### 3.1148 $$\int \frac{(a+b x+c x^2)^3}{(b d+2 c d x)^7} \, dx$$

Optimal. Leaf size=100 $\frac{\left (b^2-4 a c\right )^3}{768 c^4 d^7 (b+2 c x)^6}-\frac{3 \left (b^2-4 a c\right )^2}{512 c^4 d^7 (b+2 c x)^4}+\frac{3 \left (b^2-4 a c\right )}{256 c^4 d^7 (b+2 c x)^2}+\frac{\log (b+2 c x)}{128 c^4 d^7}$

[Out]

(b^2 - 4*a*c)^3/(768*c^4*d^7*(b + 2*c*x)^6) - (3*(b^2 - 4*a*c)^2)/(512*c^4*d^7*(b + 2*c*x)^4) + (3*(b^2 - 4*a*
c))/(256*c^4*d^7*(b + 2*c*x)^2) + Log[b + 2*c*x]/(128*c^4*d^7)

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Rubi [A]  time = 0.0828874, antiderivative size = 100, 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} $\frac{\left (b^2-4 a c\right )^3}{768 c^4 d^7 (b+2 c x)^6}-\frac{3 \left (b^2-4 a c\right )^2}{512 c^4 d^7 (b+2 c x)^4}+\frac{3 \left (b^2-4 a c\right )}{256 c^4 d^7 (b+2 c x)^2}+\frac{\log (b+2 c x)}{128 c^4 d^7}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^7,x]

[Out]

(b^2 - 4*a*c)^3/(768*c^4*d^7*(b + 2*c*x)^6) - (3*(b^2 - 4*a*c)^2)/(512*c^4*d^7*(b + 2*c*x)^4) + (3*(b^2 - 4*a*
c))/(256*c^4*d^7*(b + 2*c*x)^2) + Log[b + 2*c*x]/(128*c^4*d^7)

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 \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^7} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3}{64 c^3 d^7 (b+2 c x)^7}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^7 (b+2 c x)^5}+\frac{3 \left (-b^2+4 a c\right )}{64 c^3 d^7 (b+2 c x)^3}+\frac{1}{64 c^3 d^7 (b+2 c x)}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^3}{768 c^4 d^7 (b+2 c x)^6}-\frac{3 \left (b^2-4 a c\right )^2}{512 c^4 d^7 (b+2 c x)^4}+\frac{3 \left (b^2-4 a c\right )}{256 c^4 d^7 (b+2 c x)^2}+\frac{\log (b+2 c x)}{128 c^4 d^7}\\ \end{align*}

Mathematica [A]  time = 0.0422631, size = 78, normalized size = 0.78 $\frac{\frac{2 \left (b^2-4 a c\right )^3}{(b+2 c x)^6}-\frac{9 \left (b^2-4 a c\right )^2}{(b+2 c x)^4}+\frac{18 \left (b^2-4 a c\right )}{(b+2 c x)^2}+12 \log (b+2 c x)}{1536 c^4 d^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^7,x]

[Out]

((2*(b^2 - 4*a*c)^3)/(b + 2*c*x)^6 - (9*(b^2 - 4*a*c)^2)/(b + 2*c*x)^4 + (18*(b^2 - 4*a*c))/(b + 2*c*x)^2 + 12
*Log[b + 2*c*x])/(1536*c^4*d^7)

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Maple [B]  time = 0.069, size = 191, normalized size = 1.9 \begin{align*} -{\frac{{a}^{3}}{12\,{d}^{7}c \left ( 2\,cx+b \right ) ^{6}}}+{\frac{{b}^{2}{a}^{2}}{16\,{d}^{7}{c}^{2} \left ( 2\,cx+b \right ) ^{6}}}-{\frac{{b}^{4}a}{64\,{d}^{7}{c}^{3} \left ( 2\,cx+b \right ) ^{6}}}+{\frac{{b}^{6}}{768\,{c}^{4}{d}^{7} \left ( 2\,cx+b \right ) ^{6}}}-{\frac{3\,a}{64\,{d}^{7}{c}^{3} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{3\,{b}^{2}}{256\,{c}^{4}{d}^{7} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{3\,{a}^{2}}{32\,{d}^{7}{c}^{2} \left ( 2\,cx+b \right ) ^{4}}}+{\frac{3\,{b}^{2}a}{64\,{d}^{7}{c}^{3} \left ( 2\,cx+b \right ) ^{4}}}-{\frac{3\,{b}^{4}}{512\,{c}^{4}{d}^{7} \left ( 2\,cx+b \right ) ^{4}}}+{\frac{\ln \left ( 2\,cx+b \right ) }{128\,{c}^{4}{d}^{7}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^7,x)

[Out]

-1/12/d^7/c/(2*c*x+b)^6*a^3+1/16/d^7/c^2/(2*c*x+b)^6*a^2*b^2-1/64/d^7/c^3/(2*c*x+b)^6*a*b^4+1/768/d^7/c^4/(2*c
*x+b)^6*b^6-3/64/d^7/c^3/(2*c*x+b)^2*a+3/256/d^7/c^4/(2*c*x+b)^2*b^2-3/32/d^7/c^2/(2*c*x+b)^4*a^2+3/64/d^7/c^3
/(2*c*x+b)^4*a*b^2-3/512/d^7/c^4/(2*c*x+b)^4*b^4+1/128*ln(2*c*x+b)/c^4/d^7

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Maxima [B]  time = 1.14426, size = 321, normalized size = 3.21 \begin{align*} \frac{11 \, b^{6} - 24 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 128 \, a^{3} c^{3} + 288 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 576 \,{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 36 \,{\left (11 \, b^{4} c^{2} - 40 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 36 \,{\left (3 \, b^{5} c - 8 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x}{1536 \,{\left (64 \, c^{10} d^{7} x^{6} + 192 \, b c^{9} d^{7} x^{5} + 240 \, b^{2} c^{8} d^{7} x^{4} + 160 \, b^{3} c^{7} d^{7} x^{3} + 60 \, b^{4} c^{6} d^{7} x^{2} + 12 \, b^{5} c^{5} d^{7} x + b^{6} c^{4} d^{7}\right )}} + \frac{\log \left (2 \, c x + b\right )}{128 \, c^{4} d^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^7,x, algorithm="maxima")

[Out]

1/1536*(11*b^6 - 24*a*b^4*c - 48*a^2*b^2*c^2 - 128*a^3*c^3 + 288*(b^2*c^4 - 4*a*c^5)*x^4 + 576*(b^3*c^3 - 4*a*
b*c^4)*x^3 + 36*(11*b^4*c^2 - 40*a*b^2*c^3 - 16*a^2*c^4)*x^2 + 36*(3*b^5*c - 8*a*b^3*c^2 - 16*a^2*b*c^3)*x)/(6
4*c^10*d^7*x^6 + 192*b*c^9*d^7*x^5 + 240*b^2*c^8*d^7*x^4 + 160*b^3*c^7*d^7*x^3 + 60*b^4*c^6*d^7*x^2 + 12*b^5*c
^5*d^7*x + b^6*c^4*d^7) + 1/128*log(2*c*x + b)/(c^4*d^7)

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Fricas [B]  time = 2.04045, size = 643, normalized size = 6.43 \begin{align*} \frac{11 \, b^{6} - 24 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 128 \, a^{3} c^{3} + 288 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 576 \,{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 36 \,{\left (11 \, b^{4} c^{2} - 40 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 36 \,{\left (3 \, b^{5} c - 8 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x + 12 \,{\left (64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 240 \, b^{2} c^{4} x^{4} + 160 \, b^{3} c^{3} x^{3} + 60 \, b^{4} c^{2} x^{2} + 12 \, b^{5} c x + b^{6}\right )} \log \left (2 \, c x + b\right )}{1536 \,{\left (64 \, c^{10} d^{7} x^{6} + 192 \, b c^{9} d^{7} x^{5} + 240 \, b^{2} c^{8} d^{7} x^{4} + 160 \, b^{3} c^{7} d^{7} x^{3} + 60 \, b^{4} c^{6} d^{7} x^{2} + 12 \, b^{5} c^{5} d^{7} x + b^{6} c^{4} d^{7}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^7,x, algorithm="fricas")

[Out]

1/1536*(11*b^6 - 24*a*b^4*c - 48*a^2*b^2*c^2 - 128*a^3*c^3 + 288*(b^2*c^4 - 4*a*c^5)*x^4 + 576*(b^3*c^3 - 4*a*
b*c^4)*x^3 + 36*(11*b^4*c^2 - 40*a*b^2*c^3 - 16*a^2*c^4)*x^2 + 36*(3*b^5*c - 8*a*b^3*c^2 - 16*a^2*b*c^3)*x + 1
2*(64*c^6*x^6 + 192*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 + 60*b^4*c^2*x^2 + 12*b^5*c*x + b^6)*log(2*c
*x + b))/(64*c^10*d^7*x^6 + 192*b*c^9*d^7*x^5 + 240*b^2*c^8*d^7*x^4 + 160*b^3*c^7*d^7*x^3 + 60*b^4*c^6*d^7*x^2
+ 12*b^5*c^5*d^7*x + b^6*c^4*d^7)

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Sympy [B]  time = 14.7638, size = 245, normalized size = 2.45 \begin{align*} - \frac{128 a^{3} c^{3} + 48 a^{2} b^{2} c^{2} + 24 a b^{4} c - 11 b^{6} + x^{4} \left (1152 a c^{5} - 288 b^{2} c^{4}\right ) + x^{3} \left (2304 a b c^{4} - 576 b^{3} c^{3}\right ) + x^{2} \left (576 a^{2} c^{4} + 1440 a b^{2} c^{3} - 396 b^{4} c^{2}\right ) + x \left (576 a^{2} b c^{3} + 288 a b^{3} c^{2} - 108 b^{5} c\right )}{1536 b^{6} c^{4} d^{7} + 18432 b^{5} c^{5} d^{7} x + 92160 b^{4} c^{6} d^{7} x^{2} + 245760 b^{3} c^{7} d^{7} x^{3} + 368640 b^{2} c^{8} d^{7} x^{4} + 294912 b c^{9} d^{7} x^{5} + 98304 c^{10} d^{7} x^{6}} + \frac{\log{\left (b + 2 c x \right )}}{128 c^{4} d^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**7,x)

[Out]

-(128*a**3*c**3 + 48*a**2*b**2*c**2 + 24*a*b**4*c - 11*b**6 + x**4*(1152*a*c**5 - 288*b**2*c**4) + x**3*(2304*
a*b*c**4 - 576*b**3*c**3) + x**2*(576*a**2*c**4 + 1440*a*b**2*c**3 - 396*b**4*c**2) + x*(576*a**2*b*c**3 + 288
*a*b**3*c**2 - 108*b**5*c))/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760
*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + log(b +
2*c*x)/(128*c**4*d**7)

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Giac [A]  time = 1.18739, size = 220, normalized size = 2.2 \begin{align*} \frac{\log \left ({\left | 2 \, c x + b \right |}\right )}{128 \, c^{4} d^{7}} + \frac{11 \, b^{6} - 24 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 128 \, a^{3} c^{3} + 288 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 576 \,{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 36 \,{\left (11 \, b^{4} c^{2} - 40 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 36 \,{\left (3 \, b^{5} c - 8 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x}{1536 \,{\left (2 \, c x + b\right )}^{6} c^{4} d^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^7,x, algorithm="giac")

[Out]

1/128*log(abs(2*c*x + b))/(c^4*d^7) + 1/1536*(11*b^6 - 24*a*b^4*c - 48*a^2*b^2*c^2 - 128*a^3*c^3 + 288*(b^2*c^
4 - 4*a*c^5)*x^4 + 576*(b^3*c^3 - 4*a*b*c^4)*x^3 + 36*(11*b^4*c^2 - 40*a*b^2*c^3 - 16*a^2*c^4)*x^2 + 36*(3*b^5
*c - 8*a*b^3*c^2 - 16*a^2*b*c^3)*x)/((2*c*x + b)^6*c^4*d^7)