3.244 \(\int (d+e x)^4 (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=225 \[ \frac {1}{4} b^3 d^4 x^4+\frac {1}{3} c e^2 x^9 \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+\frac {1}{2} b d^2 x^6 \left (2 b^2 e^2+4 b c d e+c^2 d^2\right )+\frac {1}{5} b^2 d^3 x^5 (4 b e+3 c d)+\frac {1}{8} e x^8 \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )+\frac {1}{7} d x^7 \left (4 b^3 e^3+18 b^2 c d e^2+12 b c^2 d^2 e+c^3 d^3\right )+\frac {1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac {1}{11} c^3 e^4 x^{11} \]

[Out]

1/4*b^3*d^4*x^4+1/5*b^2*d^3*(4*b*e+3*c*d)*x^5+1/2*b*d^2*(2*b^2*e^2+4*b*c*d*e+c^2*d^2)*x^6+1/7*d*(4*b^3*e^3+18*
b^2*c*d*e^2+12*b*c^2*d^2*e+c^3*d^3)*x^7+1/8*e*(b^3*e^3+12*b^2*c*d*e^2+18*b*c^2*d^2*e+4*c^3*d^3)*x^8+1/3*c*e^2*
(b^2*e^2+4*b*c*d*e+2*c^2*d^2)*x^9+1/10*c^2*e^3*(3*b*e+4*c*d)*x^10+1/11*c^3*e^4*x^11

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Rubi [A]  time = 0.22, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \[ \frac {1}{3} c e^2 x^9 \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+\frac {1}{8} e x^8 \left (12 b^2 c d e^2+b^3 e^3+18 b c^2 d^2 e+4 c^3 d^3\right )+\frac {1}{7} d x^7 \left (18 b^2 c d e^2+4 b^3 e^3+12 b c^2 d^2 e+c^3 d^3\right )+\frac {1}{2} b d^2 x^6 \left (2 b^2 e^2+4 b c d e+c^2 d^2\right )+\frac {1}{5} b^2 d^3 x^5 (4 b e+3 c d)+\frac {1}{4} b^3 d^4 x^4+\frac {1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac {1}{11} c^3 e^4 x^{11} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(b*x + c*x^2)^3,x]

[Out]

(b^3*d^4*x^4)/4 + (b^2*d^3*(3*c*d + 4*b*e)*x^5)/5 + (b*d^2*(c^2*d^2 + 4*b*c*d*e + 2*b^2*e^2)*x^6)/2 + (d*(c^3*
d^3 + 12*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 4*b^3*e^3)*x^7)/7 + (e*(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 +
b^3*e^3)*x^8)/8 + (c*e^2*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*x^9)/3 + (c^2*e^3*(4*c*d + 3*b*e)*x^10)/10 + (c^3*e
^4*x^11)/11

Rule 698

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

Rubi steps

\begin {align*} \int (d+e x)^4 \left (b x+c x^2\right )^3 \, dx &=\int \left (b^3 d^4 x^3+b^2 d^3 (3 c d+4 b e) x^4+3 b d^2 \left (c^2 d^2+4 b c d e+2 b^2 e^2\right ) x^5+d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right ) x^6+e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right ) x^7+3 c e^2 \left (2 c^2 d^2+4 b c d e+b^2 e^2\right ) x^8+c^2 e^3 (4 c d+3 b e) x^9+c^3 e^4 x^{10}\right ) \, dx\\ &=\frac {1}{4} b^3 d^4 x^4+\frac {1}{5} b^2 d^3 (3 c d+4 b e) x^5+\frac {1}{2} b d^2 \left (c^2 d^2+4 b c d e+2 b^2 e^2\right ) x^6+\frac {1}{7} d \left (c^3 d^3+12 b c^2 d^2 e+18 b^2 c d e^2+4 b^3 e^3\right ) x^7+\frac {1}{8} e \left (4 c^3 d^3+18 b c^2 d^2 e+12 b^2 c d e^2+b^3 e^3\right ) x^8+\frac {1}{3} c e^2 \left (2 c^2 d^2+4 b c d e+b^2 e^2\right ) x^9+\frac {1}{10} c^2 e^3 (4 c d+3 b e) x^{10}+\frac {1}{11} c^3 e^4 x^{11}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 225, normalized size = 1.00 \[ \frac {1}{4} b^3 d^4 x^4+\frac {1}{3} c e^2 x^9 \left (b^2 e^2+4 b c d e+2 c^2 d^2\right )+\frac {1}{2} b d^2 x^6 \left (2 b^2 e^2+4 b c d e+c^2 d^2\right )+\frac {1}{5} b^2 d^3 x^5 (4 b e+3 c d)+\frac {1}{8} e x^8 \left (b^3 e^3+12 b^2 c d e^2+18 b c^2 d^2 e+4 c^3 d^3\right )+\frac {1}{7} d x^7 \left (4 b^3 e^3+18 b^2 c d e^2+12 b c^2 d^2 e+c^3 d^3\right )+\frac {1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac {1}{11} c^3 e^4 x^{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4*(b*x + c*x^2)^3,x]

[Out]

(b^3*d^4*x^4)/4 + (b^2*d^3*(3*c*d + 4*b*e)*x^5)/5 + (b*d^2*(c^2*d^2 + 4*b*c*d*e + 2*b^2*e^2)*x^6)/2 + (d*(c^3*
d^3 + 12*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 4*b^3*e^3)*x^7)/7 + (e*(4*c^3*d^3 + 18*b*c^2*d^2*e + 12*b^2*c*d*e^2 +
b^3*e^3)*x^8)/8 + (c*e^2*(2*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*x^9)/3 + (c^2*e^3*(4*c*d + 3*b*e)*x^10)/10 + (c^3*e
^4*x^11)/11

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fricas [A]  time = 1.02, size = 250, normalized size = 1.11 \[ \frac {1}{11} x^{11} e^{4} c^{3} + \frac {2}{5} x^{10} e^{3} d c^{3} + \frac {3}{10} x^{10} e^{4} c^{2} b + \frac {2}{3} x^{9} e^{2} d^{2} c^{3} + \frac {4}{3} x^{9} e^{3} d c^{2} b + \frac {1}{3} x^{9} e^{4} c b^{2} + \frac {1}{2} x^{8} e d^{3} c^{3} + \frac {9}{4} x^{8} e^{2} d^{2} c^{2} b + \frac {3}{2} x^{8} e^{3} d c b^{2} + \frac {1}{8} x^{8} e^{4} b^{3} + \frac {1}{7} x^{7} d^{4} c^{3} + \frac {12}{7} x^{7} e d^{3} c^{2} b + \frac {18}{7} x^{7} e^{2} d^{2} c b^{2} + \frac {4}{7} x^{7} e^{3} d b^{3} + \frac {1}{2} x^{6} d^{4} c^{2} b + 2 x^{6} e d^{3} c b^{2} + x^{6} e^{2} d^{2} b^{3} + \frac {3}{5} x^{5} d^{4} c b^{2} + \frac {4}{5} x^{5} e d^{3} b^{3} + \frac {1}{4} x^{4} d^{4} b^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

1/11*x^11*e^4*c^3 + 2/5*x^10*e^3*d*c^3 + 3/10*x^10*e^4*c^2*b + 2/3*x^9*e^2*d^2*c^3 + 4/3*x^9*e^3*d*c^2*b + 1/3
*x^9*e^4*c*b^2 + 1/2*x^8*e*d^3*c^3 + 9/4*x^8*e^2*d^2*c^2*b + 3/2*x^8*e^3*d*c*b^2 + 1/8*x^8*e^4*b^3 + 1/7*x^7*d
^4*c^3 + 12/7*x^7*e*d^3*c^2*b + 18/7*x^7*e^2*d^2*c*b^2 + 4/7*x^7*e^3*d*b^3 + 1/2*x^6*d^4*c^2*b + 2*x^6*e*d^3*c
*b^2 + x^6*e^2*d^2*b^3 + 3/5*x^5*d^4*c*b^2 + 4/5*x^5*e*d^3*b^3 + 1/4*x^4*d^4*b^3

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giac [A]  time = 0.17, size = 242, normalized size = 1.08 \[ \frac {1}{11} \, c^{3} x^{11} e^{4} + \frac {2}{5} \, c^{3} d x^{10} e^{3} + \frac {2}{3} \, c^{3} d^{2} x^{9} e^{2} + \frac {1}{2} \, c^{3} d^{3} x^{8} e + \frac {1}{7} \, c^{3} d^{4} x^{7} + \frac {3}{10} \, b c^{2} x^{10} e^{4} + \frac {4}{3} \, b c^{2} d x^{9} e^{3} + \frac {9}{4} \, b c^{2} d^{2} x^{8} e^{2} + \frac {12}{7} \, b c^{2} d^{3} x^{7} e + \frac {1}{2} \, b c^{2} d^{4} x^{6} + \frac {1}{3} \, b^{2} c x^{9} e^{4} + \frac {3}{2} \, b^{2} c d x^{8} e^{3} + \frac {18}{7} \, b^{2} c d^{2} x^{7} e^{2} + 2 \, b^{2} c d^{3} x^{6} e + \frac {3}{5} \, b^{2} c d^{4} x^{5} + \frac {1}{8} \, b^{3} x^{8} e^{4} + \frac {4}{7} \, b^{3} d x^{7} e^{3} + b^{3} d^{2} x^{6} e^{2} + \frac {4}{5} \, b^{3} d^{3} x^{5} e + \frac {1}{4} \, b^{3} d^{4} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

1/11*c^3*x^11*e^4 + 2/5*c^3*d*x^10*e^3 + 2/3*c^3*d^2*x^9*e^2 + 1/2*c^3*d^3*x^8*e + 1/7*c^3*d^4*x^7 + 3/10*b*c^
2*x^10*e^4 + 4/3*b*c^2*d*x^9*e^3 + 9/4*b*c^2*d^2*x^8*e^2 + 12/7*b*c^2*d^3*x^7*e + 1/2*b*c^2*d^4*x^6 + 1/3*b^2*
c*x^9*e^4 + 3/2*b^2*c*d*x^8*e^3 + 18/7*b^2*c*d^2*x^7*e^2 + 2*b^2*c*d^3*x^6*e + 3/5*b^2*c*d^4*x^5 + 1/8*b^3*x^8
*e^4 + 4/7*b^3*d*x^7*e^3 + b^3*d^2*x^6*e^2 + 4/5*b^3*d^3*x^5*e + 1/4*b^3*d^4*x^4

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maple [A]  time = 0.05, size = 232, normalized size = 1.03 \[ \frac {c^{3} e^{4} x^{11}}{11}+\frac {b^{3} d^{4} x^{4}}{4}+\frac {\left (3 e^{4} b \,c^{2}+4 d \,e^{3} c^{3}\right ) x^{10}}{10}+\frac {\left (3 e^{4} c \,b^{2}+12 d \,e^{3} b \,c^{2}+6 d^{2} e^{2} c^{3}\right ) x^{9}}{9}+\frac {\left (e^{4} b^{3}+12 d \,e^{3} c \,b^{2}+18 d^{2} e^{2} b \,c^{2}+4 d^{3} e \,c^{3}\right ) x^{8}}{8}+\frac {\left (4 d \,e^{3} b^{3}+18 d^{2} e^{2} c \,b^{2}+12 d^{3} e b \,c^{2}+d^{4} c^{3}\right ) x^{7}}{7}+\frac {\left (6 d^{2} e^{2} b^{3}+12 d^{3} e c \,b^{2}+3 d^{4} b \,c^{2}\right ) x^{6}}{6}+\frac {\left (4 d^{3} e \,b^{3}+3 d^{4} c \,b^{2}\right ) x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(c*x^2+b*x)^3,x)

[Out]

1/11*c^3*e^4*x^11+1/10*(3*b*c^2*e^4+4*c^3*d*e^3)*x^10+1/9*(3*b^2*c*e^4+12*b*c^2*d*e^3+6*c^3*d^2*e^2)*x^9+1/8*(
b^3*e^4+12*b^2*c*d*e^3+18*b*c^2*d^2*e^2+4*c^3*d^3*e)*x^8+1/7*(4*b^3*d*e^3+18*b^2*c*d^2*e^2+12*b*c^2*d^3*e+c^3*
d^4)*x^7+1/6*(6*b^3*d^2*e^2+12*b^2*c*d^3*e+3*b*c^2*d^4)*x^6+1/5*(4*b^3*d^3*e+3*b^2*c*d^4)*x^5+1/4*b^3*d^4*x^4

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maxima [A]  time = 1.39, size = 229, normalized size = 1.02 \[ \frac {1}{11} \, c^{3} e^{4} x^{11} + \frac {1}{4} \, b^{3} d^{4} x^{4} + \frac {1}{10} \, {\left (4 \, c^{3} d e^{3} + 3 \, b c^{2} e^{4}\right )} x^{10} + \frac {1}{3} \, {\left (2 \, c^{3} d^{2} e^{2} + 4 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (4 \, c^{3} d^{3} e + 18 \, b c^{2} d^{2} e^{2} + 12 \, b^{2} c d e^{3} + b^{3} e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{4} + 12 \, b c^{2} d^{3} e + 18 \, b^{2} c d^{2} e^{2} + 4 \, b^{3} d e^{3}\right )} x^{7} + \frac {1}{2} \, {\left (b c^{2} d^{4} + 4 \, b^{2} c d^{3} e + 2 \, b^{3} d^{2} e^{2}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, b^{2} c d^{4} + 4 \, b^{3} d^{3} e\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

1/11*c^3*e^4*x^11 + 1/4*b^3*d^4*x^4 + 1/10*(4*c^3*d*e^3 + 3*b*c^2*e^4)*x^10 + 1/3*(2*c^3*d^2*e^2 + 4*b*c^2*d*e
^3 + b^2*c*e^4)*x^9 + 1/8*(4*c^3*d^3*e + 18*b*c^2*d^2*e^2 + 12*b^2*c*d*e^3 + b^3*e^4)*x^8 + 1/7*(c^3*d^4 + 12*
b*c^2*d^3*e + 18*b^2*c*d^2*e^2 + 4*b^3*d*e^3)*x^7 + 1/2*(b*c^2*d^4 + 4*b^2*c*d^3*e + 2*b^3*d^2*e^2)*x^6 + 1/5*
(3*b^2*c*d^4 + 4*b^3*d^3*e)*x^5

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mupad [B]  time = 0.09, size = 213, normalized size = 0.95 \[ x^7\,\left (\frac {4\,b^3\,d\,e^3}{7}+\frac {18\,b^2\,c\,d^2\,e^2}{7}+\frac {12\,b\,c^2\,d^3\,e}{7}+\frac {c^3\,d^4}{7}\right )+x^8\,\left (\frac {b^3\,e^4}{8}+\frac {3\,b^2\,c\,d\,e^3}{2}+\frac {9\,b\,c^2\,d^2\,e^2}{4}+\frac {c^3\,d^3\,e}{2}\right )+\frac {b^3\,d^4\,x^4}{4}+\frac {c^3\,e^4\,x^{11}}{11}+\frac {b^2\,d^3\,x^5\,\left (4\,b\,e+3\,c\,d\right )}{5}+\frac {c^2\,e^3\,x^{10}\,\left (3\,b\,e+4\,c\,d\right )}{10}+\frac {b\,d^2\,x^6\,\left (2\,b^2\,e^2+4\,b\,c\,d\,e+c^2\,d^2\right )}{2}+\frac {c\,e^2\,x^9\,\left (b^2\,e^2+4\,b\,c\,d\,e+2\,c^2\,d^2\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^3*(d + e*x)^4,x)

[Out]

x^7*((c^3*d^4)/7 + (4*b^3*d*e^3)/7 + (18*b^2*c*d^2*e^2)/7 + (12*b*c^2*d^3*e)/7) + x^8*((b^3*e^4)/8 + (c^3*d^3*
e)/2 + (9*b*c^2*d^2*e^2)/4 + (3*b^2*c*d*e^3)/2) + (b^3*d^4*x^4)/4 + (c^3*e^4*x^11)/11 + (b^2*d^3*x^5*(4*b*e +
3*c*d))/5 + (c^2*e^3*x^10*(3*b*e + 4*c*d))/10 + (b*d^2*x^6*(2*b^2*e^2 + c^2*d^2 + 4*b*c*d*e))/2 + (c*e^2*x^9*(
b^2*e^2 + 2*c^2*d^2 + 4*b*c*d*e))/3

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sympy [A]  time = 0.12, size = 257, normalized size = 1.14 \[ \frac {b^{3} d^{4} x^{4}}{4} + \frac {c^{3} e^{4} x^{11}}{11} + x^{10} \left (\frac {3 b c^{2} e^{4}}{10} + \frac {2 c^{3} d e^{3}}{5}\right ) + x^{9} \left (\frac {b^{2} c e^{4}}{3} + \frac {4 b c^{2} d e^{3}}{3} + \frac {2 c^{3} d^{2} e^{2}}{3}\right ) + x^{8} \left (\frac {b^{3} e^{4}}{8} + \frac {3 b^{2} c d e^{3}}{2} + \frac {9 b c^{2} d^{2} e^{2}}{4} + \frac {c^{3} d^{3} e}{2}\right ) + x^{7} \left (\frac {4 b^{3} d e^{3}}{7} + \frac {18 b^{2} c d^{2} e^{2}}{7} + \frac {12 b c^{2} d^{3} e}{7} + \frac {c^{3} d^{4}}{7}\right ) + x^{6} \left (b^{3} d^{2} e^{2} + 2 b^{2} c d^{3} e + \frac {b c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac {4 b^{3} d^{3} e}{5} + \frac {3 b^{2} c d^{4}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4*(c*x**2+b*x)**3,x)

[Out]

b**3*d**4*x**4/4 + c**3*e**4*x**11/11 + x**10*(3*b*c**2*e**4/10 + 2*c**3*d*e**3/5) + x**9*(b**2*c*e**4/3 + 4*b
*c**2*d*e**3/3 + 2*c**3*d**2*e**2/3) + x**8*(b**3*e**4/8 + 3*b**2*c*d*e**3/2 + 9*b*c**2*d**2*e**2/4 + c**3*d**
3*e/2) + x**7*(4*b**3*d*e**3/7 + 18*b**2*c*d**2*e**2/7 + 12*b*c**2*d**3*e/7 + c**3*d**4/7) + x**6*(b**3*d**2*e
**2 + 2*b**2*c*d**3*e + b*c**2*d**4/2) + x**5*(4*b**3*d**3*e/5 + 3*b**2*c*d**4/5)

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