3.10.59 \(\int \frac {(a+b x+c x^2)^3}{(b d+2 c d x)^6} \, dx\)

Optimal. Leaf size=94 \[ \frac {\left (b^2-4 a c\right )^3}{640 c^4 d^6 (b+2 c x)^5}-\frac {\left (b^2-4 a c\right )^2}{128 c^4 d^6 (b+2 c x)^3}+\frac {3 \left (b^2-4 a c\right )}{128 c^4 d^6 (b+2 c x)}+\frac {x}{64 c^3 d^6} \]

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Rubi [A]  time = 0.09, antiderivative size = 94, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\left (b^2-4 a c\right )^3}{640 c^4 d^6 (b+2 c x)^5}-\frac {\left (b^2-4 a c\right )^2}{128 c^4 d^6 (b+2 c x)^3}+\frac {3 \left (b^2-4 a c\right )}{128 c^4 d^6 (b+2 c x)}+\frac {x}{64 c^3 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^6,x]

[Out]

x/(64*c^3*d^6) + (b^2 - 4*a*c)^3/(640*c^4*d^6*(b + 2*c*x)^5) - (b^2 - 4*a*c)^2/(128*c^4*d^6*(b + 2*c*x)^3) + (
3*(b^2 - 4*a*c))/(128*c^4*d^6*(b + 2*c*x))

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)^6} \, dx &=\int \left (\frac {1}{64 c^3 d^6}+\frac {\left (-b^2+4 a c\right )^3}{64 c^3 d^6 (b+2 c x)^6}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^6 (b+2 c x)^4}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^6 (b+2 c x)^2}\right ) \, dx\\ &=\frac {x}{64 c^3 d^6}+\frac {\left (b^2-4 a c\right )^3}{640 c^4 d^6 (b+2 c x)^5}-\frac {\left (b^2-4 a c\right )^2}{128 c^4 d^6 (b+2 c x)^3}+\frac {3 \left (b^2-4 a c\right )}{128 c^4 d^6 (b+2 c x)}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 72, normalized size = 0.77 \begin {gather*} \frac {\frac {\left (b^2-4 a c\right )^3}{(b+2 c x)^5}-\frac {5 \left (b^2-4 a c\right )^2}{(b+2 c x)^3}+\frac {15 \left (b^2-4 a c\right )}{b+2 c x}+10 c x}{640 c^4 d^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^6,x]

[Out]

(10*c*x + (b^2 - 4*a*c)^3/(b + 2*c*x)^5 - (5*(b^2 - 4*a*c)^2)/(b + 2*c*x)^3 + (15*(b^2 - 4*a*c))/(b + 2*c*x))/
(640*c^4*d^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^6,x]

[Out]

IntegrateAlgebraic[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^6, x]

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fricas [B]  time = 0.40, size = 227, normalized size = 2.41 \begin {gather*} \frac {320 \, c^{6} x^{6} + 800 \, b c^{5} x^{5} + 11 \, b^{6} - 32 \, a b^{4} c - 32 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 80 \, {\left (13 \, b^{2} c^{4} - 12 \, a c^{5}\right )} x^{4} + 80 \, {\left (11 \, b^{3} c^{3} - 24 \, a b c^{4}\right )} x^{3} + 40 \, {\left (11 \, b^{4} c^{2} - 32 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} x^{2} + 10 \, {\left (11 \, b^{5} c - 32 \, a b^{3} c^{2} - 32 \, a^{2} b c^{3}\right )} x}{640 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}} \end {gather*}

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)^6,x, algorithm="fricas")

[Out]

1/640*(320*c^6*x^6 + 800*b*c^5*x^5 + 11*b^6 - 32*a*b^4*c - 32*a^2*b^2*c^2 - 64*a^3*c^3 + 80*(13*b^2*c^4 - 12*a
*c^5)*x^4 + 80*(11*b^3*c^3 - 24*a*b*c^4)*x^3 + 40*(11*b^4*c^2 - 32*a*b^2*c^3 - 8*a^2*c^4)*x^2 + 10*(11*b^5*c -
32*a*b^3*c^2 - 32*a^2*b*c^3)*x)/(32*c^9*d^6*x^5 + 80*b*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2
+ 10*b^4*c^5*d^6*x + b^5*c^4*d^6)

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giac [A]  time = 0.18, size = 160, normalized size = 1.70 \begin {gather*} \frac {x}{64 \, c^{3} d^{6}} + \frac {240 \, b^{2} c^{4} x^{4} - 960 \, a c^{5} x^{4} + 480 \, b^{3} c^{3} x^{3} - 1920 \, a b c^{4} x^{3} + 340 \, b^{4} c^{2} x^{2} - 1280 \, a b^{2} c^{3} x^{2} - 320 \, a^{2} c^{4} x^{2} + 100 \, b^{5} c x - 320 \, a b^{3} c^{2} x - 320 \, a^{2} b c^{3} x + 11 \, b^{6} - 32 \, a b^{4} c - 32 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{640 \, {\left (2 \, c x + b\right )}^{5} c^{4} d^{6}} \end {gather*}

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)^6,x, algorithm="giac")

[Out]

1/64*x/(c^3*d^6) + 1/640*(240*b^2*c^4*x^4 - 960*a*c^5*x^4 + 480*b^3*c^3*x^3 - 1920*a*b*c^4*x^3 + 340*b^4*c^2*x
^2 - 1280*a*b^2*c^3*x^2 - 320*a^2*c^4*x^2 + 100*b^5*c*x - 320*a*b^3*c^2*x - 320*a^2*b*c^3*x + 11*b^6 - 32*a*b^
4*c - 32*a^2*b^2*c^2 - 64*a^3*c^3)/((2*c*x + b)^5*c^4*d^6)

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maple [A]  time = 0.05, size = 114, normalized size = 1.21 \begin {gather*} \frac {\frac {x}{64 c^{3}}-\frac {48 a^{2} c^{2}-24 a \,b^{2} c +3 b^{4}}{384 \left (2 c x +b \right )^{3} c^{4}}-\frac {64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{640 \left (2 c x +b \right )^{5} c^{4}}-\frac {12 a c -3 b^{2}}{128 \left (2 c x +b \right ) c^{4}}}{d^{6}} \end {gather*}

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)^6,x)

[Out]

1/d^6*(1/64/c^3*x-1/384*(48*a^2*c^2-24*a*b^2*c+3*b^4)/c^4/(2*c*x+b)^3-1/640/c^4*(64*a^3*c^3-48*a^2*b^2*c^2+12*
a*b^4*c-b^6)/(2*c*x+b)^5-1/128*(12*a*c-3*b^2)/c^4/(2*c*x+b))

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maxima [B]  time = 1.55, size = 218, normalized size = 2.32 \begin {gather*} \frac {11 \, b^{6} - 32 \, a b^{4} c - 32 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 240 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 480 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 20 \, {\left (17 \, b^{4} c^{2} - 64 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 20 \, {\left (5 \, b^{5} c - 16 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x}{640 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}} + \frac {x}{64 \, c^{3} d^{6}} \end {gather*}

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)^6,x, algorithm="maxima")

[Out]

1/640*(11*b^6 - 32*a*b^4*c - 32*a^2*b^2*c^2 - 64*a^3*c^3 + 240*(b^2*c^4 - 4*a*c^5)*x^4 + 480*(b^3*c^3 - 4*a*b*
c^4)*x^3 + 20*(17*b^4*c^2 - 64*a*b^2*c^3 - 16*a^2*c^4)*x^2 + 20*(5*b^5*c - 16*a*b^3*c^2 - 16*a^2*b*c^3)*x)/(32
*c^9*d^6*x^5 + 80*b*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2 + 10*b^4*c^5*d^6*x + b^5*c^4*d^6) +
1/64*x/(c^3*d^6)

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mupad [B]  time = 0.53, size = 129, normalized size = 1.37 \begin {gather*} -\frac {b^2\,\left (\frac {a^2\,c^2}{20}+2\,a\,c^3\,x^2-\frac {c^4\,x^4}{4}\right )+b\,\left (\frac {a^2\,c^3\,x}{2}+3\,a\,c^4\,x^3-\frac {7\,c^5\,x^5}{10}\right )+\frac {a^3\,c^3}{10}-\frac {c^6\,x^6}{2}+\frac {3\,a\,c^5\,x^4}{2}+\frac {a^2\,c^4\,x^2}{2}+\frac {a\,b^4\,c}{20}+\frac {a\,b^3\,c^2\,x}{2}}{c^4\,d^6\,{\left (b+2\,c\,x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^6,x)

[Out]

-(b^2*((a^2*c^2)/20 - (c^4*x^4)/4 + 2*a*c^3*x^2) + b*((a^2*c^3*x)/2 - (7*c^5*x^5)/10 + 3*a*c^4*x^3) + (a^3*c^3
)/10 - (c^6*x^6)/2 + (3*a*c^5*x^4)/2 + (a^2*c^4*x^2)/2 + (a*b^4*c)/20 + (a*b^3*c^2*x)/2)/(c^4*d^6*(b + 2*c*x)^
5)

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sympy [B]  time = 2.85, size = 223, normalized size = 2.37 \begin {gather*} \frac {- 64 a^{3} c^{3} - 32 a^{2} b^{2} c^{2} - 32 a b^{4} c + 11 b^{6} + x^{4} \left (- 960 a c^{5} + 240 b^{2} c^{4}\right ) + x^{3} \left (- 1920 a b c^{4} + 480 b^{3} c^{3}\right ) + x^{2} \left (- 320 a^{2} c^{4} - 1280 a b^{2} c^{3} + 340 b^{4} c^{2}\right ) + x \left (- 320 a^{2} b c^{3} - 320 a b^{3} c^{2} + 100 b^{5} c\right )}{640 b^{5} c^{4} d^{6} + 6400 b^{4} c^{5} d^{6} x + 25600 b^{3} c^{6} d^{6} x^{2} + 51200 b^{2} c^{7} d^{6} x^{3} + 51200 b c^{8} d^{6} x^{4} + 20480 c^{9} d^{6} x^{5}} + \frac {x}{64 c^{3} d^{6}} \end {gather*}

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)**6,x)

[Out]

(-64*a**3*c**3 - 32*a**2*b**2*c**2 - 32*a*b**4*c + 11*b**6 + x**4*(-960*a*c**5 + 240*b**2*c**4) + x**3*(-1920*
a*b*c**4 + 480*b**3*c**3) + x**2*(-320*a**2*c**4 - 1280*a*b**2*c**3 + 340*b**4*c**2) + x*(-320*a**2*b*c**3 - 3
20*a*b**3*c**2 + 100*b**5*c))/(640*b**5*c**4*d**6 + 6400*b**4*c**5*d**6*x + 25600*b**3*c**6*d**6*x**2 + 51200*
b**2*c**7*d**6*x**3 + 51200*b*c**8*d**6*x**4 + 20480*c**9*d**6*x**5) + x/(64*c**3*d**6)

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