\(\int (b d+2 c d x)^4 (a+b x+c x^2) \, dx\) [1109]

Optimal result

Integrand size = 22, antiderivative size = 45 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right ) \, dx=-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5}{40 c^2}+\frac {d^4 (b+2 c x)^7}{56 c^2} \]

[Out]

-1/40*(-4*a*c+b^2)*d^4*(2*c*x+b)^5/c^2+1/56*d^4*(2*c*x+b)^7/c^2

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {697} \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {d^4 (b+2 c x)^7}{56 c^2}-\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^5}{40 c^2} \]

[In]

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

[Out]

-1/40*((b^2 - 4*a*c)*d^4*(b + 2*c*x)^5)/c^2 + (d^4*(b + 2*c*x)^7)/(56*c^2)

Rule 697

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] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^4}{4 c}+\frac {(b d+2 c d x)^6}{4 c d^2}\right ) \, dx \\ & = -\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5}{40 c^2}+\frac {d^4 (b+2 c x)^7}{56 c^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(102\) vs. \(2(45)=90\).

Time = 0.02 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.27 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right ) \, dx=d^4 \left (a b^4 x+\frac {1}{2} b^3 \left (b^2+8 a c\right ) x^2+b^2 c \left (3 b^2+8 a c\right ) x^3+8 b c^2 \left (b^2+a c\right ) x^4+\frac {8}{5} c^3 \left (7 b^2+2 a c\right ) x^5+8 b c^4 x^6+\frac {16 c^5 x^7}{7}\right ) \]

[In]

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

[Out]

d^4*(a*b^4*x + (b^3*(b^2 + 8*a*c)*x^2)/2 + b^2*c*(3*b^2 + 8*a*c)*x^3 + 8*b*c^2*(b^2 + a*c)*x^4 + (8*c^3*(7*b^2
+ 2*a*c)*x^5)/5 + 8*b*c^4*x^6 + (16*c^5*x^7)/7)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(106\) vs. \(2(41)=82\).

Time = 2.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.38

method	result	size
gosper	\(\frac {x \left (160 c^{5} x^{6}+560 b \,c^{4} x^{5}+224 x^{4} a \,c^{4}+784 b^{2} c^{3} x^{4}+560 a b \,c^{3} x^{3}+560 x^{3} b^{3} c^{2}+560 a \,b^{2} c^{2} x^{2}+210 b^{4} c \,x^{2}+280 a \,b^{3} c x +35 b^{5} x +70 a \,b^{4}\right ) d^{4}}{70}\)	\(107\)
norman	\(\left (\frac {16}{5} c^{4} d^{4} a +\frac {56}{5} b^{2} d^{4} c^{3}\right ) x^{5}+\left (4 b^{3} d^{4} c a +\frac {1}{2} b^{5} d^{4}\right ) x^{2}+\left (8 b \,c^{3} d^{4} a +8 b^{3} c^{2} d^{4}\right ) x^{4}+\left (8 b^{2} d^{4} c^{2} a +3 b^{4} d^{4} c \right ) x^{3}+b^{4} d^{4} a x +\frac {16 c^{5} d^{4} x^{7}}{7}+8 b \,c^{4} d^{4} x^{6}\)	\(134\)
default	\(\frac {16 c^{5} d^{4} x^{7}}{7}+8 b \,c^{4} d^{4} x^{6}+\frac {\left (16 c^{4} d^{4} a +56 b^{2} d^{4} c^{3}\right ) x^{5}}{5}+\frac {\left (32 b \,c^{3} d^{4} a +32 b^{3} c^{2} d^{4}\right ) x^{4}}{4}+\frac {\left (24 b^{2} d^{4} c^{2} a +9 b^{4} d^{4} c \right ) x^{3}}{3}+\frac {\left (8 b^{3} d^{4} c a +b^{5} d^{4}\right ) x^{2}}{2}+b^{4} d^{4} a x\)	\(137\)
risch	\(\frac {16}{7} c^{5} d^{4} x^{7}+8 b \,c^{4} d^{4} x^{6}+\frac {16}{5} d^{4} x^{5} a \,c^{4}+\frac {56}{5} d^{4} x^{5} b^{2} c^{3}+8 d^{4} a b \,c^{3} x^{4}+8 d^{4} c^{2} b^{3} x^{4}+8 d^{4} a \,b^{2} c^{2} x^{3}+3 d^{4} b^{4} c \,x^{3}+4 d^{4} a \,b^{3} c \,x^{2}+\frac {1}{2} d^{4} b^{5} x^{2}+b^{4} d^{4} a x\)	\(138\)
parallelrisch	\(\frac {16}{7} c^{5} d^{4} x^{7}+8 b \,c^{4} d^{4} x^{6}+\frac {16}{5} d^{4} x^{5} a \,c^{4}+\frac {56}{5} d^{4} x^{5} b^{2} c^{3}+8 d^{4} a b \,c^{3} x^{4}+8 d^{4} c^{2} b^{3} x^{4}+8 d^{4} a \,b^{2} c^{2} x^{3}+3 d^{4} b^{4} c \,x^{3}+4 d^{4} a \,b^{3} c \,x^{2}+\frac {1}{2} d^{4} b^{5} x^{2}+b^{4} d^{4} a x\)	\(138\)

[In]

int((2*c*d*x+b*d)^4*(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/70*x*(160*c^5*x^6+560*b*c^4*x^5+224*a*c^4*x^4+784*b^2*c^3*x^4+560*a*b*c^3*x^3+560*b^3*c^2*x^3+560*a*b^2*c^2*
x^2+210*b^4*c*x^2+280*a*b^3*c*x+35*b^5*x+70*a*b^4)*d^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (41) = 82\).

Time = 0.32 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.67 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {16}{7} \, c^{5} d^{4} x^{7} + 8 \, b c^{4} d^{4} x^{6} + a b^{4} d^{4} x + \frac {8}{5} \, {\left (7 \, b^{2} c^{3} + 2 \, a c^{4}\right )} d^{4} x^{5} + 8 \, {\left (b^{3} c^{2} + a b c^{3}\right )} d^{4} x^{4} + {\left (3 \, b^{4} c + 8 \, a b^{2} c^{2}\right )} d^{4} x^{3} + \frac {1}{2} \, {\left (b^{5} + 8 \, a b^{3} c\right )} d^{4} x^{2} \]

[In]

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

[Out]

16/7*c^5*d^4*x^7 + 8*b*c^4*d^4*x^6 + a*b^4*d^4*x + 8/5*(7*b^2*c^3 + 2*a*c^4)*d^4*x^5 + 8*(b^3*c^2 + a*b*c^3)*d
^4*x^4 + (3*b^4*c + 8*a*b^2*c^2)*d^4*x^3 + 1/2*(b^5 + 8*a*b^3*c)*d^4*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (41) = 82\).

Time = 0.03 (sec) , antiderivative size = 143, normalized size of antiderivative = 3.18 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right ) \, dx=a b^{4} d^{4} x + 8 b c^{4} d^{4} x^{6} + \frac {16 c^{5} d^{4} x^{7}}{7} + x^{5} \cdot \left (\frac {16 a c^{4} d^{4}}{5} + \frac {56 b^{2} c^{3} d^{4}}{5}\right ) + x^{4} \cdot \left (8 a b c^{3} d^{4} + 8 b^{3} c^{2} d^{4}\right ) + x^{3} \cdot \left (8 a b^{2} c^{2} d^{4} + 3 b^{4} c d^{4}\right ) + x^{2} \cdot \left (4 a b^{3} c d^{4} + \frac {b^{5} d^{4}}{2}\right ) \]

[In]

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

[Out]

a*b**4*d**4*x + 8*b*c**4*d**4*x**6 + 16*c**5*d**4*x**7/7 + x**5*(16*a*c**4*d**4/5 + 56*b**2*c**3*d**4/5) + x**
4*(8*a*b*c**3*d**4 + 8*b**3*c**2*d**4) + x**3*(8*a*b**2*c**2*d**4 + 3*b**4*c*d**4) + x**2*(4*a*b**3*c*d**4 + b
**5*d**4/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (41) = 82\).

Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.67 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {16}{7} \, c^{5} d^{4} x^{7} + 8 \, b c^{4} d^{4} x^{6} + a b^{4} d^{4} x + \frac {8}{5} \, {\left (7 \, b^{2} c^{3} + 2 \, a c^{4}\right )} d^{4} x^{5} + 8 \, {\left (b^{3} c^{2} + a b c^{3}\right )} d^{4} x^{4} + {\left (3 \, b^{4} c + 8 \, a b^{2} c^{2}\right )} d^{4} x^{3} + \frac {1}{2} \, {\left (b^{5} + 8 \, a b^{3} c\right )} d^{4} x^{2} \]

[In]

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

[Out]

16/7*c^5*d^4*x^7 + 8*b*c^4*d^4*x^6 + a*b^4*d^4*x + 8/5*(7*b^2*c^3 + 2*a*c^4)*d^4*x^5 + 8*(b^3*c^2 + a*b*c^3)*d
^4*x^4 + (3*b^4*c + 8*a*b^2*c^2)*d^4*x^3 + 1/2*(b^5 + 8*a*b^3*c)*d^4*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (41) = 82\).

Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 3.04 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {16}{7} \, c^{5} d^{4} x^{7} + 8 \, b c^{4} d^{4} x^{6} + \frac {56}{5} \, b^{2} c^{3} d^{4} x^{5} + \frac {16}{5} \, a c^{4} d^{4} x^{5} + 8 \, b^{3} c^{2} d^{4} x^{4} + 8 \, a b c^{3} d^{4} x^{4} + 3 \, b^{4} c d^{4} x^{3} + 8 \, a b^{2} c^{2} d^{4} x^{3} + \frac {1}{2} \, b^{5} d^{4} x^{2} + 4 \, a b^{3} c d^{4} x^{2} + a b^{4} d^{4} x \]

[In]

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

[Out]

16/7*c^5*d^4*x^7 + 8*b*c^4*d^4*x^6 + 56/5*b^2*c^3*d^4*x^5 + 16/5*a*c^4*d^4*x^5 + 8*b^3*c^2*d^4*x^4 + 8*a*b*c^3
*d^4*x^4 + 3*b^4*c*d^4*x^3 + 8*a*b^2*c^2*d^4*x^3 + 1/2*b^5*d^4*x^2 + 4*a*b^3*c*d^4*x^2 + a*b^4*d^4*x

Mupad [B] (verification not implemented)

Time = 10.00 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.51 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right ) \, dx=\frac {16\,c^5\,d^4\,x^7}{7}+\frac {b^3\,d^4\,x^2\,\left (b^2+8\,a\,c\right )}{2}+8\,b\,c^4\,d^4\,x^6+\frac {8\,c^3\,d^4\,x^5\,\left (7\,b^2+2\,a\,c\right )}{5}+a\,b^4\,d^4\,x+8\,b\,c^2\,d^4\,x^4\,\left (b^2+a\,c\right )+b^2\,c\,d^4\,x^3\,\left (3\,b^2+8\,a\,c\right ) \]

[In]

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

[Out]

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