3.2306 \(\int (A+B x) (d+e x)^4 (a+b x+c x^2) \, dx\)

Optimal. Leaf size=134 \[ -\frac{(d+e x)^6 \left (A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )\right )}{6 e^4}-\frac{(d+e x)^5 (B d-A e) \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac{(d+e x)^7 (-A c e-b B e+3 B c d)}{7 e^4}+\frac{B c (d+e x)^8}{8 e^4} \]

[Out]

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

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Rubi [A]  time = 0.26251, antiderivative size = 133, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {771} \[ \frac{(d+e x)^6 \left (-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2\right )}{6 e^4}-\frac{(d+e x)^5 (B d-A e) \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac{(d+e x)^7 (-A c e-b B e+3 B c d)}{7 e^4}+\frac{B c (d+e x)^8}{8 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.147215, size = 251, normalized size = 1.87 \[ \frac{1}{6} e^2 x^6 \left (B e (a e+4 b d)+A e (b e+4 c d)+6 B c d^2\right )+\frac{1}{4} d x^4 \left (2 A e \left (e (2 a e+3 b d)+2 c d^2\right )+B d \left (6 a e^2+4 b d e+c d^2\right )\right )+\frac{1}{3} d^2 x^3 \left (6 a A e^2+4 a B d e+b d (4 A e+B d)+A c d^2\right )+\frac{1}{5} e x^5 \left (A e \left (e (a e+4 b d)+6 c d^2\right )+B \left (2 d e (2 a e+3 b d)+4 c d^3\right )\right )+\frac{1}{2} d^3 x^2 (4 a A e+a B d+A b d)+a A d^4 x+\frac{1}{7} e^3 x^7 (A c e+b B e+4 B c d)+\frac{1}{8} B c e^4 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.001, size = 283, normalized size = 2.1 \begin{align*}{\frac{B{e}^{4}c{x}^{8}}{8}}+{\frac{ \left ( \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) c+B{e}^{4}b \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) c+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) b+B{e}^{4}a \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) c+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) b+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) a \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) c+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) b+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) a \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{4}c+ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) b+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) a \right ){x}^{3}}{3}}+{\frac{ \left ( A{d}^{4}b+ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) a \right ){x}^{2}}{2}}+A{d}^{4}ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.40543, size = 329, normalized size = 2.46 \begin{align*} \frac{1}{8} \, B c e^{4} x^{8} + \frac{1}{7} \,{\left (4 \, B c d e^{3} +{\left (B b + A c\right )} e^{4}\right )} x^{7} + A a d^{4} x + \frac{1}{6} \,{\left (6 \, B c d^{2} e^{2} + 4 \,{\left (B b + A c\right )} d e^{3} +{\left (B a + A b\right )} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (4 \, B c d^{3} e + A a e^{4} + 6 \,{\left (B b + A c\right )} d^{2} e^{2} + 4 \,{\left (B a + A b\right )} d e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (B c d^{4} + 4 \, A a d e^{3} + 4 \,{\left (B b + A c\right )} d^{3} e + 6 \,{\left (B a + A b\right )} d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, A a d^{2} e^{2} +{\left (B b + A c\right )} d^{4} + 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)*(e*x+d)^4*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

1/8*B*c*e^4*x^8 + 1/7*(4*B*c*d*e^3 + (B*b + A*c)*e^4)*x^7 + A*a*d^4*x + 1/6*(6*B*c*d^2*e^2 + 4*(B*b + A*c)*d*e
^3 + (B*a + A*b)*e^4)*x^6 + 1/5*(4*B*c*d^3*e + A*a*e^4 + 6*(B*b + A*c)*d^2*e^2 + 4*(B*a + A*b)*d*e^3)*x^5 + 1/
4*(B*c*d^4 + 4*A*a*d*e^3 + 4*(B*b + A*c)*d^3*e + 6*(B*a + A*b)*d^2*e^2)*x^4 + 1/3*(6*A*a*d^2*e^2 + (B*b + A*c)
*d^4 + 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 = 1.07127, size = 757, normalized size = 5.65 \begin{align*} \frac{1}{8} x^{8} e^{4} c B + \frac{4}{7} x^{7} e^{3} d c B + \frac{1}{7} x^{7} e^{4} b B + \frac{1}{7} x^{7} e^{4} c A + x^{6} e^{2} d^{2} c B + \frac{2}{3} x^{6} e^{3} d b B + \frac{1}{6} x^{6} e^{4} a B + \frac{2}{3} x^{6} e^{3} d c A + \frac{1}{6} x^{6} e^{4} b A + \frac{4}{5} x^{5} e d^{3} c B + \frac{6}{5} x^{5} e^{2} d^{2} b B + \frac{4}{5} x^{5} e^{3} d a B + \frac{6}{5} x^{5} e^{2} d^{2} c A + \frac{4}{5} x^{5} e^{3} d b A + \frac{1}{5} x^{5} e^{4} a A + \frac{1}{4} x^{4} d^{4} c B + x^{4} e d^{3} b B + \frac{3}{2} x^{4} e^{2} d^{2} a B + x^{4} e d^{3} c A + \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{1}{3} x^{3} d^{4} c A + \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)*(e*x+d)^4*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/8*x^8*e^4*c*B + 4/7*x^7*e^3*d*c*B + 1/7*x^7*e^4*b*B + 1/7*x^7*e^4*c*A + x^6*e^2*d^2*c*B + 2/3*x^6*e^3*d*b*B
+ 1/6*x^6*e^4*a*B + 2/3*x^6*e^3*d*c*A + 1/6*x^6*e^4*b*A + 4/5*x^5*e*d^3*c*B + 6/5*x^5*e^2*d^2*b*B + 4/5*x^5*e^
3*d*a*B + 6/5*x^5*e^2*d^2*c*A + 4/5*x^5*e^3*d*b*A + 1/5*x^5*e^4*a*A + 1/4*x^4*d^4*c*B + x^4*e*d^3*b*B + 3/2*x^
4*e^2*d^2*a*B + x^4*e*d^3*c*A + 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 + 1/
3*x^3*d^4*c*A + 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 = 0.102038, size = 332, normalized size = 2.48 \begin{align*} A a d^{4} x + \frac{B c e^{4} x^{8}}{8} + x^{7} \left (\frac{A c e^{4}}{7} + \frac{B b e^{4}}{7} + \frac{4 B c d e^{3}}{7}\right ) + x^{6} \left (\frac{A b e^{4}}{6} + \frac{2 A c d e^{3}}{3} + \frac{B a e^{4}}{6} + \frac{2 B b d e^{3}}{3} + B c d^{2} e^{2}\right ) + x^{5} \left (\frac{A a e^{4}}{5} + \frac{4 A b d e^{3}}{5} + \frac{6 A c d^{2} e^{2}}{5} + \frac{4 B a d e^{3}}{5} + \frac{6 B b d^{2} e^{2}}{5} + \frac{4 B c d^{3} e}{5}\right ) + x^{4} \left (A a d e^{3} + \frac{3 A b d^{2} e^{2}}{2} + A c d^{3} e + \frac{3 B a d^{2} e^{2}}{2} + B b d^{3} e + \frac{B c d^{4}}{4}\right ) + x^{3} \left (2 A a d^{2} e^{2} + \frac{4 A b d^{3} e}{3} + \frac{A c d^{4}}{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)*(e*x+d)**4*(c*x**2+b*x+a),x)

[Out]

A*a*d**4*x + B*c*e**4*x**8/8 + x**7*(A*c*e**4/7 + B*b*e**4/7 + 4*B*c*d*e**3/7) + x**6*(A*b*e**4/6 + 2*A*c*d*e*
*3/3 + B*a*e**4/6 + 2*B*b*d*e**3/3 + B*c*d**2*e**2) + x**5*(A*a*e**4/5 + 4*A*b*d*e**3/5 + 6*A*c*d**2*e**2/5 +
4*B*a*d*e**3/5 + 6*B*b*d**2*e**2/5 + 4*B*c*d**3*e/5) + x**4*(A*a*d*e**3 + 3*A*b*d**2*e**2/2 + A*c*d**3*e + 3*B
*a*d**2*e**2/2 + B*b*d**3*e + B*c*d**4/4) + x**3*(2*A*a*d**2*e**2 + 4*A*b*d**3*e/3 + A*c*d**4/3 + 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 = 1.14815, size = 421, normalized size = 3.14 \begin{align*} \frac{1}{8} \, B c x^{8} e^{4} + \frac{4}{7} \, B c d x^{7} e^{3} + B c d^{2} x^{6} e^{2} + \frac{4}{5} \, B c d^{3} x^{5} e + \frac{1}{4} \, B c d^{4} x^{4} + \frac{1}{7} \, B b x^{7} e^{4} + \frac{1}{7} \, A c x^{7} e^{4} + \frac{2}{3} \, B b d x^{6} e^{3} + \frac{2}{3} \, A c d x^{6} e^{3} + \frac{6}{5} \, B b d^{2} x^{5} e^{2} + \frac{6}{5} \, A c d^{2} x^{5} e^{2} + B b d^{3} x^{4} e + A c d^{3} x^{4} e + \frac{1}{3} \, B b d^{4} x^{3} + \frac{1}{3} \, A c 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)*(e*x+d)^4*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

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