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

Optimal. Leaf size=37 \[ -\frac {d^3 (b+2 c x)^4}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

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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 {d^3 (b+2 c x)^4}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^3*(b + 2*c*x)^4)/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2)

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

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Mathematica [A]  time = 0.02, size = 38, normalized size = 1.03 \begin {gather*} -\frac {d^3 \left (4 c \left (a+2 c x^2\right )+b^2+8 b c x\right )}{2 (a+x (b+c x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(d^3*(b^2 + 8*b*c*x + 4*c*(a + 2*c*x^2)))/(a + x*(b + c*x))^2

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.38, size = 71, normalized size = 1.92 \begin {gather*} -\frac {8 \, c^{2} d^{3} x^{2} + 8 \, b c d^{3} x + {\left (b^{2} + 4 \, a c\right )} d^{3}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(8*c^2*d^3*x^2 + 8*b*c*d^3*x + (b^2 + 4*a*c)*d^3)/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^
2)

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giac [A]  time = 0.19, size = 50, normalized size = 1.35 \begin {gather*} -\frac {b^{2} d^{5} + 4 \, a c d^{5} + 8 \, {\left (c d x^{2} + b d x\right )} c d^{4}}{2 \, {\left (c d x^{2} + b d x + a d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^2*d^5 + 4*a*c*d^5 + 8*(c*d*x^2 + b*d*x)*c*d^4)/(c*d*x^2 + b*d*x + a*d)^2

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maple [A]  time = 0.06, size = 40, normalized size = 1.08 \begin {gather*} \frac {\left (-4 c^{2} x^{2}-4 b c x -2 a c -\frac {1}{2} b^{2}\right ) d^{3}}{\left (c \,x^{2}+b x +a \right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^3*(-4*c^2*x^2-4*b*c*x-2*a*c-1/2*b^2)/(c*x^2+b*x+a)^2

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maxima [B]  time = 1.42, size = 71, normalized size = 1.92 \begin {gather*} -\frac {8 \, c^{2} d^{3} x^{2} + 8 \, b c d^{3} x + {\left (b^{2} + 4 \, a c\right )} d^{3}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(8*c^2*d^3*x^2 + 8*b*c*d^3*x + (b^2 + 4*a*c)*d^3)/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^
2)

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mupad [B]  time = 0.07, size = 74, normalized size = 2.00 \begin {gather*} -\frac {\frac {b^2\,d^3}{2}+4\,b\,c\,d^3\,x+4\,c^2\,d^3\,x^2+2\,a\,c\,d^3}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b^2*d^3)/2 + 4*c^2*d^3*x^2 + 2*a*c*d^3 + 4*b*c*d^3*x)/(x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*
x^3)

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sympy [B]  time = 1.65, size = 80, normalized size = 2.16 \begin {gather*} \frac {- 4 a c d^{3} - b^{2} d^{3} - 8 b c d^{3} x - 8 c^{2} d^{3} x^{2}}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-4*a*c*d**3 - b**2*d**3 - 8*b*c*d**3*x - 8*c**2*d**3*x**2)/(2*a**2 + 4*a*b*x + 4*b*c*x**3 + 2*c**2*x**4 + x**
2*(4*a*c + 2*b**2))

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