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

Optimal. Leaf size=17 \[ \frac {(d+e x)^4}{4 c^2 e} \]

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \begin {gather*} \frac {(d+e x)^4}{4 c^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d + e*x)^4/(4*c^2*e)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {(d+e x)^4}{4 c^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d + e*x)^4/(4*c^2*e)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.37, size = 37, normalized size = 2.18 \begin {gather*} \frac {e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*x)/c^2

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (1/4*x^4*c^6*exp(2)^6*exp(1)^7+7/3*x^3*c
^6*exp(2)^6*d*exp(1)^6-4/3*x^3*c^6*exp(2)^5*d*exp(1)^8+21/2*x^2*c^6*exp(2)^6*d^2*exp(1)^5-15*x^2*c^6*exp(2)^5*
d^2*exp(1)^7+6*x^2*c^6*exp(2)^4*d^2*exp(1)^9+35*x*c^6*exp(2)^6*d^3*exp(1)^4-98*x*c^6*exp(2)^5*d^3*exp(1)^6+96*
x*c^6*exp(2)^4*d^3*exp(1)^8-32*x*c^6*exp(2)^3*d^3*exp(1)^10)/c^8/exp(2)^8+(-6*exp(2)^5*d^6*exp(1)+50*exp(2)^4*
d^6*exp(1)^3-146*exp(2)^3*d^6*exp(1)^5+198*exp(2)^2*d^6*exp(1)^7-128*exp(2)*d^6*exp(1)^9+32*d^6*exp(1)^11+(exp
(2)^6*d^5-27*exp(2)^5*d^5*exp(1)^2+155*exp(2)^4*d^5*exp(1)^4-377*exp(2)^3*d^5*exp(1)^6+456*exp(2)^2*d^5*exp(1)
^8-272*exp(2)*d^5*exp(1)^10+64*d^5*exp(1)^12)*x)/2/exp(2)^6/c^2/(2*exp(1)*d*x+exp(2)*x^2+d^2)+(35*exp(2)^4*d^4
*exp(1)^3-182*exp(2)^3*d^4*exp(1)^5+339*exp(2)^2*d^4*exp(1)^7-272*exp(2)*d^4*exp(1)^9+80*d^4*exp(1)^11)*1/2/c^
2/exp(2)^6*ln(x^2*exp(2)+2*x*d*exp(1)+d^2)+(exp(2)^6*d^5+15*exp(2)^5*d^5*exp(1)^2-195*exp(2)^4*d^5*exp(1)^4+64
5*exp(2)^3*d^5*exp(1)^6-930*exp(2)^2*d^5*exp(1)^8+624*exp(2)*d^5*exp(1)^10-160*d^5*exp(1)^12)/c^2/exp(2)^6*1/2
/d/sqrt(-exp(1)^2+exp(2))*atan((d*exp(1)+x*exp(2))/d/sqrt(-exp(1)^2+exp(2)))

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maple [A]  time = 0.04, size = 16, normalized size = 0.94 \begin {gather*} \frac {\left (e x +d \right )^{4}}{4 c^{2} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(e*x+d)^4/c^2/e

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maxima [B]  time = 1.36, size = 37, normalized size = 2.18 \begin {gather*} \frac {e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*x)/c^2

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mupad [B]  time = 0.05, size = 43, normalized size = 2.53 \begin {gather*} \frac {d^3\,x}{c^2}+\frac {e^3\,x^4}{4\,c^2}+\frac {3\,d^2\,e\,x^2}{2\,c^2}+\frac {d\,e^2\,x^3}{c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d^3*x)/c^2 + (e^3*x^4)/(4*c^2) + (3*d^2*e*x^2)/(2*c^2) + (d*e^2*x^3)/c^2

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sympy [B]  time = 0.12, size = 46, normalized size = 2.71 \begin {gather*} \frac {d^{3} x}{c^{2}} + \frac {3 d^{2} e x^{2}}{2 c^{2}} + \frac {d e^{2} x^{3}}{c^{2}} + \frac {e^{3} x^{4}}{4 c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**3*x/c**2 + 3*d**2*e*x**2/(2*c**2) + d*e**2*x**3/c**2 + e**3*x**4/(4*c**2)

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