3.12.39 \(\int (b d+2 c d x)^3 (a+b x+c x^2)^3 \, dx\) [1139]

Optimal. Leaf size=55 \[ \frac {1}{20} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^4+\frac {1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4 \]

[Out]

1/20*(-4*a*c+b^2)*d^3*(c*x^2+b*x+a)^4+1/5*d^3*(2*c*x+b)^2*(c*x^2+b*x+a)^4

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Rubi [A]
time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {706, 643} \begin {gather*} \frac {1}{20} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4+\frac {1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^3*(a + b*x + c*x^2)^3,x]

[Out]

((b^2 - 4*a*c)*d^3*(a + b*x + c*x^2)^4)/20 + (d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^4)/5

Rule 643

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*((a + b*x + c*x^2)^(p +
1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 706

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*d*(d + e*x)^(m - 1
)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Dist[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rubi steps

\begin {align*} \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^3 \, dx &=\frac {1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4+\frac {1}{5} \left (\left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x) \left (a+b x+c x^2\right )^3 \, dx\\ &=\frac {1}{20} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^4+\frac {1}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^4\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(132\) vs. \(2(55)=110\).
time = 0.02, size = 132, normalized size = 2.40 \begin {gather*} \frac {1}{20} d^3 x (b+c x) \left (20 a^3 \left (b^2+2 b c x+2 c^2 x^2\right )+20 a x^2 (b+c x)^2 \left (b^2+3 b c x+3 c^2 x^2\right )+x^3 (b+c x)^3 \left (5 b^2+16 b c x+16 c^2 x^2\right )+10 a^2 x \left (3 b^3+11 b^2 c x+16 b c^2 x^2+8 c^3 x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*x*(b + c*x)*(20*a^3*(b^2 + 2*b*c*x + 2*c^2*x^2) + 20*a*x^2*(b + c*x)^2*(b^2 + 3*b*c*x + 3*c^2*x^2) + x^3*
(b + c*x)^3*(5*b^2 + 16*b*c*x + 16*c^2*x^2) + 10*a^2*x*(3*b^3 + 11*b^2*c*x + 16*b*c^2*x^2 + 8*c^3*x^3)))/20

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Maple [A]
time = 0.65, size = 46, normalized size = 0.84

method	result	size
default	\(-d^{3} \left (-\frac {4 c \left (c \,x^{2}+b x +a \right )^{5}}{5}+\frac {\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{4}}{4}\right )\)	\(46\)
gosper	\(\frac {x \left (16 c^{6} x^{9}+80 b \,c^{5} x^{8}+60 x^{7} c^{5} a +165 x^{7} b^{2} c^{4}+240 a b \,c^{4} x^{6}+180 b^{3} c^{3} x^{6}+80 x^{5} a^{2} c^{4}+380 x^{5} a \,b^{2} c^{3}+110 x^{5} b^{4} c^{2}+240 a^{2} b \,c^{3} x^{4}+300 x^{4} a \,b^{3} c^{2}+36 x^{4} b^{5} c +40 x^{3} a^{3} c^{3}+270 x^{3} a^{2} b^{2} c^{2}+120 a \,b^{4} c \,x^{3}+5 x^{3} b^{6}+80 a^{3} b \,c^{2} x^{2}+140 a^{2} b^{3} c \,x^{2}+20 a \,b^{5} x^{2}+60 x \,a^{3} b^{2} c +30 a^{2} b^{4} x +20 a^{3} b^{3}\right ) d^{3}}{20}\)	\(236\)
norman	\(\left (3 a \,c^{5} d^{3}+\frac {33}{4} b^{2} c^{4} d^{3}\right ) x^{8}+\left (3 a^{3} b^{2} c \,d^{3}+\frac {3}{2} a^{2} b^{4} d^{3}\right ) x^{2}+\left (4 a^{2} c^{4} d^{3}+19 a \,b^{2} c^{3} d^{3}+\frac {11}{2} b^{4} c^{2} d^{3}\right ) x^{6}+\left (12 a^{2} b \,c^{3} d^{3}+15 a \,b^{3} c^{2} d^{3}+\frac {9}{5} b^{5} c \,d^{3}\right ) x^{5}+\left (2 a^{3} c^{3} d^{3}+\frac {27}{2} a^{2} b^{2} c^{2} d^{3}+6 a \,b^{4} c \,d^{3}+\frac {1}{4} b^{6} d^{3}\right ) x^{4}+\left (12 a b \,c^{4} d^{3}+9 b^{3} c^{3} d^{3}\right ) x^{7}+\left (4 a^{3} b \,d^{3} c^{2}+7 a^{2} b^{3} c \,d^{3}+a \,b^{5} d^{3}\right ) x^{3}+a^{3} b^{3} d^{3} x +\frac {4 c^{6} d^{3} x^{10}}{5}+4 b \,c^{5} d^{3} x^{9}\)	\(277\)
risch	\(4 b \,c^{5} d^{3} x^{9}+6 d^{3} a \,b^{4} c \,x^{4}+a^{3} b^{3} d^{3} x +\frac {4}{5} c^{6} d^{3} x^{10}+\frac {9}{5} d^{3} b^{5} c \,x^{5}+\frac {1}{4} d^{3} a^{4} b^{2}-\frac {1}{5} d^{3} c \,a^{5}+d^{3} a \,b^{5} x^{3}+\frac {3}{2} d^{3} a^{2} b^{4} x^{2}+\frac {1}{4} d^{3} b^{6} x^{4}+3 d^{3} a \,c^{5} x^{8}+\frac {33}{4} d^{3} b^{2} c^{4} x^{8}+9 d^{3} b^{3} c^{3} x^{7}+4 d^{3} a^{2} c^{4} x^{6}+\frac {11}{2} d^{3} b^{4} c^{2} x^{6}+2 d^{3} a^{3} c^{3} x^{4}+4 d^{3} a^{3} b \,c^{2} x^{3}+7 d^{3} a^{2} b^{3} c \,x^{3}+3 d^{3} a^{3} b^{2} c \,x^{2}+12 d^{3} a b \,c^{4} x^{7}+19 d^{3} a \,b^{2} c^{3} x^{6}+12 d^{3} a^{2} b \,c^{3} x^{5}+15 d^{3} a \,b^{3} c^{2} x^{5}+\frac {27}{2} d^{3} a^{2} b^{2} c^{2} x^{4}\)	\(319\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^3*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (51) = 102\).
time = 0.26, size = 243, normalized size = 4.42 \begin {gather*} \frac {4}{5} \, c^{6} d^{3} x^{10} + 4 \, b c^{5} d^{3} x^{9} + \frac {3}{4} \, {\left (11 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{3} x^{8} + 3 \, {\left (3 \, b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{3} x^{7} + a^{3} b^{3} d^{3} x + \frac {1}{2} \, {\left (11 \, b^{4} c^{2} + 38 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d^{3} x^{6} + \frac {3}{5} \, {\left (3 \, b^{5} c + 25 \, a b^{3} c^{2} + 20 \, a^{2} b c^{3}\right )} d^{3} x^{5} + \frac {1}{4} \, {\left (b^{6} + 24 \, a b^{4} c + 54 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d^{3} x^{4} + {\left (a b^{5} + 7 \, a^{2} b^{3} c + 4 \, a^{3} b c^{2}\right )} d^{3} x^{3} + \frac {3}{2} \, {\left (a^{2} b^{4} + 2 \, a^{3} b^{2} c\right )} d^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*c^6*d^3*x^10 + 4*b*c^5*d^3*x^9 + 3/4*(11*b^2*c^4 + 4*a*c^5)*d^3*x^8 + 3*(3*b^3*c^3 + 4*a*b*c^4)*d^3*x^7 +
a^3*b^3*d^3*x + 1/2*(11*b^4*c^2 + 38*a*b^2*c^3 + 8*a^2*c^4)*d^3*x^6 + 3/5*(3*b^5*c + 25*a*b^3*c^2 + 20*a^2*b*c
^3)*d^3*x^5 + 1/4*(b^6 + 24*a*b^4*c + 54*a^2*b^2*c^2 + 8*a^3*c^3)*d^3*x^4 + (a*b^5 + 7*a^2*b^3*c + 4*a^3*b*c^2
)*d^3*x^3 + 3/2*(a^2*b^4 + 2*a^3*b^2*c)*d^3*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (51) = 102\).
time = 3.32, size = 243, normalized size = 4.42 \begin {gather*} \frac {4}{5} \, c^{6} d^{3} x^{10} + 4 \, b c^{5} d^{3} x^{9} + \frac {3}{4} \, {\left (11 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{3} x^{8} + 3 \, {\left (3 \, b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{3} x^{7} + a^{3} b^{3} d^{3} x + \frac {1}{2} \, {\left (11 \, b^{4} c^{2} + 38 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d^{3} x^{6} + \frac {3}{5} \, {\left (3 \, b^{5} c + 25 \, a b^{3} c^{2} + 20 \, a^{2} b c^{3}\right )} d^{3} x^{5} + \frac {1}{4} \, {\left (b^{6} + 24 \, a b^{4} c + 54 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d^{3} x^{4} + {\left (a b^{5} + 7 \, a^{2} b^{3} c + 4 \, a^{3} b c^{2}\right )} d^{3} x^{3} + \frac {3}{2} \, {\left (a^{2} b^{4} + 2 \, a^{3} b^{2} c\right )} d^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*c^6*d^3*x^10 + 4*b*c^5*d^3*x^9 + 3/4*(11*b^2*c^4 + 4*a*c^5)*d^3*x^8 + 3*(3*b^3*c^3 + 4*a*b*c^4)*d^3*x^7 +
a^3*b^3*d^3*x + 1/2*(11*b^4*c^2 + 38*a*b^2*c^3 + 8*a^2*c^4)*d^3*x^6 + 3/5*(3*b^5*c + 25*a*b^3*c^2 + 20*a^2*b*c
^3)*d^3*x^5 + 1/4*(b^6 + 24*a*b^4*c + 54*a^2*b^2*c^2 + 8*a^3*c^3)*d^3*x^4 + (a*b^5 + 7*a^2*b^3*c + 4*a^3*b*c^2
)*d^3*x^3 + 3/2*(a^2*b^4 + 2*a^3*b^2*c)*d^3*x^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (49) = 98\).
time = 0.03, size = 299, normalized size = 5.44 \begin {gather*} a^{3} b^{3} d^{3} x + 4 b c^{5} d^{3} x^{9} + \frac {4 c^{6} d^{3} x^{10}}{5} + x^{8} \cdot \left (3 a c^{5} d^{3} + \frac {33 b^{2} c^{4} d^{3}}{4}\right ) + x^{7} \cdot \left (12 a b c^{4} d^{3} + 9 b^{3} c^{3} d^{3}\right ) + x^{6} \cdot \left (4 a^{2} c^{4} d^{3} + 19 a b^{2} c^{3} d^{3} + \frac {11 b^{4} c^{2} d^{3}}{2}\right ) + x^{5} \cdot \left (12 a^{2} b c^{3} d^{3} + 15 a b^{3} c^{2} d^{3} + \frac {9 b^{5} c d^{3}}{5}\right ) + x^{4} \cdot \left (2 a^{3} c^{3} d^{3} + \frac {27 a^{2} b^{2} c^{2} d^{3}}{2} + 6 a b^{4} c d^{3} + \frac {b^{6} d^{3}}{4}\right ) + x^{3} \cdot \left (4 a^{3} b c^{2} d^{3} + 7 a^{2} b^{3} c d^{3} + a b^{5} d^{3}\right ) + x^{2} \cdot \left (3 a^{3} b^{2} c d^{3} + \frac {3 a^{2} b^{4} d^{3}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*b**3*d**3*x + 4*b*c**5*d**3*x**9 + 4*c**6*d**3*x**10/5 + x**8*(3*a*c**5*d**3 + 33*b**2*c**4*d**3/4) + x**
7*(12*a*b*c**4*d**3 + 9*b**3*c**3*d**3) + x**6*(4*a**2*c**4*d**3 + 19*a*b**2*c**3*d**3 + 11*b**4*c**2*d**3/2)
+ x**5*(12*a**2*b*c**3*d**3 + 15*a*b**3*c**2*d**3 + 9*b**5*c*d**3/5) + x**4*(2*a**3*c**3*d**3 + 27*a**2*b**2*c
**2*d**3/2 + 6*a*b**4*c*d**3 + b**6*d**3/4) + x**3*(4*a**3*b*c**2*d**3 + 7*a**2*b**3*c*d**3 + a*b**5*d**3) + x
**2*(3*a**3*b**2*c*d**3 + 3*a**2*b**4*d**3/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (51) = 102\).
time = 1.48, size = 171, normalized size = 3.11 \begin {gather*} {\left (c d x^{2} + b d x\right )} a^{3} b^{2} d^{2} + \frac {30 \, {\left (c d x^{2} + b d x\right )}^{2} a^{2} b^{2} d^{3} + 40 \, {\left (c d x^{2} + b d x\right )}^{2} a^{3} c d^{3} + 20 \, {\left (c d x^{2} + b d x\right )}^{3} a b^{2} d^{2} + 80 \, {\left (c d x^{2} + b d x\right )}^{3} a^{2} c d^{2} + 5 \, {\left (c d x^{2} + b d x\right )}^{4} b^{2} d + 60 \, {\left (c d x^{2} + b d x\right )}^{4} a c d + 16 \, {\left (c d x^{2} + b d x\right )}^{5} c}{20 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*d*x^2 + b*d*x)*a^3*b^2*d^2 + 1/20*(30*(c*d*x^2 + b*d*x)^2*a^2*b^2*d^3 + 40*(c*d*x^2 + b*d*x)^2*a^3*c*d^3 +
20*(c*d*x^2 + b*d*x)^3*a*b^2*d^2 + 80*(c*d*x^2 + b*d*x)^3*a^2*c*d^2 + 5*(c*d*x^2 + b*d*x)^4*b^2*d + 60*(c*d*x^
2 + b*d*x)^4*a*c*d + 16*(c*d*x^2 + b*d*x)^5*c)/d^2

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Mupad [B]
time = 0.48, size = 229, normalized size = 4.16 \begin {gather*} \frac {d^3\,x^4\,\left (8\,a^3\,c^3+54\,a^2\,b^2\,c^2+24\,a\,b^4\,c+b^6\right )}{4}+\frac {4\,c^6\,d^3\,x^{10}}{5}+\frac {c^2\,d^3\,x^6\,\left (8\,a^2\,c^2+38\,a\,b^2\,c+11\,b^4\right )}{2}+a^3\,b^3\,d^3\,x+4\,b\,c^5\,d^3\,x^9+\frac {3\,c^4\,d^3\,x^8\,\left (11\,b^2+4\,a\,c\right )}{4}+\frac {3\,b\,c\,d^3\,x^5\,\left (20\,a^2\,c^2+25\,a\,b^2\,c+3\,b^4\right )}{5}+a\,b\,d^3\,x^3\,\left (4\,a^2\,c^2+7\,a\,b^2\,c+b^4\right )+\frac {3\,a^2\,b^2\,d^3\,x^2\,\left (b^2+2\,a\,c\right )}{2}+3\,b\,c^3\,d^3\,x^7\,\left (3\,b^2+4\,a\,c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d^3*x^4*(b^6 + 8*a^3*c^3 + 54*a^2*b^2*c^2 + 24*a*b^4*c))/4 + (4*c^6*d^3*x^10)/5 + (c^2*d^3*x^6*(11*b^4 + 8*a^
2*c^2 + 38*a*b^2*c))/2 + a^3*b^3*d^3*x + 4*b*c^5*d^3*x^9 + (3*c^4*d^3*x^8*(4*a*c + 11*b^2))/4 + (3*b*c*d^3*x^5
*(3*b^4 + 20*a^2*c^2 + 25*a*b^2*c))/5 + a*b*d^3*x^3*(b^4 + 4*a^2*c^2 + 7*a*b^2*c) + (3*a^2*b^2*d^3*x^2*(2*a*c
+ b^2))/2 + 3*b*c^3*d^3*x^7*(4*a*c + 3*b^2)

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