3.1103 \(\int (A+B x) (d+e x)^4 (b x+c x^2) \, dx\)

Optimal. Leaf size=118 \[ -\frac{(d+e x)^7 (-A c e-b B e+3 B c d)}{7 e^4}+\frac{(d+e x)^6 (B d (3 c d-2 b e)-A e (2 c d-b e))}{6 e^4}-\frac{d (d+e x)^5 (B d-A e) (c d-b e)}{5 e^4}+\frac{B c (d+e x)^8}{8 e^4} \]

[Out]

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

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Rubi [A]  time = 0.175077, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {771} \[ -\frac{(d+e x)^7 (-A c e-b B e+3 B c d)}{7 e^4}+\frac{(d+e x)^6 (B d (3 c d-2 b e)-A e (2 c d-b e))}{6 e^4}-\frac{d (d+e x)^5 (B d-A e) (c d-b e)}{5 e^4}+\frac{B c (d+e x)^8}{8 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*(B*d - A*e)*(c*d - b*e)*(d + e*x)^5)/(5*e^4) + ((B*d*(3*c*d - 2*b*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 (b x+c x^2\right ) \, dx &=\int \left (-\frac{d (B d-A e) (c d-b e) (d+e x)^4}{e^3}+\frac{(B d (3 c d-2 b e)-A e (2 c d-b e)) (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{d (B d-A e) (c d-b e) (d+e x)^5}{5 e^4}+\frac{(B d (3 c d-2 b e)-A e (2 c d-b e)) (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 = 0.0647882, size = 177, normalized size = 1.5 \[ \frac{1}{4} d^2 x^4 (2 A e (3 b e+2 c d)+B d (4 b e+c d))+\frac{1}{3} d^3 x^3 (4 A b e+A c d+b B d)+\frac{1}{7} e^3 x^7 (A c e+b B e+4 B c d)+\frac{1}{6} e^2 x^6 (A e (b e+4 c d)+2 B d (2 b e+3 c d))+\frac{2}{5} d e x^5 (A e (2 b e+3 c d)+B d (3 b e+2 c d))+\frac{1}{2} A b d^4 x^2+\frac{1}{8} B c e^4 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b*d^4*x^2)/2 + (d^3*(b*B*d + A*c*d + 4*A*b*e)*x^3)/3 + (d^2*(2*A*e*(2*c*d + 3*b*e) + B*d*(c*d + 4*b*e))*x^4
)/4 + (2*d*e*(A*e*(3*c*d + 2*b*e) + B*d*(2*c*d + 3*b*e))*x^5)/5 + (e^2*(A*e*(4*c*d + b*e) + 2*B*d*(3*c*d + 2*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 [A]  time = 0.001, size = 200, normalized size = 1.7 \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 \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 \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 \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{4}c+ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) b \right ){x}^{3}}{3}}+{\frac{A{d}^{4}b{x}^{2}}{2}} \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),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)*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)*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)*x^4+1/3*(A*d^4*c+(4*A*d^3*e+B*d^4)*b)*x^3+1/2*A*d^4*b*x^2

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Maxima [A]  time = 0.995372, size = 240, normalized size = 2.03 \begin{align*} \frac{1}{8} \, B c e^{4} x^{8} + \frac{1}{2} \, A b d^{4} x^{2} + \frac{1}{7} \,{\left (4 \, B c d e^{3} +{\left (B b + A c\right )} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (6 \, B c d^{2} e^{2} + A b e^{4} + 4 \,{\left (B b + A c\right )} d e^{3}\right )} x^{6} + \frac{2}{5} \,{\left (2 \, B c d^{3} e + 2 \, A b d e^{3} + 3 \,{\left (B b + A c\right )} d^{2} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B c d^{4} + 6 \, A b d^{2} e^{2} + 4 \,{\left (B b + A c\right )} d^{3} e\right )} x^{4} + \frac{1}{3} \,{\left (4 \, A b d^{3} e +{\left (B b + A c\right )} d^{4}\right )} x^{3} \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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.24413, size = 512, normalized size = 4.34 \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{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{6}{5} x^{5} e^{2} d^{2} c A + \frac{4}{5} x^{5} e^{3} d b A + \frac{1}{4} x^{4} d^{4} c B + x^{4} e d^{3} b B + x^{4} e d^{3} c A + \frac{3}{2} x^{4} e^{2} d^{2} b A + \frac{1}{3} x^{3} d^{4} b B + \frac{1}{3} x^{3} d^{4} c A + \frac{4}{3} x^{3} e d^{3} b A + \frac{1}{2} x^{2} d^{4} b 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),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
+ 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 + 6/5*x^5*e^2*d^2*c*A + 4/5*x^
5*e^3*d*b*A + 1/4*x^4*d^4*c*B + x^4*e*d^3*b*B + x^4*e*d^3*c*A + 3/2*x^4*e^2*d^2*b*A + 1/3*x^3*d^4*b*B + 1/3*x^
3*d^4*c*A + 4/3*x^3*e*d^3*b*A + 1/2*x^2*d^4*b*A

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Sympy [B]  time = 0.107564, size = 230, normalized size = 1.95 \begin{align*} \frac{A b d^{4} x^{2}}{2} + \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{2 B b d e^{3}}{3} + B c d^{2} e^{2}\right ) + x^{5} \left (\frac{4 A b d e^{3}}{5} + \frac{6 A c d^{2} e^{2}}{5} + \frac{6 B b d^{2} e^{2}}{5} + \frac{4 B c d^{3} e}{5}\right ) + x^{4} \left (\frac{3 A b d^{2} e^{2}}{2} + A c d^{3} e + B b d^{3} e + \frac{B c d^{4}}{4}\right ) + x^{3} \left (\frac{4 A b d^{3} e}{3} + \frac{A c d^{4}}{3} + \frac{B b d^{4}}{3}\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),x)

[Out]

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

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Giac [A]  time = 1.27051, size = 284, normalized size = 2.41 \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} \, A b x^{6} e^{4} + \frac{4}{5} \, A b d x^{5} e^{3} + \frac{3}{2} \, A b d^{2} x^{4} e^{2} + \frac{4}{3} \, A b d^{3} x^{3} e + \frac{1}{2} \, A b d^{4} 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),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*A*b*x^6*e^4 + 4/5*A*b*d*x^5*e^3 + 3/2*A*b*d^
2*x^4*e^2 + 4/3*A*b*d^3*x^3*e + 1/2*A*b*d^4*x^2