### 3.1137 $$\int (b d+2 c d x)^5 (a+b x+c x^2)^3 \, dx$$

Optimal. Leaf size=101 $-\frac{3 d^5 \left (b^2-4 a c\right ) (b+2 c x)^{10}}{1280 c^4}+\frac{3 d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^8}{1024 c^4}-\frac{d^5 \left (b^2-4 a c\right )^3 (b+2 c x)^6}{768 c^4}+\frac{d^5 (b+2 c x)^{12}}{1536 c^4}$

[Out]

-((b^2 - 4*a*c)^3*d^5*(b + 2*c*x)^6)/(768*c^4) + (3*(b^2 - 4*a*c)^2*d^5*(b + 2*c*x)^8)/(1024*c^4) - (3*(b^2 -
4*a*c)*d^5*(b + 2*c*x)^10)/(1280*c^4) + (d^5*(b + 2*c*x)^12)/(1536*c^4)

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Rubi [A]  time = 0.244876, 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} $-\frac{3 d^5 \left (b^2-4 a c\right ) (b+2 c x)^{10}}{1280 c^4}+\frac{3 d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^8}{1024 c^4}-\frac{d^5 \left (b^2-4 a c\right )^3 (b+2 c x)^6}{768 c^4}+\frac{d^5 (b+2 c x)^{12}}{1536 c^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^3,x]

[Out]

-((b^2 - 4*a*c)^3*d^5*(b + 2*c*x)^6)/(768*c^4) + (3*(b^2 - 4*a*c)^2*d^5*(b + 2*c*x)^8)/(1024*c^4) - (3*(b^2 -
4*a*c)*d^5*(b + 2*c*x)^10)/(1280*c^4) + (d^5*(b + 2*c*x)^12)/(1536*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)^5 \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)^5}{64 c^3}+\frac{3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^7}{64 c^3 d^2}+\frac{3 \left (-b^2+4 a c\right ) (b d+2 c d x)^9}{64 c^3 d^4}+\frac{(b d+2 c d x)^{11}}{64 c^3 d^6}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^3 d^5 (b+2 c x)^6}{768 c^4}+\frac{3 \left (b^2-4 a c\right )^2 d^5 (b+2 c x)^8}{1024 c^4}-\frac{3 \left (b^2-4 a c\right ) d^5 (b+2 c x)^{10}}{1280 c^4}+\frac{d^5 (b+2 c x)^{12}}{1536 c^4}\\ \end{align*}

Mathematica [B]  time = 0.084249, size = 224, normalized size = 2.22 $\frac{1}{60} d^5 x (b+c x) \left (20 a^3 \left (28 b^2 c^2 x^2+12 b^3 c x+3 b^4+32 b c^3 x^3+16 c^4 x^4\right )+30 a^2 x \left (56 b^3 c^2 x^2+88 b^2 c^3 x^3+19 b^4 c x+3 b^5+72 b c^4 x^4+24 c^5 x^5\right )+12 a x^2 (b+c x)^2 \left (78 b^2 c^2 x^2+30 b^3 c x+5 b^4+96 b c^3 x^3+48 c^4 x^4\right )+x^3 (b+c x)^3 \left (256 b^2 c^2 x^2+96 b^3 c x+15 b^4+320 b c^3 x^3+160 c^4 x^4\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^3,x]

[Out]

(d^5*x*(b + c*x)*(20*a^3*(3*b^4 + 12*b^3*c*x + 28*b^2*c^2*x^2 + 32*b*c^3*x^3 + 16*c^4*x^4) + 12*a*x^2*(b + c*x
)^2*(5*b^4 + 30*b^3*c*x + 78*b^2*c^2*x^2 + 96*b*c^3*x^3 + 48*c^4*x^4) + x^3*(b + c*x)^3*(15*b^4 + 96*b^3*c*x +
256*b^2*c^2*x^2 + 320*b*c^3*x^3 + 160*c^4*x^4) + 30*a^2*x*(3*b^5 + 19*b^4*c*x + 56*b^3*c^2*x^2 + 88*b^2*c^3*x
^3 + 72*b*c^4*x^4 + 24*c^5*x^5)))/60

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Maple [B]  time = 0.041, size = 810, normalized size = 8. \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)^5*(c*x^2+b*x+a)^3,x)

[Out]

8/3*c^8*d^5*x^12+16*b*d^5*c^7*x^11+1/10*(320*b^2*d^5*c^6+32*c^5*d^5*(a*c^2+2*b^2*c+c*(2*a*c+b^2)))*x^10+1/9*(2
80*b^3*d^5*c^5+80*b*d^5*c^4*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+32*c^5*d^5*(4*a*b*c+b*(2*a*c+b^2)))*x^9+1/8*(130*b^4
*d^5*c^4+80*b^2*d^5*c^3*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+80*b*d^5*c^4*(4*a*b*c+b*(2*a*c+b^2))+32*c^5*d^5*(a*(2*a*
c+b^2)+2*b^2*a+a^2*c))*x^8+1/7*(31*b^5*d^5*c^3+40*b^3*d^5*c^2*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+80*b^2*d^5*c^3*(4*
a*b*c+b*(2*a*c+b^2))+80*b*d^5*c^4*(a*(2*a*c+b^2)+2*b^2*a+a^2*c)+96*c^5*d^5*b*a^2)*x^7+1/6*(3*b^6*d^5*c^2+10*b^
4*d^5*c*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+40*b^3*d^5*c^2*(4*a*b*c+b*(2*a*c+b^2))+80*b^2*d^5*c^3*(a*(2*a*c+b^2)+2*b
^2*a+a^2*c)+240*b^2*d^5*c^4*a^2+32*c^5*d^5*a^3)*x^6+1/5*(b^5*d^5*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+10*b^4*d^5*c*(4
*a*b*c+b*(2*a*c+b^2))+40*b^3*d^5*c^2*(a*(2*a*c+b^2)+2*b^2*a+a^2*c)+240*b^3*d^5*c^3*a^2+80*b*d^5*c^4*a^3)*x^5+1
/4*(b^5*d^5*(4*a*b*c+b*(2*a*c+b^2))+10*b^4*d^5*c*(a*(2*a*c+b^2)+2*b^2*a+a^2*c)+120*b^4*d^5*c^2*a^2+80*b^2*d^5*
c^3*a^3)*x^4+1/3*(b^5*d^5*(a*(2*a*c+b^2)+2*b^2*a+a^2*c)+30*b^5*d^5*c*a^2+40*b^3*d^5*c^2*a^3)*x^3+1/2*(10*a^3*b
^4*c*d^5+3*a^2*b^6*d^5)*x^2+b^5*d^5*a^3*x

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Maxima [B]  time = 1.1377, size = 462, normalized size = 4.57 \begin{align*} \frac{8}{3} \, c^{8} d^{5} x^{12} + 16 \, b c^{7} d^{5} x^{11} + \frac{16}{5} \,{\left (13 \, b^{2} c^{6} + 3 \, a c^{7}\right )} d^{5} x^{10} + \frac{8}{3} \,{\left (23 \, b^{3} c^{5} + 18 \, a b c^{6}\right )} d^{5} x^{9} + a^{3} b^{5} d^{5} x + \frac{3}{4} \,{\left (75 \, b^{4} c^{4} + 136 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{5} x^{8} + 3 \,{\left (11 \, b^{5} c^{3} + 40 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{5} x^{7} + \frac{1}{6} \,{\left (73 \, b^{6} c^{2} + 510 \, a b^{4} c^{3} + 480 \, a^{2} b^{2} c^{4} + 32 \, a^{3} c^{5}\right )} d^{5} x^{6} + \frac{1}{5} \,{\left (13 \, b^{7} c + 183 \, a b^{5} c^{2} + 360 \, a^{2} b^{3} c^{3} + 80 \, a^{3} b c^{4}\right )} d^{5} x^{5} + \frac{1}{4} \,{\left (b^{8} + 36 \, a b^{6} c + 150 \, a^{2} b^{4} c^{2} + 80 \, a^{3} b^{2} c^{3}\right )} d^{5} x^{4} + \frac{1}{3} \,{\left (3 \, a b^{7} + 33 \, a^{2} b^{5} c + 40 \, a^{3} b^{3} c^{2}\right )} d^{5} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b^{6} + 10 \, a^{3} b^{4} c\right )} d^{5} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

8/3*c^8*d^5*x^12 + 16*b*c^7*d^5*x^11 + 16/5*(13*b^2*c^6 + 3*a*c^7)*d^5*x^10 + 8/3*(23*b^3*c^5 + 18*a*b*c^6)*d^
5*x^9 + a^3*b^5*d^5*x + 3/4*(75*b^4*c^4 + 136*a*b^2*c^5 + 16*a^2*c^6)*d^5*x^8 + 3*(11*b^5*c^3 + 40*a*b^3*c^4 +
16*a^2*b*c^5)*d^5*x^7 + 1/6*(73*b^6*c^2 + 510*a*b^4*c^3 + 480*a^2*b^2*c^4 + 32*a^3*c^5)*d^5*x^6 + 1/5*(13*b^7
*c + 183*a*b^5*c^2 + 360*a^2*b^3*c^3 + 80*a^3*b*c^4)*d^5*x^5 + 1/4*(b^8 + 36*a*b^6*c + 150*a^2*b^4*c^2 + 80*a^
3*b^2*c^3)*d^5*x^4 + 1/3*(3*a*b^7 + 33*a^2*b^5*c + 40*a^3*b^3*c^2)*d^5*x^3 + 1/2*(3*a^2*b^6 + 10*a^3*b^4*c)*d^
5*x^2

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Fricas [B]  time = 1.72415, size = 907, normalized size = 8.98 \begin{align*} \frac{8}{3} x^{12} d^{5} c^{8} + 16 x^{11} d^{5} c^{7} b + \frac{208}{5} x^{10} d^{5} c^{6} b^{2} + \frac{48}{5} x^{10} d^{5} c^{7} a + \frac{184}{3} x^{9} d^{5} c^{5} b^{3} + 48 x^{9} d^{5} c^{6} b a + \frac{225}{4} x^{8} d^{5} c^{4} b^{4} + 102 x^{8} d^{5} c^{5} b^{2} a + 12 x^{8} d^{5} c^{6} a^{2} + 33 x^{7} d^{5} c^{3} b^{5} + 120 x^{7} d^{5} c^{4} b^{3} a + 48 x^{7} d^{5} c^{5} b a^{2} + \frac{73}{6} x^{6} d^{5} c^{2} b^{6} + 85 x^{6} d^{5} c^{3} b^{4} a + 80 x^{6} d^{5} c^{4} b^{2} a^{2} + \frac{16}{3} x^{6} d^{5} c^{5} a^{3} + \frac{13}{5} x^{5} d^{5} c b^{7} + \frac{183}{5} x^{5} d^{5} c^{2} b^{5} a + 72 x^{5} d^{5} c^{3} b^{3} a^{2} + 16 x^{5} d^{5} c^{4} b a^{3} + \frac{1}{4} x^{4} d^{5} b^{8} + 9 x^{4} d^{5} c b^{6} a + \frac{75}{2} x^{4} d^{5} c^{2} b^{4} a^{2} + 20 x^{4} d^{5} c^{3} b^{2} a^{3} + x^{3} d^{5} b^{7} a + 11 x^{3} d^{5} c b^{5} a^{2} + \frac{40}{3} x^{3} d^{5} c^{2} b^{3} a^{3} + \frac{3}{2} x^{2} d^{5} b^{6} a^{2} + 5 x^{2} d^{5} c b^{4} a^{3} + x d^{5} b^{5} a^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

8/3*x^12*d^5*c^8 + 16*x^11*d^5*c^7*b + 208/5*x^10*d^5*c^6*b^2 + 48/5*x^10*d^5*c^7*a + 184/3*x^9*d^5*c^5*b^3 +
48*x^9*d^5*c^6*b*a + 225/4*x^8*d^5*c^4*b^4 + 102*x^8*d^5*c^5*b^2*a + 12*x^8*d^5*c^6*a^2 + 33*x^7*d^5*c^3*b^5 +
120*x^7*d^5*c^4*b^3*a + 48*x^7*d^5*c^5*b*a^2 + 73/6*x^6*d^5*c^2*b^6 + 85*x^6*d^5*c^3*b^4*a + 80*x^6*d^5*c^4*b
^2*a^2 + 16/3*x^6*d^5*c^5*a^3 + 13/5*x^5*d^5*c*b^7 + 183/5*x^5*d^5*c^2*b^5*a + 72*x^5*d^5*c^3*b^3*a^2 + 16*x^5
*d^5*c^4*b*a^3 + 1/4*x^4*d^5*b^8 + 9*x^4*d^5*c*b^6*a + 75/2*x^4*d^5*c^2*b^4*a^2 + 20*x^4*d^5*c^3*b^2*a^3 + x^3
*d^5*b^7*a + 11*x^3*d^5*c*b^5*a^2 + 40/3*x^3*d^5*c^2*b^3*a^3 + 3/2*x^2*d^5*b^6*a^2 + 5*x^2*d^5*c*b^4*a^3 + x*d
^5*b^5*a^3

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Sympy [B]  time = 0.127715, size = 428, normalized size = 4.24 \begin{align*} a^{3} b^{5} d^{5} x + 16 b c^{7} d^{5} x^{11} + \frac{8 c^{8} d^{5} x^{12}}{3} + x^{10} \left (\frac{48 a c^{7} d^{5}}{5} + \frac{208 b^{2} c^{6} d^{5}}{5}\right ) + x^{9} \left (48 a b c^{6} d^{5} + \frac{184 b^{3} c^{5} d^{5}}{3}\right ) + x^{8} \left (12 a^{2} c^{6} d^{5} + 102 a b^{2} c^{5} d^{5} + \frac{225 b^{4} c^{4} d^{5}}{4}\right ) + x^{7} \left (48 a^{2} b c^{5} d^{5} + 120 a b^{3} c^{4} d^{5} + 33 b^{5} c^{3} d^{5}\right ) + x^{6} \left (\frac{16 a^{3} c^{5} d^{5}}{3} + 80 a^{2} b^{2} c^{4} d^{5} + 85 a b^{4} c^{3} d^{5} + \frac{73 b^{6} c^{2} d^{5}}{6}\right ) + x^{5} \left (16 a^{3} b c^{4} d^{5} + 72 a^{2} b^{3} c^{3} d^{5} + \frac{183 a b^{5} c^{2} d^{5}}{5} + \frac{13 b^{7} c d^{5}}{5}\right ) + x^{4} \left (20 a^{3} b^{2} c^{3} d^{5} + \frac{75 a^{2} b^{4} c^{2} d^{5}}{2} + 9 a b^{6} c d^{5} + \frac{b^{8} d^{5}}{4}\right ) + x^{3} \left (\frac{40 a^{3} b^{3} c^{2} d^{5}}{3} + 11 a^{2} b^{5} c d^{5} + a b^{7} d^{5}\right ) + x^{2} \left (5 a^{3} b^{4} c d^{5} + \frac{3 a^{2} b^{6} d^{5}}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**5*(c*x**2+b*x+a)**3,x)

[Out]

a**3*b**5*d**5*x + 16*b*c**7*d**5*x**11 + 8*c**8*d**5*x**12/3 + x**10*(48*a*c**7*d**5/5 + 208*b**2*c**6*d**5/5
) + x**9*(48*a*b*c**6*d**5 + 184*b**3*c**5*d**5/3) + x**8*(12*a**2*c**6*d**5 + 102*a*b**2*c**5*d**5 + 225*b**4
*c**4*d**5/4) + x**7*(48*a**2*b*c**5*d**5 + 120*a*b**3*c**4*d**5 + 33*b**5*c**3*d**5) + x**6*(16*a**3*c**5*d**
5/3 + 80*a**2*b**2*c**4*d**5 + 85*a*b**4*c**3*d**5 + 73*b**6*c**2*d**5/6) + x**5*(16*a**3*b*c**4*d**5 + 72*a**
2*b**3*c**3*d**5 + 183*a*b**5*c**2*d**5/5 + 13*b**7*c*d**5/5) + x**4*(20*a**3*b**2*c**3*d**5 + 75*a**2*b**4*c*
*2*d**5/2 + 9*a*b**6*c*d**5 + b**8*d**5/4) + x**3*(40*a**3*b**3*c**2*d**5/3 + 11*a**2*b**5*c*d**5 + a*b**7*d**
5) + x**2*(5*a**3*b**4*c*d**5 + 3*a**2*b**6*d**5/2)

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Giac [B]  time = 1.23557, size = 572, normalized size = 5.66 \begin{align*} \frac{8}{3} \, c^{8} d^{5} x^{12} + 16 \, b c^{7} d^{5} x^{11} + \frac{208}{5} \, b^{2} c^{6} d^{5} x^{10} + \frac{48}{5} \, a c^{7} d^{5} x^{10} + \frac{184}{3} \, b^{3} c^{5} d^{5} x^{9} + 48 \, a b c^{6} d^{5} x^{9} + \frac{225}{4} \, b^{4} c^{4} d^{5} x^{8} + 102 \, a b^{2} c^{5} d^{5} x^{8} + 12 \, a^{2} c^{6} d^{5} x^{8} + 33 \, b^{5} c^{3} d^{5} x^{7} + 120 \, a b^{3} c^{4} d^{5} x^{7} + 48 \, a^{2} b c^{5} d^{5} x^{7} + \frac{73}{6} \, b^{6} c^{2} d^{5} x^{6} + 85 \, a b^{4} c^{3} d^{5} x^{6} + 80 \, a^{2} b^{2} c^{4} d^{5} x^{6} + \frac{16}{3} \, a^{3} c^{5} d^{5} x^{6} + \frac{13}{5} \, b^{7} c d^{5} x^{5} + \frac{183}{5} \, a b^{5} c^{2} d^{5} x^{5} + 72 \, a^{2} b^{3} c^{3} d^{5} x^{5} + 16 \, a^{3} b c^{4} d^{5} x^{5} + \frac{1}{4} \, b^{8} d^{5} x^{4} + 9 \, a b^{6} c d^{5} x^{4} + \frac{75}{2} \, a^{2} b^{4} c^{2} d^{5} x^{4} + 20 \, a^{3} b^{2} c^{3} d^{5} x^{4} + a b^{7} d^{5} x^{3} + 11 \, a^{2} b^{5} c d^{5} x^{3} + \frac{40}{3} \, a^{3} b^{3} c^{2} d^{5} x^{3} + \frac{3}{2} \, a^{2} b^{6} d^{5} x^{2} + 5 \, a^{3} b^{4} c d^{5} x^{2} + a^{3} b^{5} d^{5} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

8/3*c^8*d^5*x^12 + 16*b*c^7*d^5*x^11 + 208/5*b^2*c^6*d^5*x^10 + 48/5*a*c^7*d^5*x^10 + 184/3*b^3*c^5*d^5*x^9 +
48*a*b*c^6*d^5*x^9 + 225/4*b^4*c^4*d^5*x^8 + 102*a*b^2*c^5*d^5*x^8 + 12*a^2*c^6*d^5*x^8 + 33*b^5*c^3*d^5*x^7 +
120*a*b^3*c^4*d^5*x^7 + 48*a^2*b*c^5*d^5*x^7 + 73/6*b^6*c^2*d^5*x^6 + 85*a*b^4*c^3*d^5*x^6 + 80*a^2*b^2*c^4*d
^5*x^6 + 16/3*a^3*c^5*d^5*x^6 + 13/5*b^7*c*d^5*x^5 + 183/5*a*b^5*c^2*d^5*x^5 + 72*a^2*b^3*c^3*d^5*x^5 + 16*a^3
*b*c^4*d^5*x^5 + 1/4*b^8*d^5*x^4 + 9*a*b^6*c*d^5*x^4 + 75/2*a^2*b^4*c^2*d^5*x^4 + 20*a^3*b^2*c^3*d^5*x^4 + a*b
^7*d^5*x^3 + 11*a^2*b^5*c*d^5*x^3 + 40/3*a^3*b^3*c^2*d^5*x^3 + 3/2*a^2*b^6*d^5*x^2 + 5*a^3*b^4*c*d^5*x^2 + a^3
*b^5*d^5*x