3.1023 \(\int (a+b x)^2 (A+B x) (d+e x)^3 \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.153629, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{b (d+e x)^6 (-2 a B e-A b e+3 b B d)}{6 e^4}+\frac{(d+e x)^5 (b d-a e) (-a B e-2 A b e+3 b B d)}{5 e^4}-\frac{(d+e x)^4 (b d-a e)^2 (B d-A e)}{4 e^4}+\frac{b^2 B (d+e x)^7}{7 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(A + B*x)*(d + e*x)^3,x]

[Out]

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

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

Mathematica [A]  time = 0.0695251, size = 216, normalized size = 1.8 \[ \frac{1}{4} x^4 \left (a^2 e^2 (A e+3 B d)+6 a b d e (A e+B d)+b^2 d^2 (3 A e+B d)\right )+\frac{1}{3} d x^3 \left (A \left (3 a^2 e^2+6 a b d e+b^2 d^2\right )+a B d (3 a e+2 b d)\right )+\frac{1}{5} e x^5 \left (a^2 B e^2+2 a b e (A e+3 B d)+3 b^2 d (A e+B d)\right )+a^2 A d^3 x+\frac{1}{2} a d^2 x^2 (3 a A e+a B d+2 A b d)+\frac{1}{6} b e^2 x^6 (2 a B e+A b e+3 b B d)+\frac{1}{7} b^2 B e^3 x^7 \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2*(A + B*x)*(d + e*x)^3,x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0., size = 237, normalized size = 2. \begin{align*}{\frac{{b}^{2}B{e}^{3}{x}^{7}}{7}}+{\frac{ \left ( \left ( A{b}^{2}+2\,Bba \right ){e}^{3}+3\,{b}^{2}Bd{e}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 2\,Aba+B{a}^{2} \right ){e}^{3}+3\, \left ( A{b}^{2}+2\,Bba \right ) d{e}^{2}+3\,{b}^{2}B{d}^{2}e \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{2}A{e}^{3}+3\, \left ( 2\,Aba+B{a}^{2} \right ) d{e}^{2}+3\, \left ( A{b}^{2}+2\,Bba \right ){d}^{2}e+{b}^{2}B{d}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{a}^{2}Ad{e}^{2}+3\, \left ( 2\,Aba+B{a}^{2} \right ){d}^{2}e+ \left ( A{b}^{2}+2\,Bba \right ){d}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{a}^{2}A{d}^{2}e+ \left ( 2\,Aba+B{a}^{2} \right ){d}^{3} \right ){x}^{2}}{2}}+{a}^{2}A{d}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(B*x+A)*(e*x+d)^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.01993, size = 319, normalized size = 2.66 \begin{align*} \frac{1}{7} \, B b^{2} e^{3} x^{7} + A a^{2} d^{3} x + \frac{1}{6} \,{\left (3 \, B b^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (3 \, B b^{2} d^{2} e + 3 \,{\left (2 \, B a b + A b^{2}\right )} d e^{2} +{\left (B a^{2} + 2 \, A a b\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{2} d^{3} + A a^{2} e^{3} + 3 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e + 3 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, A a^{2} d e^{2} +{\left (2 \, B a b + A b^{2}\right )} d^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, A a^{2} d^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d^{3}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)*(e*x+d)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.77486, size = 652, normalized size = 5.43 \begin{align*} \frac{1}{7} x^{7} e^{3} b^{2} B + \frac{1}{2} x^{6} e^{2} d b^{2} B + \frac{1}{3} x^{6} e^{3} b a B + \frac{1}{6} x^{6} e^{3} b^{2} A + \frac{3}{5} x^{5} e d^{2} b^{2} B + \frac{6}{5} x^{5} e^{2} d b a B + \frac{1}{5} x^{5} e^{3} a^{2} B + \frac{3}{5} x^{5} e^{2} d b^{2} A + \frac{2}{5} x^{5} e^{3} b a A + \frac{1}{4} x^{4} d^{3} b^{2} B + \frac{3}{2} x^{4} e d^{2} b a B + \frac{3}{4} x^{4} e^{2} d a^{2} B + \frac{3}{4} x^{4} e d^{2} b^{2} A + \frac{3}{2} x^{4} e^{2} d b a A + \frac{1}{4} x^{4} e^{3} a^{2} A + \frac{2}{3} x^{3} d^{3} b a B + x^{3} e d^{2} a^{2} B + \frac{1}{3} x^{3} d^{3} b^{2} A + 2 x^{3} e d^{2} b a A + x^{3} e^{2} d a^{2} A + \frac{1}{2} x^{2} d^{3} a^{2} B + x^{2} d^{3} b a A + \frac{3}{2} x^{2} e d^{2} a^{2} A + x d^{3} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)*(e*x+d)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.185716, size = 296, normalized size = 2.47 \begin{align*} A a^{2} d^{3} x + \frac{B b^{2} e^{3} x^{7}}{7} + x^{6} \left (\frac{A b^{2} e^{3}}{6} + \frac{B a b e^{3}}{3} + \frac{B b^{2} d e^{2}}{2}\right ) + x^{5} \left (\frac{2 A a b e^{3}}{5} + \frac{3 A b^{2} d e^{2}}{5} + \frac{B a^{2} e^{3}}{5} + \frac{6 B a b d e^{2}}{5} + \frac{3 B b^{2} d^{2} e}{5}\right ) + x^{4} \left (\frac{A a^{2} e^{3}}{4} + \frac{3 A a b d e^{2}}{2} + \frac{3 A b^{2} d^{2} e}{4} + \frac{3 B a^{2} d e^{2}}{4} + \frac{3 B a b d^{2} e}{2} + \frac{B b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{2} d e^{2} + 2 A a b d^{2} e + \frac{A b^{2} d^{3}}{3} + B a^{2} d^{2} e + \frac{2 B a b d^{3}}{3}\right ) + x^{2} \left (\frac{3 A a^{2} d^{2} e}{2} + A a b d^{3} + \frac{B a^{2} d^{3}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(B*x+A)*(e*x+d)**3,x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 2.49638, size = 379, normalized size = 3.16 \begin{align*} \frac{1}{7} \, B b^{2} x^{7} e^{3} + \frac{1}{2} \, B b^{2} d x^{6} e^{2} + \frac{3}{5} \, B b^{2} d^{2} x^{5} e + \frac{1}{4} \, B b^{2} d^{3} x^{4} + \frac{1}{3} \, B a b x^{6} e^{3} + \frac{1}{6} \, A b^{2} x^{6} e^{3} + \frac{6}{5} \, B a b d x^{5} e^{2} + \frac{3}{5} \, A b^{2} d x^{5} e^{2} + \frac{3}{2} \, B a b d^{2} x^{4} e + \frac{3}{4} \, A b^{2} d^{2} x^{4} e + \frac{2}{3} \, B a b d^{3} x^{3} + \frac{1}{3} \, A b^{2} d^{3} x^{3} + \frac{1}{5} \, B a^{2} x^{5} e^{3} + \frac{2}{5} \, A a b x^{5} e^{3} + \frac{3}{4} \, B a^{2} d x^{4} e^{2} + \frac{3}{2} \, A a b d x^{4} e^{2} + B a^{2} d^{2} x^{3} e + 2 \, A a b d^{2} x^{3} e + \frac{1}{2} \, B a^{2} d^{3} x^{2} + A a b d^{3} x^{2} + \frac{1}{4} \, A a^{2} x^{4} e^{3} + A a^{2} d x^{3} e^{2} + \frac{3}{2} \, A a^{2} d^{2} x^{2} e + A a^{2} d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)*(e*x+d)^3,x, algorithm="giac")

[Out]

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