3.1119 \(\int \frac{a+b x+c x^2}{(b d+2 c d x)^7} \, dx\)

Optimal. Leaf size=45 \[ \frac{b^2-4 a c}{48 c^2 d^7 (b+2 c x)^6}-\frac{1}{32 c^2 d^7 (b+2 c x)^4} \]

[Out]

(b^2 - 4*a*c)/(48*c^2*d^7*(b + 2*c*x)^6) - 1/(32*c^2*d^7*(b + 2*c*x)^4)

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Rubi [A]  time = 0.0325614, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {683} \[ \frac{b^2-4 a c}{48 c^2 d^7 (b+2 c x)^6}-\frac{1}{32 c^2 d^7 (b+2 c x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2 - 4*a*c)/(48*c^2*d^7*(b + 2*c*x)^6) - 1/(32*c^2*d^7*(b + 2*c*x)^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 \frac{a+b x+c x^2}{(b d+2 c d x)^7} \, dx &=\int \left (\frac{-b^2+4 a c}{4 c d^7 (b+2 c x)^7}+\frac{1}{4 c d^7 (b+2 c x)^5}\right ) \, dx\\ &=\frac{b^2-4 a c}{48 c^2 d^7 (b+2 c x)^6}-\frac{1}{32 c^2 d^7 (b+2 c x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0138337, size = 43, normalized size = 0.96 \[ \frac{\frac{b^2-4 a c}{48 c^2 (b+2 c x)^6}-\frac{1}{32 c^2 (b+2 c x)^4}}{d^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2 - 4*a*c)/(48*c^2*(b + 2*c*x)^6) - 1/(32*c^2*(b + 2*c*x)^4))/d^7

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Maple [A]  time = 0.045, size = 42, normalized size = 0.9 \begin{align*}{\frac{1}{{d}^{7}} \left ( -{\frac{4\,ac-{b}^{2}}{48\,{c}^{2} \left ( 2\,cx+b \right ) ^{6}}}-{\frac{1}{32\,{c}^{2} \left ( 2\,cx+b \right ) ^{4}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^7*(-1/48*(4*a*c-b^2)/c^2/(2*c*x+b)^6-1/32/c^2/(2*c*x+b)^4)

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Maxima [B]  time = 1.14208, size = 153, normalized size = 3.4 \begin{align*} -\frac{12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c}{96 \,{\left (64 \, c^{8} d^{7} x^{6} + 192 \, b c^{7} d^{7} x^{5} + 240 \, b^{2} c^{6} d^{7} x^{4} + 160 \, b^{3} c^{5} d^{7} x^{3} + 60 \, b^{4} c^{4} d^{7} x^{2} + 12 \, b^{5} c^{3} d^{7} x + b^{6} c^{2} d^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(12*c^2*x^2 + 12*b*c*x + b^2 + 8*a*c)/(64*c^8*d^7*x^6 + 192*b*c^7*d^7*x^5 + 240*b^2*c^6*d^7*x^4 + 160*b^
3*c^5*d^7*x^3 + 60*b^4*c^4*d^7*x^2 + 12*b^5*c^3*d^7*x + b^6*c^2*d^7)

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Fricas [B]  time = 2.36044, size = 243, normalized size = 5.4 \begin{align*} -\frac{12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c}{96 \,{\left (64 \, c^{8} d^{7} x^{6} + 192 \, b c^{7} d^{7} x^{5} + 240 \, b^{2} c^{6} d^{7} x^{4} + 160 \, b^{3} c^{5} d^{7} x^{3} + 60 \, b^{4} c^{4} d^{7} x^{2} + 12 \, b^{5} c^{3} d^{7} x + b^{6} c^{2} d^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(12*c^2*x^2 + 12*b*c*x + b^2 + 8*a*c)/(64*c^8*d^7*x^6 + 192*b*c^7*d^7*x^5 + 240*b^2*c^6*d^7*x^4 + 160*b^
3*c^5*d^7*x^3 + 60*b^4*c^4*d^7*x^2 + 12*b^5*c^3*d^7*x + b^6*c^2*d^7)

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Sympy [B]  time = 1.48938, size = 121, normalized size = 2.69 \begin{align*} - \frac{8 a c + b^{2} + 12 b c x + 12 c^{2} x^{2}}{96 b^{6} c^{2} d^{7} + 1152 b^{5} c^{3} d^{7} x + 5760 b^{4} c^{4} d^{7} x^{2} + 15360 b^{3} c^{5} d^{7} x^{3} + 23040 b^{2} c^{6} d^{7} x^{4} + 18432 b c^{7} d^{7} x^{5} + 6144 c^{8} d^{7} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(8*a*c + b**2 + 12*b*c*x + 12*c**2*x**2)/(96*b**6*c**2*d**7 + 1152*b**5*c**3*d**7*x + 5760*b**4*c**4*d**7*x**
2 + 15360*b**3*c**5*d**7*x**3 + 23040*b**2*c**6*d**7*x**4 + 18432*b*c**7*d**7*x**5 + 6144*c**8*d**7*x**6)

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Giac [A]  time = 1.20721, size = 50, normalized size = 1.11 \begin{align*} -\frac{12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c}{96 \,{\left (2 \, c x + b\right )}^{6} c^{2} d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(12*c^2*x^2 + 12*b*c*x + b^2 + 8*a*c)/((2*c*x + b)^6*c^2*d^7)