3.10.62 \(\int \frac {(a+b x+c x^2)^3}{(b d+2 c d x)^9} \, dx\)

Optimal. Leaf size=37 \[ \frac {\left (a+b x+c x^2\right )^4}{4 d^9 \left (b^2-4 a c\right ) (b+2 c x)^8} \]

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {682} \begin {gather*} \frac {\left (a+b x+c x^2\right )^4}{4 d^9 \left (b^2-4 a c\right ) (b+2 c x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^9,x]

[Out]

(a + b*x + c*x^2)^4/(4*(b^2 - 4*a*c)*d^9*(b + 2*c*x)^8)

Rule 682

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^9} \, dx &=\frac {\left (a+b x+c x^2\right )^4}{4 \left (b^2-4 a c\right ) d^9 (b+2 c x)^8}\\ \end {align*}

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Mathematica [B]  time = 0.05, size = 96, normalized size = 2.59 \begin {gather*} \frac {-64 a^3 c^3+48 a^2 b^2 c^2-12 a b^4 c+6 \left (b^2-4 a c\right ) (b+2 c x)^4-4 \left (b^2-4 a c\right )^2 (b+2 c x)^2+b^6-4 (b+2 c x)^6}{1024 c^4 d^9 (b+2 c x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^9,x]

[Out]

(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3 - 4*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 6*(b^2 - 4*a*c)*(b + 2*c*x
)^4 - 4*(b + 2*c*x)^6)/(1024*c^4*d^9*(b + 2*c*x)^8)

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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)^9} \, 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)^9,x]

[Out]

IntegrateAlgebraic[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^9, x]

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fricas [B]  time = 0.40, size = 266, normalized size = 7.19 \begin {gather*} -\frac {256 \, c^{6} x^{6} + 768 \, b c^{5} x^{5} + b^{6} + 4 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3} + 96 \, {\left (9 \, b^{2} c^{4} + 4 \, a c^{5}\right )} x^{4} + 64 \, {\left (7 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} x^{3} + 16 \, {\left (7 \, b^{4} c^{2} + 28 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 16 \, {\left (b^{5} c + 4 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{1024 \, {\left (256 \, c^{12} d^{9} x^{8} + 1024 \, b c^{11} d^{9} x^{7} + 1792 \, b^{2} c^{10} d^{9} x^{6} + 1792 \, b^{3} c^{9} d^{9} x^{5} + 1120 \, b^{4} c^{8} d^{9} x^{4} + 448 \, b^{5} c^{7} d^{9} x^{3} + 112 \, b^{6} c^{6} d^{9} x^{2} + 16 \, b^{7} c^{5} d^{9} x + b^{8} c^{4} d^{9}\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)^9,x, algorithm="fricas")

[Out]

-1/1024*(256*c^6*x^6 + 768*b*c^5*x^5 + b^6 + 4*a*b^4*c + 16*a^2*b^2*c^2 + 64*a^3*c^3 + 96*(9*b^2*c^4 + 4*a*c^5
)*x^4 + 64*(7*b^3*c^3 + 12*a*b*c^4)*x^3 + 16*(7*b^4*c^2 + 28*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 16*(b^5*c + 4*a*b^3
*c^2 + 16*a^2*b*c^3)*x)/(256*c^12*d^9*x^8 + 1024*b*c^11*d^9*x^7 + 1792*b^2*c^10*d^9*x^6 + 1792*b^3*c^9*d^9*x^5
+ 1120*b^4*c^8*d^9*x^4 + 448*b^5*c^7*d^9*x^3 + 112*b^6*c^6*d^9*x^2 + 16*b^7*c^5*d^9*x + b^8*c^4*d^9)

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giac [B]  time = 0.16, size = 165, normalized size = 4.46 \begin {gather*} -\frac {256 \, c^{6} x^{6} + 768 \, b c^{5} x^{5} + 864 \, b^{2} c^{4} x^{4} + 384 \, a c^{5} x^{4} + 448 \, b^{3} c^{3} x^{3} + 768 \, a b c^{4} x^{3} + 112 \, b^{4} c^{2} x^{2} + 448 \, a b^{2} c^{3} x^{2} + 256 \, a^{2} c^{4} x^{2} + 16 \, b^{5} c x + 64 \, a b^{3} c^{2} x + 256 \, a^{2} b c^{3} x + b^{6} + 4 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3}}{1024 \, {\left (2 \, c x + b\right )}^{8} c^{4} d^{9}} \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)^9,x, algorithm="giac")

[Out]

-1/1024*(256*c^6*x^6 + 768*b*c^5*x^5 + 864*b^2*c^4*x^4 + 384*a*c^5*x^4 + 448*b^3*c^3*x^3 + 768*a*b*c^4*x^3 + 1
12*b^4*c^2*x^2 + 448*a*b^2*c^3*x^2 + 256*a^2*c^4*x^2 + 16*b^5*c*x + 64*a*b^3*c^2*x + 256*a^2*b*c^3*x + b^6 + 4
*a*b^4*c + 16*a^2*b^2*c^2 + 64*a^3*c^3)/((2*c*x + b)^8*c^4*d^9)

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maple [B]  time = 0.05, size = 121, normalized size = 3.27 \begin {gather*} \frac {-\frac {12 a c -3 b^{2}}{512 \left (2 c x +b \right )^{4} c^{4}}-\frac {64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{1024 \left (2 c x +b \right )^{8} c^{4}}-\frac {1}{256 \left (2 c x +b \right )^{2} c^{4}}-\frac {48 a^{2} c^{2}-24 a \,b^{2} c +3 b^{4}}{768 \left (2 c x +b \right )^{6} c^{4}}}{d^{9}} \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)^9,x)

[Out]

1/d^9*(-1/512*(12*a*c-3*b^2)/c^4/(2*c*x+b)^4-1/1024*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^4/(2*c*x+b)^8
-1/256/c^4/(2*c*x+b)^2-1/768*(48*a^2*c^2-24*a*b^2*c+3*b^4)/c^4/(2*c*x+b)^6)

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maxima [B]  time = 1.57, size = 266, normalized size = 7.19 \begin {gather*} -\frac {256 \, c^{6} x^{6} + 768 \, b c^{5} x^{5} + b^{6} + 4 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3} + 96 \, {\left (9 \, b^{2} c^{4} + 4 \, a c^{5}\right )} x^{4} + 64 \, {\left (7 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} x^{3} + 16 \, {\left (7 \, b^{4} c^{2} + 28 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 16 \, {\left (b^{5} c + 4 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{1024 \, {\left (256 \, c^{12} d^{9} x^{8} + 1024 \, b c^{11} d^{9} x^{7} + 1792 \, b^{2} c^{10} d^{9} x^{6} + 1792 \, b^{3} c^{9} d^{9} x^{5} + 1120 \, b^{4} c^{8} d^{9} x^{4} + 448 \, b^{5} c^{7} d^{9} x^{3} + 112 \, b^{6} c^{6} d^{9} x^{2} + 16 \, b^{7} c^{5} d^{9} x + b^{8} c^{4} d^{9}\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)^9,x, algorithm="maxima")

[Out]

-1/1024*(256*c^6*x^6 + 768*b*c^5*x^5 + b^6 + 4*a*b^4*c + 16*a^2*b^2*c^2 + 64*a^3*c^3 + 96*(9*b^2*c^4 + 4*a*c^5
)*x^4 + 64*(7*b^3*c^3 + 12*a*b*c^4)*x^3 + 16*(7*b^4*c^2 + 28*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 16*(b^5*c + 4*a*b^3
*c^2 + 16*a^2*b*c^3)*x)/(256*c^12*d^9*x^8 + 1024*b*c^11*d^9*x^7 + 1792*b^2*c^10*d^9*x^6 + 1792*b^3*c^9*d^9*x^5
+ 1120*b^4*c^8*d^9*x^4 + 448*b^5*c^7*d^9*x^3 + 112*b^6*c^6*d^9*x^2 + 16*b^7*c^5*d^9*x + b^8*c^4*d^9)

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mupad [B]  time = 0.55, size = 254, normalized size = 6.86 \begin {gather*} -\frac {\frac {64\,a^3\,c^3+16\,a^2\,b^2\,c^2+4\,a\,b^4\,c+b^6}{1024\,c^4}+x^4\,\left (\frac {27\,b^2}{32}+\frac {3\,a\,c}{8}\right )+\frac {c^2\,x^6}{4}+\frac {x^2\,\left (16\,a^2\,c^2+28\,a\,b^2\,c+7\,b^4\right )}{64\,c^2}+\frac {3\,b\,c\,x^5}{4}+\frac {x^3\,\left (7\,b^3+12\,a\,c\,b\right )}{16\,c}+\frac {b\,x\,\left (16\,a^2\,c^2+4\,a\,b^2\,c+b^4\right )}{64\,c^3}}{b^8\,d^9+16\,b^7\,c\,d^9\,x+112\,b^6\,c^2\,d^9\,x^2+448\,b^5\,c^3\,d^9\,x^3+1120\,b^4\,c^4\,d^9\,x^4+1792\,b^3\,c^5\,d^9\,x^5+1792\,b^2\,c^6\,d^9\,x^6+1024\,b\,c^7\,d^9\,x^7+256\,c^8\,d^9\,x^8} \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)^9,x)

[Out]

-((b^6 + 64*a^3*c^3 + 16*a^2*b^2*c^2 + 4*a*b^4*c)/(1024*c^4) + x^4*((3*a*c)/8 + (27*b^2)/32) + (c^2*x^6)/4 + (
x^2*(7*b^4 + 16*a^2*c^2 + 28*a*b^2*c))/(64*c^2) + (3*b*c*x^5)/4 + (x^3*(7*b^3 + 12*a*b*c))/(16*c) + (b*x*(b^4
+ 16*a^2*c^2 + 4*a*b^2*c))/(64*c^3))/(b^8*d^9 + 256*c^8*d^9*x^8 + 1024*b*c^7*d^9*x^7 + 112*b^6*c^2*d^9*x^2 + 4
48*b^5*c^3*d^9*x^3 + 1120*b^4*c^4*d^9*x^4 + 1792*b^3*c^5*d^9*x^5 + 1792*b^2*c^6*d^9*x^6 + 16*b^7*c*d^9*x)

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sympy [B]  time = 10.11, size = 282, normalized size = 7.62 \begin {gather*} \frac {- 64 a^{3} c^{3} - 16 a^{2} b^{2} c^{2} - 4 a b^{4} c - b^{6} - 768 b c^{5} x^{5} - 256 c^{6} x^{6} + x^{4} \left (- 384 a c^{5} - 864 b^{2} c^{4}\right ) + x^{3} \left (- 768 a b c^{4} - 448 b^{3} c^{3}\right ) + x^{2} \left (- 256 a^{2} c^{4} - 448 a b^{2} c^{3} - 112 b^{4} c^{2}\right ) + x \left (- 256 a^{2} b c^{3} - 64 a b^{3} c^{2} - 16 b^{5} c\right )}{1024 b^{8} c^{4} d^{9} + 16384 b^{7} c^{5} d^{9} x + 114688 b^{6} c^{6} d^{9} x^{2} + 458752 b^{5} c^{7} d^{9} x^{3} + 1146880 b^{4} c^{8} d^{9} x^{4} + 1835008 b^{3} c^{9} d^{9} x^{5} + 1835008 b^{2} c^{10} d^{9} x^{6} + 1048576 b c^{11} d^{9} x^{7} + 262144 c^{12} d^{9} x^{8}} \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)**9,x)

[Out]

(-64*a**3*c**3 - 16*a**2*b**2*c**2 - 4*a*b**4*c - b**6 - 768*b*c**5*x**5 - 256*c**6*x**6 + x**4*(-384*a*c**5 -
864*b**2*c**4) + x**3*(-768*a*b*c**4 - 448*b**3*c**3) + x**2*(-256*a**2*c**4 - 448*a*b**2*c**3 - 112*b**4*c**
2) + x*(-256*a**2*b*c**3 - 64*a*b**3*c**2 - 16*b**5*c))/(1024*b**8*c**4*d**9 + 16384*b**7*c**5*d**9*x + 114688
*b**6*c**6*d**9*x**2 + 458752*b**5*c**7*d**9*x**3 + 1146880*b**4*c**8*d**9*x**4 + 1835008*b**3*c**9*d**9*x**5
+ 1835008*b**2*c**10*d**9*x**6 + 1048576*b*c**11*d**9*x**7 + 262144*c**12*d**9*x**8)

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