3.1115 \(\int (A+B x) (d+e x)^3 (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=228 \[ \frac {1}{7} e x^7 \left (A c e (2 b e+3 c d)+B \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )\right )+\frac {1}{6} x^6 \left (A e \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+B d \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac {1}{5} d x^5 \left (3 b^2 e (A e+B d)+2 b c d (3 A e+B d)+A c^2 d^2\right )+\frac {1}{3} A b^2 d^3 x^3+\frac {1}{4} b d^2 x^4 (3 A b e+2 A c d+b B d)+\frac {1}{8} c e^2 x^8 (A c e+2 b B e+3 B c d)+\frac {1}{9} B c^2 e^3 x^9 \]

[Out]

1/3*A*b^2*d^3*x^3+1/4*b*d^2*(3*A*b*e+2*A*c*d+B*b*d)*x^4+1/5*d*(A*c^2*d^2+3*b^2*e*(A*e+B*d)+2*b*c*d*(3*A*e+B*d)
)*x^5+1/6*(A*e*(b^2*e^2+6*b*c*d*e+3*c^2*d^2)+B*d*(3*b^2*e^2+6*b*c*d*e+c^2*d^2))*x^6+1/7*e*(A*c*e*(2*b*e+3*c*d)
+B*(b^2*e^2+6*b*c*d*e+3*c^2*d^2))*x^7+1/8*c*e^2*(A*c*e+2*B*b*e+3*B*c*d)*x^8+1/9*B*c^2*e^3*x^9

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Rubi [A]  time = 0.25, antiderivative size = 228, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{7} e x^7 \left (A c e (2 b e+3 c d)+B \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )\right )+\frac {1}{6} x^6 \left (A e \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+B d \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac {1}{5} d x^5 \left (3 b^2 e (A e+B d)+2 b c d (3 A e+B d)+A c^2 d^2\right )+\frac {1}{3} A b^2 d^3 x^3+\frac {1}{4} b d^2 x^4 (3 A b e+2 A c d+b B d)+\frac {1}{8} c e^2 x^8 (A c e+2 b B e+3 B c d)+\frac {1}{9} B c^2 e^3 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^2*d^3*x^3)/3 + (b*d^2*(b*B*d + 2*A*c*d + 3*A*b*e)*x^4)/4 + (d*(A*c^2*d^2 + 3*b^2*e*(B*d + A*e) + 2*b*c*d*
(B*d + 3*A*e))*x^5)/5 + ((A*e*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2) + B*d*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2))*x^6)/
6 + (e*(A*c*e*(3*c*d + 2*b*e) + B*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2))*x^7)/7 + (c*e^2*(3*B*c*d + 2*b*B*e + A*c*
e)*x^8)/8 + (B*c^2*e^3*x^9)/9

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)^3 \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 d^3 x^2+b d^2 (b B d+2 A c d+3 A b e) x^3+d \left (A c^2 d^2+3 b^2 e (B d+A e)+2 b c d (B d+3 A e)\right ) x^4+\left (A e \left (3 c^2 d^2+6 b c d e+b^2 e^2\right )+B d \left (c^2 d^2+6 b c d e+3 b^2 e^2\right )\right ) x^5+e \left (A c e (3 c d+2 b e)+B \left (3 c^2 d^2+6 b c d e+b^2 e^2\right )\right ) x^6+c e^2 (3 B c d+2 b B e+A c e) x^7+B c^2 e^3 x^8\right ) \, dx\\ &=\frac {1}{3} A b^2 d^3 x^3+\frac {1}{4} b d^2 (b B d+2 A c d+3 A b e) x^4+\frac {1}{5} d \left (A c^2 d^2+3 b^2 e (B d+A e)+2 b c d (B d+3 A e)\right ) x^5+\frac {1}{6} \left (A e \left (3 c^2 d^2+6 b c d e+b^2 e^2\right )+B d \left (c^2 d^2+6 b c d e+3 b^2 e^2\right )\right ) x^6+\frac {1}{7} e \left (A c e (3 c d+2 b e)+B \left (3 c^2 d^2+6 b c d e+b^2 e^2\right )\right ) x^7+\frac {1}{8} c e^2 (3 B c d+2 b B e+A c e) x^8+\frac {1}{9} B c^2 e^3 x^9\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 228, normalized size = 1.00 \[ \frac {1}{7} e x^7 \left (A c e (2 b e+3 c d)+B \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )\right )+\frac {1}{6} x^6 \left (A e \left (b^2 e^2+6 b c d e+3 c^2 d^2\right )+B d \left (3 b^2 e^2+6 b c d e+c^2 d^2\right )\right )+\frac {1}{5} d x^5 \left (3 b^2 e (A e+B d)+2 b c d (3 A e+B d)+A c^2 d^2\right )+\frac {1}{3} A b^2 d^3 x^3+\frac {1}{4} b d^2 x^4 (3 A b e+2 A c d+b B d)+\frac {1}{8} c e^2 x^8 (A c e+2 b B e+3 B c d)+\frac {1}{9} B c^2 e^3 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^2*d^3*x^3)/3 + (b*d^2*(b*B*d + 2*A*c*d + 3*A*b*e)*x^4)/4 + (d*(A*c^2*d^2 + 3*b^2*e*(B*d + A*e) + 2*b*c*d*
(B*d + 3*A*e))*x^5)/5 + ((A*e*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2) + B*d*(c^2*d^2 + 6*b*c*d*e + 3*b^2*e^2))*x^6)/
6 + (e*(A*c*e*(3*c*d + 2*b*e) + B*(3*c^2*d^2 + 6*b*c*d*e + b^2*e^2))*x^7)/7 + (c*e^2*(3*B*c*d + 2*b*B*e + A*c*
e)*x^8)/8 + (B*c^2*e^3*x^9)/9

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fricas [A]  time = 0.75, size = 291, normalized size = 1.28 \[ \frac {1}{9} x^{9} e^{3} c^{2} B + \frac {3}{8} x^{8} e^{2} d c^{2} B + \frac {1}{4} x^{8} e^{3} c b B + \frac {1}{8} x^{8} e^{3} c^{2} A + \frac {3}{7} x^{7} e d^{2} c^{2} B + \frac {6}{7} x^{7} e^{2} d c b B + \frac {1}{7} x^{7} e^{3} b^{2} B + \frac {3}{7} x^{7} e^{2} d c^{2} A + \frac {2}{7} x^{7} e^{3} c b A + \frac {1}{6} x^{6} d^{3} c^{2} B + x^{6} e d^{2} c b B + \frac {1}{2} x^{6} e^{2} d b^{2} B + \frac {1}{2} x^{6} e d^{2} c^{2} A + x^{6} e^{2} d c b A + \frac {1}{6} x^{6} e^{3} b^{2} A + \frac {2}{5} x^{5} d^{3} c b B + \frac {3}{5} x^{5} e d^{2} b^{2} B + \frac {1}{5} x^{5} d^{3} c^{2} A + \frac {6}{5} x^{5} e d^{2} c b A + \frac {3}{5} x^{5} e^{2} d b^{2} A + \frac {1}{4} x^{4} d^{3} b^{2} B + \frac {1}{2} x^{4} d^{3} c b A + \frac {3}{4} x^{4} e d^{2} b^{2} A + \frac {1}{3} x^{3} d^{3} b^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*e^3*c^2*B + 3/8*x^8*e^2*d*c^2*B + 1/4*x^8*e^3*c*b*B + 1/8*x^8*e^3*c^2*A + 3/7*x^7*e*d^2*c^2*B + 6/7*x^
7*e^2*d*c*b*B + 1/7*x^7*e^3*b^2*B + 3/7*x^7*e^2*d*c^2*A + 2/7*x^7*e^3*c*b*A + 1/6*x^6*d^3*c^2*B + x^6*e*d^2*c*
b*B + 1/2*x^6*e^2*d*b^2*B + 1/2*x^6*e*d^2*c^2*A + x^6*e^2*d*c*b*A + 1/6*x^6*e^3*b^2*A + 2/5*x^5*d^3*c*b*B + 3/
5*x^5*e*d^2*b^2*B + 1/5*x^5*d^3*c^2*A + 6/5*x^5*e*d^2*c*b*A + 3/5*x^5*e^2*d*b^2*A + 1/4*x^4*d^3*b^2*B + 1/2*x^
4*d^3*c*b*A + 3/4*x^4*e*d^2*b^2*A + 1/3*x^3*d^3*b^2*A

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giac [A]  time = 0.16, size = 285, normalized size = 1.25 \[ \frac {1}{9} \, B c^{2} x^{9} e^{3} + \frac {3}{8} \, B c^{2} d x^{8} e^{2} + \frac {3}{7} \, B c^{2} d^{2} x^{7} e + \frac {1}{6} \, B c^{2} d^{3} x^{6} + \frac {1}{4} \, B b c x^{8} e^{3} + \frac {1}{8} \, A c^{2} x^{8} e^{3} + \frac {6}{7} \, B b c d x^{7} e^{2} + \frac {3}{7} \, A c^{2} d x^{7} e^{2} + B b c d^{2} x^{6} e + \frac {1}{2} \, A c^{2} d^{2} x^{6} e + \frac {2}{5} \, B b c d^{3} x^{5} + \frac {1}{5} \, A c^{2} d^{3} x^{5} + \frac {1}{7} \, B b^{2} x^{7} e^{3} + \frac {2}{7} \, A b c x^{7} e^{3} + \frac {1}{2} \, B b^{2} d x^{6} e^{2} + A b c d x^{6} e^{2} + \frac {3}{5} \, B b^{2} d^{2} x^{5} e + \frac {6}{5} \, A b c d^{2} x^{5} e + \frac {1}{4} \, B b^{2} d^{3} x^{4} + \frac {1}{2} \, A b c d^{3} x^{4} + \frac {1}{6} \, A b^{2} x^{6} e^{3} + \frac {3}{5} \, A b^{2} d x^{5} e^{2} + \frac {3}{4} \, A b^{2} d^{2} x^{4} e + \frac {1}{3} \, A b^{2} d^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*c^2*x^9*e^3 + 3/8*B*c^2*d*x^8*e^2 + 3/7*B*c^2*d^2*x^7*e + 1/6*B*c^2*d^3*x^6 + 1/4*B*b*c*x^8*e^3 + 1/8*A*
c^2*x^8*e^3 + 6/7*B*b*c*d*x^7*e^2 + 3/7*A*c^2*d*x^7*e^2 + B*b*c*d^2*x^6*e + 1/2*A*c^2*d^2*x^6*e + 2/5*B*b*c*d^
3*x^5 + 1/5*A*c^2*d^3*x^5 + 1/7*B*b^2*x^7*e^3 + 2/7*A*b*c*x^7*e^3 + 1/2*B*b^2*d*x^6*e^2 + A*b*c*d*x^6*e^2 + 3/
5*B*b^2*d^2*x^5*e + 6/5*A*b*c*d^2*x^5*e + 1/4*B*b^2*d^3*x^4 + 1/2*A*b*c*d^3*x^4 + 1/6*A*b^2*x^6*e^3 + 3/5*A*b^
2*d*x^5*e^2 + 3/4*A*b^2*d^2*x^4*e + 1/3*A*b^2*d^3*x^3

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maple [A]  time = 0.04, size = 247, normalized size = 1.08 \[ \frac {B \,c^{2} e^{3} x^{9}}{9}+\frac {A \,b^{2} d^{3} x^{3}}{3}+\frac {\left (2 B b c \,e^{3}+\left (A \,e^{3}+3 B d \,e^{2}\right ) c^{2}\right ) x^{8}}{8}+\frac {\left (B \,b^{2} e^{3}+2 \left (A \,e^{3}+3 B d \,e^{2}\right ) b c +\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) c^{2}\right ) x^{7}}{7}+\frac {\left (\left (A \,e^{3}+3 B d \,e^{2}\right ) b^{2}+2 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) b c +\left (3 A \,d^{2} e +B \,d^{3}\right ) c^{2}\right ) x^{6}}{6}+\frac {\left (A \,c^{2} d^{3}+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) b^{2}+2 \left (3 A \,d^{2} e +B \,d^{3}\right ) b c \right ) x^{5}}{5}+\frac {\left (2 A b c \,d^{3}+\left (3 A \,d^{2} e +B \,d^{3}\right ) b^{2}\right ) x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*c^2*e^3*x^9+1/8*((A*e^3+3*B*d*e^2)*c^2+2*B*e^3*b*c)*x^8+1/7*((3*A*d*e^2+3*B*d^2*e)*c^2+2*(A*e^3+3*B*d*e^
2)*b*c+B*e^3*b^2)*x^7+1/6*((3*A*d^2*e+B*d^3)*c^2+2*(3*A*d*e^2+3*B*d^2*e)*b*c+(A*e^3+3*B*d*e^2)*b^2)*x^6+1/5*(A
*d^3*c^2+2*(3*A*d^2*e+B*d^3)*b*c+(3*A*d*e^2+3*B*d^2*e)*b^2)*x^5+1/4*(2*A*d^3*b*c+(3*A*d^2*e+B*d^3)*b^2)*x^4+1/
3*A*b^2*d^3*x^3

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maxima [A]  time = 0.58, size = 239, normalized size = 1.05 \[ \frac {1}{9} \, B c^{2} e^{3} x^{9} + \frac {1}{3} \, A b^{2} d^{3} x^{3} + \frac {1}{8} \, {\left (3 \, B c^{2} d e^{2} + {\left (2 \, B b c + A c^{2}\right )} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, B c^{2} d^{2} e + 3 \, {\left (2 \, B b c + A c^{2}\right )} d e^{2} + {\left (B b^{2} + 2 \, A b c\right )} e^{3}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{3} + A b^{2} e^{3} + 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, A b^{2} d e^{2} + {\left (2 \, B b c + A c^{2}\right )} d^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e\right )} x^{5} + \frac {1}{4} \, {\left (3 \, A b^{2} d^{2} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3}\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*B*c^2*e^3*x^9 + 1/3*A*b^2*d^3*x^3 + 1/8*(3*B*c^2*d*e^2 + (2*B*b*c + A*c^2)*e^3)*x^8 + 1/7*(3*B*c^2*d^2*e +
3*(2*B*b*c + A*c^2)*d*e^2 + (B*b^2 + 2*A*b*c)*e^3)*x^7 + 1/6*(B*c^2*d^3 + A*b^2*e^3 + 3*(2*B*b*c + A*c^2)*d^2
*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*x^6 + 1/5*(3*A*b^2*d*e^2 + (2*B*b*c + A*c^2)*d^3 + 3*(B*b^2 + 2*A*b*c)*d^2*e)*
x^5 + 1/4*(3*A*b^2*d^2*e + (B*b^2 + 2*A*b*c)*d^3)*x^4

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mupad [B]  time = 0.11, size = 234, normalized size = 1.03 \[ x^6\,\left (\frac {B\,b^2\,d\,e^2}{2}+\frac {A\,b^2\,e^3}{6}+B\,b\,c\,d^2\,e+A\,b\,c\,d\,e^2+\frac {B\,c^2\,d^3}{6}+\frac {A\,c^2\,d^2\,e}{2}\right )+x^5\,\left (\frac {3\,B\,b^2\,d^2\,e}{5}+\frac {3\,A\,b^2\,d\,e^2}{5}+\frac {2\,B\,b\,c\,d^3}{5}+\frac {6\,A\,b\,c\,d^2\,e}{5}+\frac {A\,c^2\,d^3}{5}\right )+x^7\,\left (\frac {B\,b^2\,e^3}{7}+\frac {6\,B\,b\,c\,d\,e^2}{7}+\frac {2\,A\,b\,c\,e^3}{7}+\frac {3\,B\,c^2\,d^2\,e}{7}+\frac {3\,A\,c^2\,d\,e^2}{7}\right )+\frac {b\,d^2\,x^4\,\left (3\,A\,b\,e+2\,A\,c\,d+B\,b\,d\right )}{4}+\frac {c\,e^2\,x^8\,\left (A\,c\,e+2\,B\,b\,e+3\,B\,c\,d\right )}{8}+\frac {A\,b^2\,d^3\,x^3}{3}+\frac {B\,c^2\,e^3\,x^9}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^6*((A*b^2*e^3)/6 + (B*c^2*d^3)/6 + (A*c^2*d^2*e)/2 + (B*b^2*d*e^2)/2 + A*b*c*d*e^2 + B*b*c*d^2*e) + x^5*((A*
c^2*d^3)/5 + (2*B*b*c*d^3)/5 + (3*A*b^2*d*e^2)/5 + (3*B*b^2*d^2*e)/5 + (6*A*b*c*d^2*e)/5) + x^7*((B*b^2*e^3)/7
+ (2*A*b*c*e^3)/7 + (3*A*c^2*d*e^2)/7 + (3*B*c^2*d^2*e)/7 + (6*B*b*c*d*e^2)/7) + (b*d^2*x^4*(3*A*b*e + 2*A*c*
d + B*b*d))/4 + (c*e^2*x^8*(A*c*e + 2*B*b*e + 3*B*c*d))/8 + (A*b^2*d^3*x^3)/3 + (B*c^2*e^3*x^9)/9

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sympy [A]  time = 0.11, size = 301, normalized size = 1.32 \[ \frac {A b^{2} d^{3} x^{3}}{3} + \frac {B c^{2} e^{3} x^{9}}{9} + x^{8} \left (\frac {A c^{2} e^{3}}{8} + \frac {B b c e^{3}}{4} + \frac {3 B c^{2} d e^{2}}{8}\right ) + x^{7} \left (\frac {2 A b c e^{3}}{7} + \frac {3 A c^{2} d e^{2}}{7} + \frac {B b^{2} e^{3}}{7} + \frac {6 B b c d e^{2}}{7} + \frac {3 B c^{2} d^{2} e}{7}\right ) + x^{6} \left (\frac {A b^{2} e^{3}}{6} + A b c d e^{2} + \frac {A c^{2} d^{2} e}{2} + \frac {B b^{2} d e^{2}}{2} + B b c d^{2} e + \frac {B c^{2} d^{3}}{6}\right ) + x^{5} \left (\frac {3 A b^{2} d e^{2}}{5} + \frac {6 A b c d^{2} e}{5} + \frac {A c^{2} d^{3}}{5} + \frac {3 B b^{2} d^{2} e}{5} + \frac {2 B b c d^{3}}{5}\right ) + x^{4} \left (\frac {3 A b^{2} d^{2} e}{4} + \frac {A b c d^{3}}{2} + \frac {B b^{2} d^{3}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b**2*d**3*x**3/3 + B*c**2*e**3*x**9/9 + x**8*(A*c**2*e**3/8 + B*b*c*e**3/4 + 3*B*c**2*d*e**2/8) + x**7*(2*A*
b*c*e**3/7 + 3*A*c**2*d*e**2/7 + B*b**2*e**3/7 + 6*B*b*c*d*e**2/7 + 3*B*c**2*d**2*e/7) + x**6*(A*b**2*e**3/6 +
A*b*c*d*e**2 + A*c**2*d**2*e/2 + B*b**2*d*e**2/2 + B*b*c*d**2*e + B*c**2*d**3/6) + x**5*(3*A*b**2*d*e**2/5 +
6*A*b*c*d**2*e/5 + A*c**2*d**3/5 + 3*B*b**2*d**2*e/5 + 2*B*b*c*d**3/5) + x**4*(3*A*b**2*d**2*e/4 + A*b*c*d**3/
2 + B*b**2*d**3/4)

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