3.9.31 \(\int \frac {(d+e x)^7}{(c d^2+2 c d e x+c e^2 x^2)^3} \, dx\)

Optimal. Leaf size=17 \[ \frac {(d+e x)^2}{2 c^3 e} \]

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {27, 9} \begin {gather*} \frac {(d+e x)^2}{2 c^3 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)^3,x]

[Out]

(d + e*x)^2/(2*c^3*e)

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, 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]

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.94 \begin {gather*} \frac {d x+\frac {e x^2}{2}}{c^3} \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)^3,x]

[Out]

(d*x + (e*x^2)/2)/c^3

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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 )^3} \, 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)^3,x]

[Out]

IntegrateAlgebraic[(d + e*x)^7/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3, x]

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fricas [A]  time = 0.38, size = 15, normalized size = 0.88 \begin {gather*} \frac {e x^{2} + 2 \, d x}{2 \, c^{3}} \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)^3,x, algorithm="fricas")

[Out]

1/2*(e*x^2 + 2*d*x)/c^3

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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)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (1/2*x^2*c^3*exp(2)^3*exp(1)^7+7*x*c^3*e
xp(2)^3*d*exp(1)^6-6*x*c^3*exp(2)^2*d*exp(1)^8)/c^6/exp(2)^6+((3*exp(2)^6*d^3+6*exp(2)^5*d^3*exp(1)^2-229*exp(
2)^4*d^3*exp(1)^4+684*exp(2)^3*d^3*exp(1)^6-720*exp(2)^2*d^3*exp(1)^8+256*exp(2)*d^3*exp(1)^10)*x^3+(9*exp(2)^
5*d^4*exp(1)-122*exp(2)^4*d^4*exp(1)^3+41*exp(2)^3*d^4*exp(1)^5+696*exp(2)^2*d^4*exp(1)^7-1072*exp(2)*d^4*exp(
1)^9+448*d^4*exp(1)^11)*x^2+(5*exp(2)^5*d^5-42*exp(2)^4*d^5*exp(1)^2-187*exp(2)^3*d^5*exp(1)^4+928*exp(2)^2*d^
5*exp(1)^6-1152*exp(2)*d^5*exp(1)^8+448*d^5*exp(1)^10)*x-9*exp(2)^4*d^6*exp(1)-34*exp(2)^3*d^6*exp(1)^3+207*ex
p(2)^2*d^6*exp(1)^5-276*exp(2)*d^6*exp(1)^7+112*d^6*exp(1)^9)/8/exp(2)^5/c^3/(2*exp(1)*d*x+exp(2)*x^2+d^2)^2-(
-21*exp(2)^2*d^2*exp(1)^5+45*exp(2)*d^2*exp(1)^7-24*d^2*exp(1)^9)*1/2/c^3/exp(2)^5*ln(x^2*exp(2)+2*x*d*exp(1)+
d^2)-(-3*exp(2)^5*d^3-6*exp(2)^4*d^3*exp(1)^2-51*exp(2)^3*d^3*exp(1)^4+324*exp(2)^2*d^3*exp(1)^6-456*exp(2)*d^
3*exp(1)^8+192*d^3*exp(1)^10)*1/4/c^3/exp(2)^5*1/2/d/sqrt(-exp(1)^2+exp(2))*atan((d*exp(1)+x*exp(2))/d/sqrt(-e
xp(1)^2+exp(2)))

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maple [A]  time = 0.04, size = 15, normalized size = 0.88 \begin {gather*} \frac {\frac {1}{2} e \,x^{2}+d x}{c^{3}} \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)^3,x)

[Out]

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

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maxima [A]  time = 1.36, size = 15, normalized size = 0.88 \begin {gather*} \frac {e x^{2} + 2 \, d x}{2 \, c^{3}} \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)^3,x, algorithm="maxima")

[Out]

1/2*(e*x^2 + 2*d*x)/c^3

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mupad [B]  time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {x\,\left (2\,d+e\,x\right )}{2\,c^3} \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)^3,x)

[Out]

(x*(2*d + e*x))/(2*c^3)

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sympy [A]  time = 0.13, size = 15, normalized size = 0.88 \begin {gather*} \frac {d x}{c^{3}} + \frac {e x^{2}}{2 c^{3}} \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)**3,x)

[Out]

d*x/c**3 + e*x**2/(2*c**3)

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