)^4 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (A+B x) (d+e x)^4 \, dx\\ &=\int \left (\frac{(-b d+a e)^2 (-B d+A e) (d+e x)^4}{e^3}+\frac{(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^5}{e^3}+\frac{b (-3 b B d+A b e+2 a B e) (d+e x)^6}{e^3}+\frac{b^2 B (d+e x)^7}{e^3}\right ) \, dx\\ &=-\frac{(b d-a e)^2 (B d-A e) (d+e x)^5}{5 e^4}+\frac{(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^6}{6 e^4}-\frac{b (3 b B d-A b e-2 a B e) (d+e x)^7}{7 e^4}+\frac{b^2 B (d+e x)^8}{8 e^4}\\ \end{align*}

Mathematica [B]  time = 0.0987672, size = 283, normalized size = 2.36 \[ \frac{1}{5} e x^5 \left (a^2 e^2 (A e+4 B d)+4 a b d e (2 A e+3 B d)+2 b^2 d^2 (3 A e+2 B d)\right )+\frac{1}{4} d x^4 \left (2 a^2 e^2 (2 A e+3 B d)+4 a b d e (3 A e+2 B d)+b^2 d^2 (4 A e+B d)\right )+\frac{1}{3} d^2 x^3 \left (A \left (6 a^2 e^2+8 a b d e+b^2 d^2\right )+2 a B d (2 a e+b d)\right )+\frac{1}{6} e^2 x^6 \left (a^2 B e^2+2 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )+a^2 A d^4 x+\frac{1}{2} a d^3 x^2 (4 a A e+a B d+2 A b d)+\frac{1}{7} b e^3 x^7 (2 a B e+A b e+4 b B d)+\frac{1}{8} b^2 B e^4 x^8 \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

a^2*A*d^4*x + (a*d^3*(2*A*b*d + a*B*d + 4*a*A*e)*x^2)/2 + (d^2*(2*a*B*d*(b*d + 2*a*e) + A*(b^2*d^2 + 8*a*b*d*e
+ 6*a^2*e^2))*x^3)/3 + (d*(2*a^2*e^2*(3*B*d + 2*A*e) + 4*a*b*d*e*(2*B*d + 3*A*e) + b^2*d^2*(B*d + 4*A*e))*x^4
)/4 + (e*(a^2*e^2*(4*B*d + A*e) + 4*a*b*d*e*(3*B*d + 2*A*e) + 2*b^2*d^2*(2*B*d + 3*A*e))*x^5)/5 + (e^2*(a^2*B*
e^2 + 2*a*b*e*(4*B*d + A*e) + 2*b^2*d*(3*B*d + 2*A*e))*x^6)/6 + (b*e^3*(4*b*B*d + A*b*e + 2*a*B*e)*x^7)/7 + (b
^2*B*e^4*x^8)/8

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Maple [B]  time = 0.002, size = 319, normalized size = 2.7 \begin{align*}{\frac{B{e}^{4}{b}^{2}{x}^{8}}{8}}+{\frac{ \left ( \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){b}^{2}+2\,B{e}^{4}ab \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){b}^{2}+2\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) ab+B{a}^{2}{e}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){b}^{2}+2\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) ab+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){b}^{2}+2\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) ab+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{4}{b}^{2}+2\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) ab+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,A{d}^{4}ab+ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){a}^{2} \right ){x}^{2}}{2}}+A{d}^{4}{a}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

1/8*B*e^4*b^2*x^8+1/7*((A*e^4+4*B*d*e^3)*b^2+2*B*e^4*a*b)*x^7+1/6*((4*A*d*e^3+6*B*d^2*e^2)*b^2+2*(A*e^4+4*B*d*
e^3)*a*b+B*a^2*e^4)*x^6+1/5*((6*A*d^2*e^2+4*B*d^3*e)*b^2+2*(4*A*d*e^3+6*B*d^2*e^2)*a*b+(A*e^4+4*B*d*e^3)*a^2)*
x^5+1/4*((4*A*d^3*e+B*d^4)*b^2+2*(6*A*d^2*e^2+4*B*d^3*e)*a*b+(4*A*d*e^3+6*B*d^2*e^2)*a^2)*x^4+1/3*(A*d^4*b^2+2
*(4*A*d^3*e+B*d^4)*a*b+(6*A*d^2*e^2+4*B*d^3*e)*a^2)*x^3+1/2*(2*A*d^4*a*b+(4*A*d^3*e+B*d^4)*a^2)*x^2+A*d^4*a^2*
x

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Maxima [B]  time = 1.37971, size = 410, normalized size = 3.42 \begin{align*} \frac{1}{8} \, B b^{2} e^{4} x^{8} + A a^{2} d^{4} x + \frac{1}{7} \,{\left (4 \, B b^{2} d e^{3} +{\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (6 \, B b^{2} d^{2} e^{2} + 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{3} +{\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (4 \, B b^{2} d^{3} e + A a^{2} e^{4} + 6 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{2} d^{4} + 4 \, A a^{2} d e^{3} + 4 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e + 6 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, A a^{2} d^{2} e^{2} +{\left (2 \, B a b + A b^{2}\right )} d^{4} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, A a^{2} d^{3} e +{\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

1/8*B*b^2*e^4*x^8 + A*a^2*d^4*x + 1/7*(4*B*b^2*d*e^3 + (2*B*a*b + A*b^2)*e^4)*x^7 + 1/6*(6*B*b^2*d^2*e^2 + 4*(
2*B*a*b + A*b^2)*d*e^3 + (B*a^2 + 2*A*a*b)*e^4)*x^6 + 1/5*(4*B*b^2*d^3*e + A*a^2*e^4 + 6*(2*B*a*b + A*b^2)*d^2
*e^2 + 4*(B*a^2 + 2*A*a*b)*d*e^3)*x^5 + 1/4*(B*b^2*d^4 + 4*A*a^2*d*e^3 + 4*(2*B*a*b + A*b^2)*d^3*e + 6*(B*a^2
+ 2*A*a*b)*d^2*e^2)*x^4 + 1/3*(6*A*a^2*d^2*e^2 + (2*B*a*b + A*b^2)*d^4 + 4*(B*a^2 + 2*A*a*b)*d^3*e)*x^3 + 1/2*
(4*A*a^2*d^3*e + (B*a^2 + 2*A*a*b)*d^4)*x^2

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Fricas [B]  time = 1.29567, size = 834, normalized size = 6.95 \begin{align*} \frac{1}{8} x^{8} e^{4} b^{2} B + \frac{4}{7} x^{7} e^{3} d b^{2} B + \frac{2}{7} x^{7} e^{4} b a B + \frac{1}{7} x^{7} e^{4} b^{2} A + x^{6} e^{2} d^{2} b^{2} B + \frac{4}{3} x^{6} e^{3} d b a B + \frac{1}{6} x^{6} e^{4} a^{2} B + \frac{2}{3} x^{6} e^{3} d b^{2} A + \frac{1}{3} x^{6} e^{4} b a A + \frac{4}{5} x^{5} e d^{3} b^{2} B + \frac{12}{5} x^{5} e^{2} d^{2} b a B + \frac{4}{5} x^{5} e^{3} d a^{2} B + \frac{6}{5} x^{5} e^{2} d^{2} b^{2} A + \frac{8}{5} x^{5} e^{3} d b a A + \frac{1}{5} x^{5} e^{4} a^{2} A + \frac{1}{4} x^{4} d^{4} b^{2} B + 2 x^{4} e d^{3} b a B + \frac{3}{2} x^{4} e^{2} d^{2} a^{2} B + x^{4} e d^{3} b^{2} A + 3 x^{4} e^{2} d^{2} b a A + x^{4} e^{3} d a^{2} A + \frac{2}{3} x^{3} d^{4} b a B + \frac{4}{3} x^{3} e d^{3} a^{2} B + \frac{1}{3} x^{3} d^{4} b^{2} A + \frac{8}{3} x^{3} e d^{3} b a A + 2 x^{3} e^{2} d^{2} a^{2} A + \frac{1}{2} x^{2} d^{4} a^{2} B + x^{2} d^{4} b a A + 2 x^{2} e d^{3} a^{2} A + x d^{4} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

1/8*x^8*e^4*b^2*B + 4/7*x^7*e^3*d*b^2*B + 2/7*x^7*e^4*b*a*B + 1/7*x^7*e^4*b^2*A + x^6*e^2*d^2*b^2*B + 4/3*x^6*
e^3*d*b*a*B + 1/6*x^6*e^4*a^2*B + 2/3*x^6*e^3*d*b^2*A + 1/3*x^6*e^4*b*a*A + 4/5*x^5*e*d^3*b^2*B + 12/5*x^5*e^2
*d^2*b*a*B + 4/5*x^5*e^3*d*a^2*B + 6/5*x^5*e^2*d^2*b^2*A + 8/5*x^5*e^3*d*b*a*A + 1/5*x^5*e^4*a^2*A + 1/4*x^4*d
^4*b^2*B + 2*x^4*e*d^3*b*a*B + 3/2*x^4*e^2*d^2*a^2*B + x^4*e*d^3*b^2*A + 3*x^4*e^2*d^2*b*a*A + x^4*e^3*d*a^2*A
+ 2/3*x^3*d^4*b*a*B + 4/3*x^3*e*d^3*a^2*B + 1/3*x^3*d^4*b^2*A + 8/3*x^3*e*d^3*b*a*A + 2*x^3*e^2*d^2*a^2*A + 1
/2*x^2*d^4*a^2*B + x^2*d^4*b*a*A + 2*x^2*e*d^3*a^2*A + x*d^4*a^2*A

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Sympy [B]  time = 0.112377, size = 384, normalized size = 3.2 \begin{align*} A a^{2} d^{4} x + \frac{B b^{2} e^{4} x^{8}}{8} + x^{7} \left (\frac{A b^{2} e^{4}}{7} + \frac{2 B a b e^{4}}{7} + \frac{4 B b^{2} d e^{3}}{7}\right ) + x^{6} \left (\frac{A a b e^{4}}{3} + \frac{2 A b^{2} d e^{3}}{3} + \frac{B a^{2} e^{4}}{6} + \frac{4 B a b d e^{3}}{3} + B b^{2} d^{2} e^{2}\right ) + x^{5} \left (\frac{A a^{2} e^{4}}{5} + \frac{8 A a b d e^{3}}{5} + \frac{6 A b^{2} d^{2} e^{2}}{5} + \frac{4 B a^{2} d e^{3}}{5} + \frac{12 B a b d^{2} e^{2}}{5} + \frac{4 B b^{2} d^{3} e}{5}\right ) + x^{4} \left (A a^{2} d e^{3} + 3 A a b d^{2} e^{2} + A b^{2} d^{3} e + \frac{3 B a^{2} d^{2} e^{2}}{2} + 2 B a b d^{3} e + \frac{B b^{2} d^{4}}{4}\right ) + x^{3} \left (2 A a^{2} d^{2} e^{2} + \frac{8 A a b d^{3} e}{3} + \frac{A b^{2} d^{4}}{3} + \frac{4 B a^{2} d^{3} e}{3} + \frac{2 B a b d^{4}}{3}\right ) + x^{2} \left (2 A a^{2} d^{3} e + A a b d^{4} + \frac{B a^{2} d^{4}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

A*a**2*d**4*x + B*b**2*e**4*x**8/8 + x**7*(A*b**2*e**4/7 + 2*B*a*b*e**4/7 + 4*B*b**2*d*e**3/7) + x**6*(A*a*b*e
**4/3 + 2*A*b**2*d*e**3/3 + B*a**2*e**4/6 + 4*B*a*b*d*e**3/3 + B*b**2*d**2*e**2) + x**5*(A*a**2*e**4/5 + 8*A*a
*b*d*e**3/5 + 6*A*b**2*d**2*e**2/5 + 4*B*a**2*d*e**3/5 + 12*B*a*b*d**2*e**2/5 + 4*B*b**2*d**3*e/5) + x**4*(A*a
**2*d*e**3 + 3*A*a*b*d**2*e**2 + A*b**2*d**3*e + 3*B*a**2*d**2*e**2/2 + 2*B*a*b*d**3*e + B*b**2*d**4/4) + x**3
*(2*A*a**2*d**2*e**2 + 8*A*a*b*d**3*e/3 + A*b**2*d**4/3 + 4*B*a**2*d**3*e/3 + 2*B*a*b*d**4/3) + x**2*(2*A*a**2
*d**3*e + A*a*b*d**4 + B*a**2*d**4/2)

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Giac [B]  time = 1.18285, size = 489, normalized size = 4.08 \begin{align*} \frac{1}{8} \, B b^{2} x^{8} e^{4} + \frac{4}{7} \, B b^{2} d x^{7} e^{3} + B b^{2} d^{2} x^{6} e^{2} + \frac{4}{5} \, B b^{2} d^{3} x^{5} e + \frac{1}{4} \, B b^{2} d^{4} x^{4} + \frac{2}{7} \, B a b x^{7} e^{4} + \frac{1}{7} \, A b^{2} x^{7} e^{4} + \frac{4}{3} \, B a b d x^{6} e^{3} + \frac{2}{3} \, A b^{2} d x^{6} e^{3} + \frac{12}{5} \, B a b d^{2} x^{5} e^{2} + \frac{6}{5} \, A b^{2} d^{2} x^{5} e^{2} + 2 \, B a b d^{3} x^{4} e + A b^{2} d^{3} x^{4} e + \frac{2}{3} \, B a b d^{4} x^{3} + \frac{1}{3} \, A b^{2} d^{4} x^{3} + \frac{1}{6} \, B a^{2} x^{6} e^{4} + \frac{1}{3} \, A a b x^{6} e^{4} + \frac{4}{5} \, B a^{2} d x^{5} e^{3} + \frac{8}{5} \, A a b d x^{5} e^{3} + \frac{3}{2} \, B a^{2} d^{2} x^{4} e^{2} + 3 \, A a b d^{2} x^{4} e^{2} + \frac{4}{3} \, B a^{2} d^{3} x^{3} e + \frac{8}{3} \, A a b d^{3} x^{3} e + \frac{1}{2} \, B a^{2} d^{4} x^{2} + A a b d^{4} x^{2} + \frac{1}{5} \, A a^{2} x^{5} e^{4} + A a^{2} d x^{4} e^{3} + 2 \, A a^{2} d^{2} x^{3} e^{2} + 2 \, A a^{2} d^{3} x^{2} e + A a^{2} d^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

1/8*B*b^2*x^8*e^4 + 4/7*B*b^2*d*x^7*e^3 + B*b^2*d^2*x^6*e^2 + 4/5*B*b^2*d^3*x^5*e + 1/4*B*b^2*d^4*x^4 + 2/7*B*
a*b*x^7*e^4 + 1/7*A*b^2*x^7*e^4 + 4/3*B*a*b*d*x^6*e^3 + 2/3*A*b^2*d*x^6*e^3 + 12/5*B*a*b*d^2*x^5*e^2 + 6/5*A*b
^2*d^2*x^5*e^2 + 2*B*a*b*d^3*x^4*e + A*b^2*d^3*x^4*e + 2/3*B*a*b*d^4*x^3 + 1/3*A*b^2*d^4*x^3 + 1/6*B*a^2*x^6*e
^4 + 1/3*A*a*b*x^6*e^4 + 4/5*B*a^2*d*x^5*e^3 + 8/5*A*a*b*d*x^5*e^3 + 3/2*B*a^2*d^2*x^4*e^2 + 3*A*a*b*d^2*x^4*e
^2 + 4/3*B*a^2*d^3*x^3*e + 8/3*A*a*b*d^3*x^3*e + 1/2*B*a^2*d^4*x^2 + A*a*b*d^4*x^2 + 1/5*A*a^2*x^5*e^4 + A*a^2
*d*x^4*e^3 + 2*A*a^2*d^2*x^3*e^2 + 2*A*a^2*d^3*x^2*e + A*a^2*d^4*x