3.1011 \(\int (a+b x) (A+B x) (d+e x)^4 \, dx\)

Optimal. Leaf size=77 \[ -\frac{(d+e x)^6 (-a B e-A b e+2 b B d)}{6 e^3}+\frac{(d+e x)^5 (b d-a e) (B d-A e)}{5 e^3}+\frac{b B (d+e x)^7}{7 e^3} \]

[Out]

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

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Rubi [A]  time = 0.142731, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ -\frac{(d+e x)^6 (-a B e-A b e+2 b B d)}{6 e^3}+\frac{(d+e x)^5 (b d-a e) (B d-A e)}{5 e^3}+\frac{b B (d+e x)^7}{7 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [B]  time = 0.060307, size = 172, normalized size = 2.23 \[ \frac{1}{3} d^2 x^3 (2 a e (3 A e+2 B d)+b d (4 A e+B d))+\frac{1}{2} d^3 x^2 (4 a A e+a B d+A b d)+\frac{1}{6} e^3 x^6 (a B e+A b e+4 b B d)+\frac{1}{5} e^2 x^5 (a e (A e+4 B d)+2 b d (2 A e+3 B d))+\frac{1}{2} d e x^4 (a e (2 A e+3 B d)+b d (3 A e+2 B d))+a A d^4 x+\frac{1}{7} b B e^4 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.001, size = 176, normalized size = 2.3 \begin{align*}{\frac{bB{e}^{4}{x}^{7}}{7}}+{\frac{ \left ( \left ( Ab+Ba \right ){e}^{4}+4\,bBd{e}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( aA{e}^{4}+4\, \left ( Ab+Ba \right ) d{e}^{3}+6\,bB{d}^{2}{e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,aAd{e}^{3}+6\, \left ( Ab+Ba \right ){d}^{2}{e}^{2}+4\,bB{d}^{3}e \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,aA{d}^{2}{e}^{2}+4\, \left ( Ab+Ba \right ){d}^{3}e+bB{d}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,aA{d}^{3}e+ \left ( Ab+Ba \right ){d}^{4} \right ){x}^{2}}{2}}+aA{d}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.15035, size = 236, normalized size = 3.06 \begin{align*} \frac{1}{7} \, B b e^{4} x^{7} + A a d^{4} x + \frac{1}{6} \,{\left (4 \, B b d e^{3} +{\left (B a + A b\right )} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (6 \, B b d^{2} e^{2} + A a e^{4} + 4 \,{\left (B a + A b\right )} d e^{3}\right )} x^{5} + \frac{1}{2} \,{\left (2 \, B b d^{3} e + 2 \, A a d e^{3} + 3 \,{\left (B a + A b\right )} d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b d^{4} + 6 \, A a d^{2} e^{2} + 4 \,{\left (B a + A b\right )} d^{3} e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, A a d^{3} e +{\left (B 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)*(B*x+A)*(e*x+d)^4,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.59138, size = 504, normalized size = 6.55 \begin{align*} \frac{1}{7} x^{7} e^{4} b B + \frac{2}{3} x^{6} e^{3} d b B + \frac{1}{6} x^{6} e^{4} a B + \frac{1}{6} x^{6} e^{4} b A + \frac{6}{5} x^{5} e^{2} d^{2} b B + \frac{4}{5} x^{5} e^{3} d a B + \frac{4}{5} x^{5} e^{3} d b A + \frac{1}{5} x^{5} e^{4} a A + x^{4} e d^{3} b B + \frac{3}{2} x^{4} e^{2} d^{2} a B + \frac{3}{2} x^{4} e^{2} d^{2} b A + x^{4} e^{3} d a A + \frac{1}{3} x^{3} d^{4} b B + \frac{4}{3} x^{3} e d^{3} a B + \frac{4}{3} x^{3} e d^{3} b A + 2 x^{3} e^{2} d^{2} a A + \frac{1}{2} x^{2} d^{4} a B + \frac{1}{2} x^{2} d^{4} b A + 2 x^{2} e d^{3} a A + x d^{4} a A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.129041, size = 226, normalized size = 2.94 \begin{align*} A a d^{4} x + \frac{B b e^{4} x^{7}}{7} + x^{6} \left (\frac{A b e^{4}}{6} + \frac{B a e^{4}}{6} + \frac{2 B b d e^{3}}{3}\right ) + x^{5} \left (\frac{A a e^{4}}{5} + \frac{4 A b d e^{3}}{5} + \frac{4 B a d e^{3}}{5} + \frac{6 B b d^{2} e^{2}}{5}\right ) + x^{4} \left (A a d e^{3} + \frac{3 A b d^{2} e^{2}}{2} + \frac{3 B a d^{2} e^{2}}{2} + B b d^{3} e\right ) + x^{3} \left (2 A a d^{2} e^{2} + \frac{4 A b d^{3} e}{3} + \frac{4 B a d^{3} e}{3} + \frac{B b d^{4}}{3}\right ) + x^{2} \left (2 A a d^{3} e + \frac{A b d^{4}}{2} + \frac{B a d^{4}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.33093, size = 281, normalized size = 3.65 \begin{align*} \frac{1}{7} \, B b x^{7} e^{4} + \frac{2}{3} \, B b d x^{6} e^{3} + \frac{6}{5} \, B b d^{2} x^{5} e^{2} + B b d^{3} x^{4} e + \frac{1}{3} \, B b d^{4} x^{3} + \frac{1}{6} \, B a x^{6} e^{4} + \frac{1}{6} \, A b x^{6} e^{4} + \frac{4}{5} \, B a d x^{5} e^{3} + \frac{4}{5} \, A b d x^{5} e^{3} + \frac{3}{2} \, B a d^{2} x^{4} e^{2} + \frac{3}{2} \, A b d^{2} x^{4} e^{2} + \frac{4}{3} \, B a d^{3} x^{3} e + \frac{4}{3} \, A b d^{3} x^{3} e + \frac{1}{2} \, B a d^{4} x^{2} + \frac{1}{2} \, A b d^{4} x^{2} + \frac{1}{5} \, A a x^{5} e^{4} + A a d x^{4} e^{3} + 2 \, A a d^{2} x^{3} e^{2} + 2 \, A a d^{3} x^{2} e + A a d^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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