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

Optimal. Leaf size=69 \[ -\frac {a e^2-b d e+c d^2}{5 e^3 (d+e x)^5}+\frac {2 c d-b e}{4 e^3 (d+e x)^4}-\frac {c}{3 e^3 (d+e x)^3} \]

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Rubi [A]  time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {698} \begin {gather*} -\frac {a e^2-b d e+c d^2}{5 e^3 (d+e x)^5}+\frac {2 c d-b e}{4 e^3 (d+e x)^4}-\frac {c}{3 e^3 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)/(d + e*x)^6,x]

[Out]

-(c*d^2 - b*d*e + a*e^2)/(5*e^3*(d + e*x)^5) + (2*c*d - b*e)/(4*e^3*(d + e*x)^4) - c/(3*e^3*(d + e*x)^3)

Rule 698

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] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 51, normalized size = 0.74 \begin {gather*} -\frac {3 e (4 a e+b (d+5 e x))+2 c \left (d^2+5 d e x+10 e^2 x^2\right )}{60 e^3 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/60*(2*c*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*e*(4*a*e + b*(d + 5*e*x)))/(e^3*(d + e*x)^5)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.37, size = 101, normalized size = 1.46 \begin {gather*} -\frac {20 \, c e^{2} x^{2} + 2 \, c d^{2} + 3 \, b d e + 12 \, a e^{2} + 5 \, {\left (2 \, c d e + 3 \, b e^{2}\right )} x}{60 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/60*(20*c*e^2*x^2 + 2*c*d^2 + 3*b*d*e + 12*a*e^2 + 5*(2*c*d*e + 3*b*e^2)*x)/(e^8*x^5 + 5*d*e^7*x^4 + 10*d^2*
e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)

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giac [A]  time = 0.15, size = 51, normalized size = 0.74 \begin {gather*} -\frac {{\left (20 \, c x^{2} e^{2} + 10 \, c d x e + 2 \, c d^{2} + 15 \, b x e^{2} + 3 \, b d e + 12 \, a e^{2}\right )} e^{\left (-3\right )}}{60 \, {\left (x e + d\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/60*(20*c*x^2*e^2 + 10*c*d*x*e + 2*c*d^2 + 15*b*x*e^2 + 3*b*d*e + 12*a*e^2)*e^(-3)/(x*e + d)^5

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maple [A]  time = 0.05, size = 63, normalized size = 0.91 \begin {gather*} -\frac {c}{3 \left (e x +d \right )^{3} e^{3}}-\frac {a \,e^{2}-b d e +c \,d^{2}}{5 \left (e x +d \right )^{5} e^{3}}-\frac {b e -2 c d}{4 \left (e x +d \right )^{4} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(a*e^2-b*d*e+c*d^2)/e^3/(e*x+d)^5-1/3*c/e^3/(e*x+d)^3-1/4*(b*e-2*c*d)/e^3/(e*x+d)^4

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maxima [A]  time = 1.19, size = 101, normalized size = 1.46 \begin {gather*} -\frac {20 \, c e^{2} x^{2} + 2 \, c d^{2} + 3 \, b d e + 12 \, a e^{2} + 5 \, {\left (2 \, c d e + 3 \, b e^{2}\right )} x}{60 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/60*(20*c*e^2*x^2 + 2*c*d^2 + 3*b*d*e + 12*a*e^2 + 5*(2*c*d*e + 3*b*e^2)*x)/(e^8*x^5 + 5*d*e^7*x^4 + 10*d^2*
e^6*x^3 + 10*d^3*e^5*x^2 + 5*d^4*e^4*x + d^5*e^3)

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mupad [B]  time = 0.07, size = 101, normalized size = 1.46 \begin {gather*} -\frac {\frac {2\,c\,d^2+3\,b\,d\,e+12\,a\,e^2}{60\,e^3}+\frac {x\,\left (3\,b\,e+2\,c\,d\right )}{12\,e^2}+\frac {c\,x^2}{3\,e}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)/(d + e*x)^6,x)

[Out]

-((12*a*e^2 + 2*c*d^2 + 3*b*d*e)/(60*e^3) + (x*(3*b*e + 2*c*d))/(12*e^2) + (c*x^2)/(3*e))/(d^5 + e^5*x^5 + 5*d
*e^4*x^4 + 10*d^3*e^2*x^2 + 10*d^2*e^3*x^3 + 5*d^4*e*x)

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sympy [A]  time = 2.17, size = 107, normalized size = 1.55 \begin {gather*} \frac {- 12 a e^{2} - 3 b d e - 2 c d^{2} - 20 c e^{2} x^{2} + x \left (- 15 b e^{2} - 10 c d e\right )}{60 d^{5} e^{3} + 300 d^{4} e^{4} x + 600 d^{3} e^{5} x^{2} + 600 d^{2} e^{6} x^{3} + 300 d e^{7} x^{4} + 60 e^{8} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)/(e*x+d)**6,x)

[Out]

(-12*a*e**2 - 3*b*d*e - 2*c*d**2 - 20*c*e**2*x**2 + x*(-15*b*e**2 - 10*c*d*e))/(60*d**5*e**3 + 300*d**4*e**4*x
+ 600*d**3*e**5*x**2 + 600*d**2*e**6*x**3 + 300*d*e**7*x**4 + 60*e**8*x**5)

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