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

Optimal. Leaf size=103 \[ -\frac {3 c d^2-e (2 b d-a e)}{5 e^4 (d+e x)^5}+\frac {d \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac {3 c d-b e}{4 e^4 (d+e x)^4}-\frac {c}{3 e^4 (d+e x)^3} \]

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Rubi [A]  time = 0.07, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {771} \begin {gather*} -\frac {3 c d^2-e (2 b d-a e)}{5 e^4 (d+e x)^5}+\frac {d \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac {3 c d-b e}{4 e^4 (d+e x)^4}-\frac {c}{3 e^4 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^7} \, dx &=\int \left (-\frac {d \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^7}+\frac {3 c d^2-e (2 b d-a e)}{e^3 (d+e x)^6}+\frac {-3 c d+b e}{e^3 (d+e x)^5}+\frac {c}{e^3 (d+e x)^4}\right ) \, dx\\ &=\frac {d \left (c d^2-b d e+a e^2\right )}{6 e^4 (d+e x)^6}-\frac {3 c d^2-e (2 b d-a e)}{5 e^4 (d+e x)^5}+\frac {3 c d-b e}{4 e^4 (d+e x)^4}-\frac {c}{3 e^4 (d+e x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 77, normalized size = 0.75 \begin {gather*} -\frac {e \left (2 a e (d+6 e x)+b \left (d^2+6 d e x+15 e^2 x^2\right )\right )+c \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )}{60 e^4 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/60*(c*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + e*(2*a*e*(d + 6*e*x) + b*(d^2 + 6*d*e*x + 15*e^2*x^2)
))/(e^4*(d + e*x)^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.37, size = 137, normalized size = 1.33 \begin {gather*} -\frac {20 \, c e^{3} x^{3} + c d^{3} + b d^{2} e + 2 \, a d e^{2} + 15 \, {\left (c d e^{2} + b e^{3}\right )} x^{2} + 6 \, {\left (c d^{2} e + b d e^{2} + 2 \, a e^{3}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(20*c*e^3*x^3 + c*d^3 + b*d^2*e + 2*a*d*e^2 + 15*(c*d*e^2 + b*e^3)*x^2 + 6*(c*d^2*e + b*d*e^2 + 2*a*e^3)
*x)/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^8*x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2 + 6*d^5*e^5*x + d^6*e^4)

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giac [A]  time = 0.15, size = 78, normalized size = 0.76 \begin {gather*} -\frac {{\left (20 \, c x^{3} e^{3} + 15 \, c d x^{2} e^{2} + 6 \, c d^{2} x e + c d^{3} + 15 \, b x^{2} e^{3} + 6 \, b d x e^{2} + b d^{2} e + 12 \, a x e^{3} + 2 \, a d e^{2}\right )} e^{\left (-4\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(20*c*x^3*e^3 + 15*c*d*x^2*e^2 + 6*c*d^2*x*e + c*d^3 + 15*b*x^2*e^3 + 6*b*d*x*e^2 + b*d^2*e + 12*a*x*e^3
+ 2*a*d*e^2)*e^(-4)/(x*e + d)^6

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maple [A]  time = 0.05, size = 93, normalized size = 0.90 \begin {gather*} -\frac {c}{3 \left (e x +d \right )^{3} e^{4}}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d}{6 \left (e x +d \right )^{6} e^{4}}-\frac {b e -3 c d}{4 \left (e x +d \right )^{4} e^{4}}-\frac {a \,e^{2}-2 b d e +3 c \,d^{2}}{5 \left (e x +d \right )^{5} e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b*e-3*c*d)/e^4/(e*x+d)^4+1/6*d*(a*e^2-b*d*e+c*d^2)/e^4/(e*x+d)^6-1/5*(a*e^2-2*b*d*e+3*c*d^2)/e^4/(e*x+d)
^5-1/3*c/e^4/(e*x+d)^3

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maxima [A]  time = 0.62, size = 137, normalized size = 1.33 \begin {gather*} -\frac {20 \, c e^{3} x^{3} + c d^{3} + b d^{2} e + 2 \, a d e^{2} + 15 \, {\left (c d e^{2} + b e^{3}\right )} x^{2} + 6 \, {\left (c d^{2} e + b d e^{2} + 2 \, a e^{3}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(20*c*e^3*x^3 + c*d^3 + b*d^2*e + 2*a*d*e^2 + 15*(c*d*e^2 + b*e^3)*x^2 + 6*(c*d^2*e + b*d*e^2 + 2*a*e^3)
*x)/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^8*x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2 + 6*d^5*e^5*x + d^6*e^4)

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mupad [B]  time = 2.35, size = 133, normalized size = 1.29 \begin {gather*} -\frac {\frac {c\,x^3}{3\,e}+\frac {d\,\left (c\,d^2+b\,d\,e+2\,a\,e^2\right )}{60\,e^4}+\frac {x\,\left (c\,d^2+b\,d\,e+2\,a\,e^2\right )}{10\,e^3}+\frac {x^2\,\left (b\,e+c\,d\right )}{4\,e^2}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c*x^3)/(3*e) + (d*(2*a*e^2 + c*d^2 + b*d*e))/(60*e^4) + (x*(2*a*e^2 + c*d^2 + b*d*e))/(10*e^3) + (x^2*(b*e
+ c*d))/(4*e^2))/(d^6 + e^6*x^6 + 6*d*e^5*x^5 + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 15*d^2*e^4*x^4 + 6*d^5*e*x)

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sympy [A]  time = 5.83, size = 150, normalized size = 1.46 \begin {gather*} \frac {- 2 a d e^{2} - b d^{2} e - c d^{3} - 20 c e^{3} x^{3} + x^{2} \left (- 15 b e^{3} - 15 c d e^{2}\right ) + x \left (- 12 a e^{3} - 6 b d e^{2} - 6 c d^{2} e\right )}{60 d^{6} e^{4} + 360 d^{5} e^{5} x + 900 d^{4} e^{6} x^{2} + 1200 d^{3} e^{7} x^{3} + 900 d^{2} e^{8} x^{4} + 360 d e^{9} x^{5} + 60 e^{10} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*a*d*e**2 - b*d**2*e - c*d**3 - 20*c*e**3*x**3 + x**2*(-15*b*e**3 - 15*c*d*e**2) + x*(-12*a*e**3 - 6*b*d*e*
*2 - 6*c*d**2*e))/(60*d**6*e**4 + 360*d**5*e**5*x + 900*d**4*e**6*x**2 + 1200*d**3*e**7*x**3 + 900*d**2*e**8*x
**4 + 360*d*e**9*x**5 + 60*e**10*x**6)

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