\(\int (b d+2 c d x)^2 (a+b x+c x^2)^2 \, dx\) [1124]

Optimal result

Integrand size = 24, antiderivative size = 73 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^2 \, dx=\frac {\left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3}{96 c^3}-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^5}{80 c^3}+\frac {d^2 (b+2 c x)^7}{224 c^3} \]

[Out]

1/96*(-4*a*c+b^2)^2*d^2*(2*c*x+b)^3/c^3-1/80*(-4*a*c+b^2)*d^2*(2*c*x+b)^5/c^3+1/224*d^2*(2*c*x+b)^7/c^3

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^2 \, dx=-\frac {d^2 \left (b^2-4 a c\right ) (b+2 c x)^5}{80 c^3}+\frac {d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3}{96 c^3}+\frac {d^2 (b+2 c x)^7}{224 c^3} \]

[In]

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

[Out]

((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)^3)/(96*c^3) - ((b^2 - 4*a*c)*d^2*(b + 2*c*x)^5)/(80*c^3) + (d^2*(b + 2*c*x)^7
)/(224*c^3)

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 )^2 (b d+2 c d x)^2}{16 c^2}+\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^4}{8 c^2 d^2}+\frac {(b d+2 c d x)^6}{16 c^2 d^4}\right ) \, dx \\ & = \frac {\left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3}{96 c^3}-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^5}{80 c^3}+\frac {d^2 (b+2 c x)^7}{224 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.52 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^2 \, dx=d^2 \left (a^2 b^2 x+a b \left (b^2+2 a c\right ) x^2+\frac {1}{3} \left (b^4+10 a b^2 c+4 a^2 c^2\right ) x^3+\frac {1}{2} b c \left (3 b^2+8 a c\right ) x^4+\frac {1}{5} c^2 \left (13 b^2+8 a c\right ) x^5+2 b c^3 x^6+\frac {4 c^4 x^7}{7}\right ) \]

[In]

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

[Out]

d^2*(a^2*b^2*x + a*b*(b^2 + 2*a*c)*x^2 + ((b^4 + 10*a*b^2*c + 4*a^2*c^2)*x^3)/3 + (b*c*(3*b^2 + 8*a*c)*x^4)/2
+ (c^2*(13*b^2 + 8*a*c)*x^5)/5 + 2*b*c^3*x^6 + (4*c^4*x^7)/7)

Maple [A] (verified)

Time = 2.74 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.59

method	result	size
gosper	\(\frac {x \left (120 c^{4} x^{6}+420 b \,c^{3} x^{5}+336 x^{4} a \,c^{3}+546 b^{2} c^{2} x^{4}+840 a b \,c^{2} x^{3}+315 b^{3} c \,x^{3}+280 a^{2} c^{2} x^{2}+700 a \,b^{2} c \,x^{2}+70 b^{4} x^{2}+420 a^{2} b c x +210 a \,b^{3} x +210 a^{2} b^{2}\right ) d^{2}}{210}\)	\(116\)
norman	\(\left (\frac {8}{5} a \,c^{3} d^{2}+\frac {13}{5} b^{2} c^{2} d^{2}\right ) x^{5}+\left (4 a b \,c^{2} d^{2}+\frac {3}{2} b^{3} c \,d^{2}\right ) x^{4}+\left (\frac {4}{3} a^{2} c^{2} d^{2}+\frac {10}{3} a \,b^{2} c \,d^{2}+\frac {1}{3} b^{4} d^{2}\right ) x^{3}+\left (2 a^{2} b c \,d^{2}+b^{3} d^{2} a \right ) x^{2}+b^{2} d^{2} a^{2} x +\frac {4 c^{4} d^{2} x^{7}}{7}+2 b \,c^{3} d^{2} x^{6}\)	\(142\)
risch	\(\frac {4}{7} c^{4} d^{2} x^{7}+2 b \,c^{3} d^{2} x^{6}+\frac {8}{5} d^{2} a \,c^{3} x^{5}+\frac {13}{5} d^{2} x^{5} b^{2} c^{2}+4 d^{2} x^{4} b \,c^{2} a +\frac {3}{2} b^{3} c \,d^{2} x^{4}+\frac {4}{3} a^{2} c^{2} d^{2} x^{3}+\frac {10}{3} a \,b^{2} c \,d^{2} x^{3}+\frac {1}{3} d^{2} b^{4} x^{3}+2 a^{2} b c \,d^{2} x^{2}+d^{2} a \,b^{3} x^{2}+b^{2} d^{2} a^{2} x\)	\(149\)
parallelrisch	\(\frac {4}{7} c^{4} d^{2} x^{7}+2 b \,c^{3} d^{2} x^{6}+\frac {8}{5} d^{2} a \,c^{3} x^{5}+\frac {13}{5} d^{2} x^{5} b^{2} c^{2}+4 d^{2} x^{4} b \,c^{2} a +\frac {3}{2} b^{3} c \,d^{2} x^{4}+\frac {4}{3} a^{2} c^{2} d^{2} x^{3}+\frac {10}{3} a \,b^{2} c \,d^{2} x^{3}+\frac {1}{3} d^{2} b^{4} x^{3}+2 a^{2} b c \,d^{2} x^{2}+d^{2} a \,b^{3} x^{2}+b^{2} d^{2} a^{2} x\)	\(149\)
default	\(\frac {4 c^{4} d^{2} x^{7}}{7}+2 b \,c^{3} d^{2} x^{6}+\frac {\left (9 b^{2} c^{2} d^{2}+4 c^{2} d^{2} \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 b^{3} c \,d^{2}+4 b c \,d^{2} \left (2 a c +b^{2}\right )+8 a b \,c^{2} d^{2}\right ) x^{4}}{4}+\frac {\left (b^{2} d^{2} \left (2 a c +b^{2}\right )+8 a \,b^{2} c \,d^{2}+4 a^{2} c^{2} d^{2}\right ) x^{3}}{3}+\frac {\left (4 a^{2} b c \,d^{2}+2 b^{3} d^{2} a \right ) x^{2}}{2}+b^{2} d^{2} a^{2} x\)	\(176\)

[In]

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

[Out]

1/210*x*(120*c^4*x^6+420*b*c^3*x^5+336*a*c^3*x^4+546*b^2*c^2*x^4+840*a*b*c^2*x^3+315*b^3*c*x^3+280*a^2*c^2*x^2
+700*a*b^2*c*x^2+70*b^4*x^2+420*a^2*b*c*x+210*a*b^3*x+210*a^2*b^2)*d^2

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.74 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^2 \, dx=\frac {4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac {1}{5} \, {\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{5} + a^{2} b^{2} d^{2} x + \frac {1}{2} \, {\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2} x^{4} + \frac {1}{3} \, {\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x^{3} + {\left (a b^{3} + 2 \, a^{2} b c\right )} d^{2} x^{2} \]

[In]

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

[Out]

4/7*c^4*d^2*x^7 + 2*b*c^3*d^2*x^6 + 1/5*(13*b^2*c^2 + 8*a*c^3)*d^2*x^5 + a^2*b^2*d^2*x + 1/2*(3*b^3*c + 8*a*b*
c^2)*d^2*x^4 + 1/3*(b^4 + 10*a*b^2*c + 4*a^2*c^2)*d^2*x^3 + (a*b^3 + 2*a^2*b*c)*d^2*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (68) = 136\).

Time = 0.03 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.14 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^2 \, dx=a^{2} b^{2} d^{2} x + 2 b c^{3} d^{2} x^{6} + \frac {4 c^{4} d^{2} x^{7}}{7} + x^{5} \cdot \left (\frac {8 a c^{3} d^{2}}{5} + \frac {13 b^{2} c^{2} d^{2}}{5}\right ) + x^{4} \cdot \left (4 a b c^{2} d^{2} + \frac {3 b^{3} c d^{2}}{2}\right ) + x^{3} \cdot \left (\frac {4 a^{2} c^{2} d^{2}}{3} + \frac {10 a b^{2} c d^{2}}{3} + \frac {b^{4} d^{2}}{3}\right ) + x^{2} \cdot \left (2 a^{2} b c d^{2} + a b^{3} d^{2}\right ) \]

[In]

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

[Out]

a**2*b**2*d**2*x + 2*b*c**3*d**2*x**6 + 4*c**4*d**2*x**7/7 + x**5*(8*a*c**3*d**2/5 + 13*b**2*c**2*d**2/5) + x*
*4*(4*a*b*c**2*d**2 + 3*b**3*c*d**2/2) + x**3*(4*a**2*c**2*d**2/3 + 10*a*b**2*c*d**2/3 + b**4*d**2/3) + x**2*(
2*a**2*b*c*d**2 + a*b**3*d**2)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.74 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^2 \, dx=\frac {4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac {1}{5} \, {\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{5} + a^{2} b^{2} d^{2} x + \frac {1}{2} \, {\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2} x^{4} + \frac {1}{3} \, {\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x^{3} + {\left (a b^{3} + 2 \, a^{2} b c\right )} d^{2} x^{2} \]

[In]

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

[Out]

4/7*c^4*d^2*x^7 + 2*b*c^3*d^2*x^6 + 1/5*(13*b^2*c^2 + 8*a*c^3)*d^2*x^5 + a^2*b^2*d^2*x + 1/2*(3*b^3*c + 8*a*b*
c^2)*d^2*x^4 + 1/3*(b^4 + 10*a*b^2*c + 4*a^2*c^2)*d^2*x^3 + (a*b^3 + 2*a^2*b*c)*d^2*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (67) = 134\).

Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.03 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^2 \, dx=\frac {4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac {13}{5} \, b^{2} c^{2} d^{2} x^{5} + \frac {8}{5} \, a c^{3} d^{2} x^{5} + \frac {3}{2} \, b^{3} c d^{2} x^{4} + 4 \, a b c^{2} d^{2} x^{4} + \frac {1}{3} \, b^{4} d^{2} x^{3} + \frac {10}{3} \, a b^{2} c d^{2} x^{3} + \frac {4}{3} \, a^{2} c^{2} d^{2} x^{3} + a b^{3} d^{2} x^{2} + 2 \, a^{2} b c d^{2} x^{2} + a^{2} b^{2} d^{2} x \]

[In]

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

[Out]

4/7*c^4*d^2*x^7 + 2*b*c^3*d^2*x^6 + 13/5*b^2*c^2*d^2*x^5 + 8/5*a*c^3*d^2*x^5 + 3/2*b^3*c*d^2*x^4 + 4*a*b*c^2*d
^2*x^4 + 1/3*b^4*d^2*x^3 + 10/3*a*b^2*c*d^2*x^3 + 4/3*a^2*c^2*d^2*x^3 + a*b^3*d^2*x^2 + 2*a^2*b*c*d^2*x^2 + a^
2*b^2*d^2*x

Mupad [B] (verification not implemented)

Time = 9.91 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.64 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^2 \, dx=\frac {4\,c^4\,d^2\,x^7}{7}+\frac {d^2\,x^3\,\left (4\,a^2\,c^2+10\,a\,b^2\,c+b^4\right )}{3}+a^2\,b^2\,d^2\,x+2\,b\,c^3\,d^2\,x^6+\frac {c^2\,d^2\,x^5\,\left (13\,b^2+8\,a\,c\right )}{5}+a\,b\,d^2\,x^2\,\left (b^2+2\,a\,c\right )+\frac {b\,c\,d^2\,x^4\,\left (3\,b^2+8\,a\,c\right )}{2} \]

[In]

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

[Out]

(4*c^4*d^2*x^7)/7 + (d^2*x^3*(b^4 + 4*a^2*c^2 + 10*a*b^2*c))/3 + a^2*b^2*d^2*x + 2*b*c^3*d^2*x^6 + (c^2*d^2*x^
5*(8*a*c + 13*b^2))/5 + a*b*d^2*x^2*(2*a*c + b^2) + (b*c*d^2*x^4*(8*a*c + 3*b^2))/2