3.1116 \(\int (A+B x) (d+e x)^2 (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=162 \[ \frac{1}{6} x^6 \left (2 A c e (b e+c d)+B \left (b^2 e^2+4 b c d e+c^2 d^2\right )\right )+\frac{1}{5} x^5 \left (b^2 e (A e+2 B d)+2 b c d (2 A e+B d)+A c^2 d^2\right )+\frac{1}{3} A b^2 d^2 x^3+\frac{1}{7} c e x^7 (A c e+2 B (b e+c d))+\frac{1}{4} b d x^4 (2 A b e+2 A c d+b B d)+\frac{1}{8} B c^2 e^2 x^8 \]

[Out]

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

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Rubi [A]  time = 0.223913, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{1}{6} x^6 \left (2 A c e (b e+c d)+B \left (b^2 e^2+4 b c d e+c^2 d^2\right )\right )+\frac{1}{5} x^5 \left (b^2 e (A e+2 B d)+2 b c d (2 A e+B d)+A c^2 d^2\right )+\frac{1}{3} A b^2 d^2 x^3+\frac{1}{7} c e x^7 (A c e+2 B (b e+c d))+\frac{1}{4} b d x^4 (2 A b e+2 A c d+b B d)+\frac{1}{8} B c^2 e^2 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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)^2 \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 d^2 x^2+b d (b B d+2 A c d+2 A b e) x^3+\left (A c^2 d^2+b^2 e (2 B d+A e)+2 b c d (B d+2 A e)\right ) x^4+\left (2 A c e (c d+b e)+B \left (c^2 d^2+4 b c d e+b^2 e^2\right )\right ) x^5+c e (A c e+2 B (c d+b e)) x^6+B c^2 e^2 x^7\right ) \, dx\\ &=\frac{1}{3} A b^2 d^2 x^3+\frac{1}{4} b d (b B d+2 A c d+2 A b e) x^4+\frac{1}{5} \left (A c^2 d^2+b^2 e (2 B d+A e)+2 b c d (B d+2 A e)\right ) x^5+\frac{1}{6} \left (2 A c e (c d+b e)+B \left (c^2 d^2+4 b c d e+b^2 e^2\right )\right ) x^6+\frac{1}{7} c e (A c e+2 B (c d+b e)) x^7+\frac{1}{8} B c^2 e^2 x^8\\ \end{align*}

Mathematica [A]  time = 0.0579121, size = 162, normalized size = 1. \[ \frac{1}{6} x^6 \left (2 A c e (b e+c d)+B \left (b^2 e^2+4 b c d e+c^2 d^2\right )\right )+\frac{1}{5} x^5 \left (b^2 e (A e+2 B d)+2 b c d (2 A e+B d)+A c^2 d^2\right )+\frac{1}{3} A b^2 d^2 x^3+\frac{1}{7} c e x^7 (A c e+2 B (b e+c d))+\frac{1}{4} b d x^4 (2 A b e+2 A c d+b B d)+\frac{1}{8} B c^2 e^2 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 172, normalized size = 1.1 \begin{align*}{\frac{B{c}^{2}{e}^{2}{x}^{8}}{8}}+{\frac{ \left ( \left ( A{e}^{2}+2\,Bde \right ){c}^{2}+2\,B{e}^{2}bc \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ){c}^{2}+2\, \left ( A{e}^{2}+2\,Bde \right ) bc+B{e}^{2}{b}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( A{c}^{2}{d}^{2}+2\, \left ( 2\,Ade+B{d}^{2} \right ) bc+ \left ( A{e}^{2}+2\,Bde \right ){b}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,A{d}^{2}bc+ \left ( 2\,Ade+B{d}^{2} \right ){b}^{2} \right ){x}^{4}}{4}}+{\frac{A{b}^{2}{d}^{2}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08374, size = 231, normalized size = 1.43 \begin{align*} \frac{1}{8} \, B c^{2} e^{2} x^{8} + \frac{1}{3} \, A b^{2} d^{2} x^{3} + \frac{1}{7} \,{\left (2 \, B c^{2} d e +{\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B c^{2} d^{2} + 2 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (A b^{2} e^{2} +{\left (2 \, B b c + A c^{2}\right )} d^{2} + 2 \,{\left (B b^{2} + 2 \, A b c\right )} d e\right )} x^{5} + \frac{1}{4} \,{\left (2 \, A b^{2} d e +{\left (B b^{2} + 2 \, A b c\right )} d^{2}\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.23306, size = 485, normalized size = 2.99 \begin{align*} \frac{1}{8} x^{8} e^{2} c^{2} B + \frac{2}{7} x^{7} e d c^{2} B + \frac{2}{7} x^{7} e^{2} c b B + \frac{1}{7} x^{7} e^{2} c^{2} A + \frac{1}{6} x^{6} d^{2} c^{2} B + \frac{2}{3} x^{6} e d c b B + \frac{1}{6} x^{6} e^{2} b^{2} B + \frac{1}{3} x^{6} e d c^{2} A + \frac{1}{3} x^{6} e^{2} c b A + \frac{2}{5} x^{5} d^{2} c b B + \frac{2}{5} x^{5} e d b^{2} B + \frac{1}{5} x^{5} d^{2} c^{2} A + \frac{4}{5} x^{5} e d c b A + \frac{1}{5} x^{5} e^{2} b^{2} A + \frac{1}{4} x^{4} d^{2} b^{2} B + \frac{1}{2} x^{4} d^{2} c b A + \frac{1}{2} x^{4} e d b^{2} A + \frac{1}{3} x^{3} d^{2} b^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.240419, size = 212, normalized size = 1.31 \begin{align*} \frac{A b^{2} d^{2} x^{3}}{3} + \frac{B c^{2} e^{2} x^{8}}{8} + x^{7} \left (\frac{A c^{2} e^{2}}{7} + \frac{2 B b c e^{2}}{7} + \frac{2 B c^{2} d e}{7}\right ) + x^{6} \left (\frac{A b c e^{2}}{3} + \frac{A c^{2} d e}{3} + \frac{B b^{2} e^{2}}{6} + \frac{2 B b c d e}{3} + \frac{B c^{2} d^{2}}{6}\right ) + x^{5} \left (\frac{A b^{2} e^{2}}{5} + \frac{4 A b c d e}{5} + \frac{A c^{2} d^{2}}{5} + \frac{2 B b^{2} d e}{5} + \frac{2 B b c d^{2}}{5}\right ) + x^{4} \left (\frac{A b^{2} d e}{2} + \frac{A b c d^{2}}{2} + \frac{B b^{2} d^{2}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.29711, size = 277, normalized size = 1.71 \begin{align*} \frac{1}{8} \, B c^{2} x^{8} e^{2} + \frac{2}{7} \, B c^{2} d x^{7} e + \frac{1}{6} \, B c^{2} d^{2} x^{6} + \frac{2}{7} \, B b c x^{7} e^{2} + \frac{1}{7} \, A c^{2} x^{7} e^{2} + \frac{2}{3} \, B b c d x^{6} e + \frac{1}{3} \, A c^{2} d x^{6} e + \frac{2}{5} \, B b c d^{2} x^{5} + \frac{1}{5} \, A c^{2} d^{2} x^{5} + \frac{1}{6} \, B b^{2} x^{6} e^{2} + \frac{1}{3} \, A b c x^{6} e^{2} + \frac{2}{5} \, B b^{2} d x^{5} e + \frac{4}{5} \, A b c d x^{5} e + \frac{1}{4} \, B b^{2} d^{2} x^{4} + \frac{1}{2} \, A b c d^{2} x^{4} + \frac{1}{5} \, A b^{2} x^{5} e^{2} + \frac{1}{2} \, A b^{2} d x^{4} e + \frac{1}{3} \, A b^{2} d^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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