### 3.1138 $$\int (b d+2 c d x)^4 (a+b x+c x^2)^3 \, dx$$

Optimal. Leaf size=101 $-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^9}{384 c^4}+\frac{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^7}{896 c^4}-\frac{d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^5}{640 c^4}+\frac{d^4 (b+2 c x)^{11}}{1408 c^4}$

[Out]

-((b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^5)/(640*c^4) + (3*(b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^7)/(896*c^4) - ((b^2 - 4*a
*c)*d^4*(b + 2*c*x)^9)/(384*c^4) + (d^4*(b + 2*c*x)^11)/(1408*c^4)

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Rubi [A]  time = 0.191634, antiderivative size = 101, normalized size of antiderivative = 1., 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 = {683} $-\frac{d^4 \left (b^2-4 a c\right ) (b+2 c x)^9}{384 c^4}+\frac{3 d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^7}{896 c^4}-\frac{d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^5}{640 c^4}+\frac{d^4 (b+2 c x)^{11}}{1408 c^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^3,x]

[Out]

-((b^2 - 4*a*c)^3*d^4*(b + 2*c*x)^5)/(640*c^4) + (3*(b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^7)/(896*c^4) - ((b^2 - 4*a
*c)*d^4*(b + 2*c*x)^9)/(384*c^4) + (d^4*(b + 2*c*x)^11)/(1408*c^4)

Rule 683

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*} \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3 (b d+2 c d x)^4}{64 c^3}+\frac{3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^6}{64 c^3 d^2}+\frac{3 \left (-b^2+4 a c\right ) (b d+2 c d x)^8}{64 c^3 d^4}+\frac{(b d+2 c d x)^{10}}{64 c^3 d^6}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^5}{640 c^4}+\frac{3 \left (b^2-4 a c\right )^2 d^4 (b+2 c x)^7}{896 c^4}-\frac{\left (b^2-4 a c\right ) d^4 (b+2 c x)^9}{384 c^4}+\frac{d^4 (b+2 c x)^{11}}{1408 c^4}\\ \end{align*}

Mathematica [B]  time = 0.042036, size = 259, normalized size = 2.56 $d^4 \left (\frac{3}{7} c^3 x^7 \left (16 a^2 c^2+104 a b^2 c+43 b^4\right )+\frac{1}{2} b c^2 x^6 \left (48 a^2 c^2+88 a b^2 c+17 b^4\right )+\frac{1}{5} c x^5 \left (168 a^2 b^2 c^2+16 a^3 c^3+123 a b^4 c+11 b^6\right )+\frac{1}{4} b x^4 \left (96 a^2 b^2 c^2+32 a^3 c^3+30 a b^4 c+b^6\right )+a b^2 x^3 \left (8 a^2 c^2+9 a b^2 c+b^4\right )+\frac{1}{2} a^2 b^3 x^2 \left (8 a c+3 b^2\right )+a^3 b^4 x+\frac{8}{3} c^5 x^9 \left (2 a c+7 b^2\right )+24 b c^4 x^8 \left (a c+b^2\right )+8 b c^6 x^{10}+\frac{16 c^7 x^{11}}{11}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

d^4*(a^3*b^4*x + (a^2*b^3*(3*b^2 + 8*a*c)*x^2)/2 + a*b^2*(b^4 + 9*a*b^2*c + 8*a^2*c^2)*x^3 + (b*(b^6 + 30*a*b^
4*c + 96*a^2*b^2*c^2 + 32*a^3*c^3)*x^4)/4 + (c*(11*b^6 + 123*a*b^4*c + 168*a^2*b^2*c^2 + 16*a^3*c^3)*x^5)/5 +
(b*c^2*(17*b^4 + 88*a*b^2*c + 48*a^2*c^2)*x^6)/2 + (3*c^3*(43*b^4 + 104*a*b^2*c + 16*a^2*c^2)*x^7)/7 + 24*b*c^
4*(b^2 + a*c)*x^8 + (8*c^5*(7*b^2 + 2*a*c)*x^9)/3 + 8*b*c^6*x^10 + (16*c^7*x^11)/11)

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Maple [B]  time = 0.038, size = 672, normalized size = 6.7 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16/11*c^7*d^4*x^11+8*b*d^4*c^6*x^10+1/9*(120*b^2*d^4*c^5+16*c^4*d^4*(a*c^2+2*b^2*c+c*(2*a*c+b^2)))*x^9+1/8*(80
*b^3*d^4*c^4+32*b*d^4*c^3*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+16*c^4*d^4*(4*a*b*c+b*(2*a*c+b^2)))*x^8+1/7*(25*b^4*d^
4*c^3+24*b^2*d^4*c^2*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+32*b*d^4*c^3*(4*a*b*c+b*(2*a*c+b^2))+16*c^4*d^4*(a*(2*a*c+b
^2)+2*b^2*a+a^2*c))*x^7+1/6*(3*b^5*d^4*c^2+8*b^3*d^4*c*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+24*b^2*d^4*c^2*(4*a*b*c+b
*(2*a*c+b^2))+32*b*d^4*c^3*(a*(2*a*c+b^2)+2*b^2*a+a^2*c)+48*c^4*d^4*b*a^2)*x^6+1/5*(b^4*d^4*(a*c^2+2*b^2*c+c*(
2*a*c+b^2))+8*b^3*d^4*c*(4*a*b*c+b*(2*a*c+b^2))+24*b^2*d^4*c^2*(a*(2*a*c+b^2)+2*b^2*a+a^2*c)+96*b^2*d^4*c^3*a^
2+16*c^4*d^4*a^3)*x^5+1/4*(b^4*d^4*(4*a*b*c+b*(2*a*c+b^2))+8*b^3*d^4*c*(a*(2*a*c+b^2)+2*b^2*a+a^2*c)+72*b^3*d^
4*c^2*a^2+32*b*d^4*c^3*a^3)*x^4+1/3*(b^4*d^4*(a*(2*a*c+b^2)+2*b^2*a+a^2*c)+24*b^4*d^4*c*a^2+24*b^2*d^4*c^2*a^3
)*x^3+1/2*(8*a^3*b^3*c*d^4+3*a^2*b^5*d^4)*x^2+b^4*d^4*a^3*x

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Maxima [B]  time = 1.16763, size = 392, normalized size = 3.88 \begin{align*} \frac{16}{11} \, c^{7} d^{4} x^{11} + 8 \, b c^{6} d^{4} x^{10} + \frac{8}{3} \,{\left (7 \, b^{2} c^{5} + 2 \, a c^{6}\right )} d^{4} x^{9} + 24 \,{\left (b^{3} c^{4} + a b c^{5}\right )} d^{4} x^{8} + a^{3} b^{4} d^{4} x + \frac{3}{7} \,{\left (43 \, b^{4} c^{3} + 104 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{7} + \frac{1}{2} \,{\left (17 \, b^{5} c^{2} + 88 \, a b^{3} c^{3} + 48 \, a^{2} b c^{4}\right )} d^{4} x^{6} + \frac{1}{5} \,{\left (11 \, b^{6} c + 123 \, a b^{4} c^{2} + 168 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}\right )} d^{4} x^{5} + \frac{1}{4} \,{\left (b^{7} + 30 \, a b^{5} c + 96 \, a^{2} b^{3} c^{2} + 32 \, a^{3} b c^{3}\right )} d^{4} x^{4} +{\left (a b^{6} + 9 \, a^{2} b^{4} c + 8 \, a^{3} b^{2} c^{2}\right )} d^{4} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b^{5} + 8 \, a^{3} b^{3} c\right )} d^{4} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16/11*c^7*d^4*x^11 + 8*b*c^6*d^4*x^10 + 8/3*(7*b^2*c^5 + 2*a*c^6)*d^4*x^9 + 24*(b^3*c^4 + a*b*c^5)*d^4*x^8 + a
^3*b^4*d^4*x + 3/7*(43*b^4*c^3 + 104*a*b^2*c^4 + 16*a^2*c^5)*d^4*x^7 + 1/2*(17*b^5*c^2 + 88*a*b^3*c^3 + 48*a^2
*b*c^4)*d^4*x^6 + 1/5*(11*b^6*c + 123*a*b^4*c^2 + 168*a^2*b^2*c^3 + 16*a^3*c^4)*d^4*x^5 + 1/4*(b^7 + 30*a*b^5*
c + 96*a^2*b^3*c^2 + 32*a^3*b*c^3)*d^4*x^4 + (a*b^6 + 9*a^2*b^4*c + 8*a^3*b^2*c^2)*d^4*x^3 + 1/2*(3*a^2*b^5 +
8*a^3*b^3*c)*d^4*x^2

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Fricas [B]  time = 1.70301, size = 776, normalized size = 7.68 \begin{align*} \frac{16}{11} x^{11} d^{4} c^{7} + 8 x^{10} d^{4} c^{6} b + \frac{56}{3} x^{9} d^{4} c^{5} b^{2} + \frac{16}{3} x^{9} d^{4} c^{6} a + 24 x^{8} d^{4} c^{4} b^{3} + 24 x^{8} d^{4} c^{5} b a + \frac{129}{7} x^{7} d^{4} c^{3} b^{4} + \frac{312}{7} x^{7} d^{4} c^{4} b^{2} a + \frac{48}{7} x^{7} d^{4} c^{5} a^{2} + \frac{17}{2} x^{6} d^{4} c^{2} b^{5} + 44 x^{6} d^{4} c^{3} b^{3} a + 24 x^{6} d^{4} c^{4} b a^{2} + \frac{11}{5} x^{5} d^{4} c b^{6} + \frac{123}{5} x^{5} d^{4} c^{2} b^{4} a + \frac{168}{5} x^{5} d^{4} c^{3} b^{2} a^{2} + \frac{16}{5} x^{5} d^{4} c^{4} a^{3} + \frac{1}{4} x^{4} d^{4} b^{7} + \frac{15}{2} x^{4} d^{4} c b^{5} a + 24 x^{4} d^{4} c^{2} b^{3} a^{2} + 8 x^{4} d^{4} c^{3} b a^{3} + x^{3} d^{4} b^{6} a + 9 x^{3} d^{4} c b^{4} a^{2} + 8 x^{3} d^{4} c^{2} b^{2} a^{3} + \frac{3}{2} x^{2} d^{4} b^{5} a^{2} + 4 x^{2} d^{4} c b^{3} a^{3} + x d^{4} b^{4} a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16/11*x^11*d^4*c^7 + 8*x^10*d^4*c^6*b + 56/3*x^9*d^4*c^5*b^2 + 16/3*x^9*d^4*c^6*a + 24*x^8*d^4*c^4*b^3 + 24*x^
8*d^4*c^5*b*a + 129/7*x^7*d^4*c^3*b^4 + 312/7*x^7*d^4*c^4*b^2*a + 48/7*x^7*d^4*c^5*a^2 + 17/2*x^6*d^4*c^2*b^5
+ 44*x^6*d^4*c^3*b^3*a + 24*x^6*d^4*c^4*b*a^2 + 11/5*x^5*d^4*c*b^6 + 123/5*x^5*d^4*c^2*b^4*a + 168/5*x^5*d^4*c
^3*b^2*a^2 + 16/5*x^5*d^4*c^4*a^3 + 1/4*x^4*d^4*b^7 + 15/2*x^4*d^4*c*b^5*a + 24*x^4*d^4*c^2*b^3*a^2 + 8*x^4*d^
4*c^3*b*a^3 + x^3*d^4*b^6*a + 9*x^3*d^4*c*b^4*a^2 + 8*x^3*d^4*c^2*b^2*a^3 + 3/2*x^2*d^4*b^5*a^2 + 4*x^2*d^4*c*
b^3*a^3 + x*d^4*b^4*a^3

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Sympy [B]  time = 0.126409, size = 371, normalized size = 3.67 \begin{align*} a^{3} b^{4} d^{4} x + 8 b c^{6} d^{4} x^{10} + \frac{16 c^{7} d^{4} x^{11}}{11} + x^{9} \left (\frac{16 a c^{6} d^{4}}{3} + \frac{56 b^{2} c^{5} d^{4}}{3}\right ) + x^{8} \left (24 a b c^{5} d^{4} + 24 b^{3} c^{4} d^{4}\right ) + x^{7} \left (\frac{48 a^{2} c^{5} d^{4}}{7} + \frac{312 a b^{2} c^{4} d^{4}}{7} + \frac{129 b^{4} c^{3} d^{4}}{7}\right ) + x^{6} \left (24 a^{2} b c^{4} d^{4} + 44 a b^{3} c^{3} d^{4} + \frac{17 b^{5} c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac{16 a^{3} c^{4} d^{4}}{5} + \frac{168 a^{2} b^{2} c^{3} d^{4}}{5} + \frac{123 a b^{4} c^{2} d^{4}}{5} + \frac{11 b^{6} c d^{4}}{5}\right ) + x^{4} \left (8 a^{3} b c^{3} d^{4} + 24 a^{2} b^{3} c^{2} d^{4} + \frac{15 a b^{5} c d^{4}}{2} + \frac{b^{7} d^{4}}{4}\right ) + x^{3} \left (8 a^{3} b^{2} c^{2} d^{4} + 9 a^{2} b^{4} c d^{4} + a b^{6} d^{4}\right ) + x^{2} \left (4 a^{3} b^{3} c d^{4} + \frac{3 a^{2} b^{5} d^{4}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*b**4*d**4*x + 8*b*c**6*d**4*x**10 + 16*c**7*d**4*x**11/11 + x**9*(16*a*c**6*d**4/3 + 56*b**2*c**5*d**4/3)
+ x**8*(24*a*b*c**5*d**4 + 24*b**3*c**4*d**4) + x**7*(48*a**2*c**5*d**4/7 + 312*a*b**2*c**4*d**4/7 + 129*b**4
*c**3*d**4/7) + x**6*(24*a**2*b*c**4*d**4 + 44*a*b**3*c**3*d**4 + 17*b**5*c**2*d**4/2) + x**5*(16*a**3*c**4*d*
*4/5 + 168*a**2*b**2*c**3*d**4/5 + 123*a*b**4*c**2*d**4/5 + 11*b**6*c*d**4/5) + x**4*(8*a**3*b*c**3*d**4 + 24*
a**2*b**3*c**2*d**4 + 15*a*b**5*c*d**4/2 + b**7*d**4/4) + x**3*(8*a**3*b**2*c**2*d**4 + 9*a**2*b**4*c*d**4 + a
*b**6*d**4) + x**2*(4*a**3*b**3*c*d**4 + 3*a**2*b**5*d**4/2)

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Giac [B]  time = 1.28277, size = 487, normalized size = 4.82 \begin{align*} \frac{16}{11} \, c^{7} d^{4} x^{11} + 8 \, b c^{6} d^{4} x^{10} + \frac{56}{3} \, b^{2} c^{5} d^{4} x^{9} + \frac{16}{3} \, a c^{6} d^{4} x^{9} + 24 \, b^{3} c^{4} d^{4} x^{8} + 24 \, a b c^{5} d^{4} x^{8} + \frac{129}{7} \, b^{4} c^{3} d^{4} x^{7} + \frac{312}{7} \, a b^{2} c^{4} d^{4} x^{7} + \frac{48}{7} \, a^{2} c^{5} d^{4} x^{7} + \frac{17}{2} \, b^{5} c^{2} d^{4} x^{6} + 44 \, a b^{3} c^{3} d^{4} x^{6} + 24 \, a^{2} b c^{4} d^{4} x^{6} + \frac{11}{5} \, b^{6} c d^{4} x^{5} + \frac{123}{5} \, a b^{4} c^{2} d^{4} x^{5} + \frac{168}{5} \, a^{2} b^{2} c^{3} d^{4} x^{5} + \frac{16}{5} \, a^{3} c^{4} d^{4} x^{5} + \frac{1}{4} \, b^{7} d^{4} x^{4} + \frac{15}{2} \, a b^{5} c d^{4} x^{4} + 24 \, a^{2} b^{3} c^{2} d^{4} x^{4} + 8 \, a^{3} b c^{3} d^{4} x^{4} + a b^{6} d^{4} x^{3} + 9 \, a^{2} b^{4} c d^{4} x^{3} + 8 \, a^{3} b^{2} c^{2} d^{4} x^{3} + \frac{3}{2} \, a^{2} b^{5} d^{4} x^{2} + 4 \, a^{3} b^{3} c d^{4} x^{2} + a^{3} b^{4} d^{4} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16/11*c^7*d^4*x^11 + 8*b*c^6*d^4*x^10 + 56/3*b^2*c^5*d^4*x^9 + 16/3*a*c^6*d^4*x^9 + 24*b^3*c^4*d^4*x^8 + 24*a*
b*c^5*d^4*x^8 + 129/7*b^4*c^3*d^4*x^7 + 312/7*a*b^2*c^4*d^4*x^7 + 48/7*a^2*c^5*d^4*x^7 + 17/2*b^5*c^2*d^4*x^6
+ 44*a*b^3*c^3*d^4*x^6 + 24*a^2*b*c^4*d^4*x^6 + 11/5*b^6*c*d^4*x^5 + 123/5*a*b^4*c^2*d^4*x^5 + 168/5*a^2*b^2*c
^3*d^4*x^5 + 16/5*a^3*c^4*d^4*x^5 + 1/4*b^7*d^4*x^4 + 15/2*a*b^5*c*d^4*x^4 + 24*a^2*b^3*c^2*d^4*x^4 + 8*a^3*b*
c^3*d^4*x^4 + a*b^6*d^4*x^3 + 9*a^2*b^4*c*d^4*x^3 + 8*a^3*b^2*c^2*d^4*x^3 + 3/2*a^2*b^5*d^4*x^2 + 4*a^3*b^3*c*
d^4*x^2 + a^3*b^4*d^4*x