3.11.93 \(\int \frac {(d+e x)^m}{(c d^2+2 c d e x+c e^2 x^2)^3} \, dx\) [1093]

Optimal. Leaf size=24 \[ -\frac {(d+e x)^{-5+m}}{c^3 e (5-m)} \]

[Out]

-(e*x+d)^(-5+m)/c^3/e/(5-m)

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Rubi [A]
time = 0.01, antiderivative size = 24, 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)^{m-5}}{c^3 e (5-m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

-((d + e*x)^(-5 + m)/(c^3*e*(5 - m)))

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

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Mathematica [A]
time = 0.02, size = 21, normalized size = 0.88 \begin {gather*} \frac {(d+e x)^{-5+m}}{c^3 e (-5+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

(d + e*x)^(-5 + m)/(c^3*e*(-5 + m))

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Maple [A]
time = 0.60, size = 27, normalized size = 1.12

method	result	size
risch	\(\frac {\left (e x +d \right )^{m}}{c^{3} e \left (-5+m \right ) \left (e x +d \right )^{5}}\)	\(27\)
norman	\(\frac {{\mathrm e}^{m \ln \left (e x +d \right )}}{c^{3} e \left (-5+m \right ) \left (e x +d \right )^{5}}\)	\(29\)
gosper	\(\frac {\left (e x +d \right )^{-1+m}}{\left (e^{2} x^{2}+2 d x e +d^{2}\right )^{2} c^{3} e \left (-5+m \right )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/c^3/e/(-5+m)*(e*x+d)^m/(e*x+d)^5

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (21) = 42\).
time = 0.30, size = 96, normalized size = 4.00 \begin {gather*} \frac {{\left (x e + d\right )}^{m}}{c^{3} {\left (m - 5\right )} x^{5} e^{6} + 5 \, c^{3} d {\left (m - 5\right )} x^{4} e^{5} + 10 \, c^{3} d^{2} {\left (m - 5\right )} x^{3} e^{4} + 10 \, c^{3} d^{3} {\left (m - 5\right )} x^{2} e^{3} + 5 \, c^{3} d^{4} {\left (m - 5\right )} x e^{2} + c^{3} d^{5} {\left (m - 5\right )} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="maxima")

[Out]

(x*e + d)^m/(c^3*(m - 5)*x^5*e^6 + 5*c^3*d*(m - 5)*x^4*e^5 + 10*c^3*d^2*(m - 5)*x^3*e^4 + 10*c^3*d^3*(m - 5)*x
^2*e^3 + 5*c^3*d^4*(m - 5)*x*e^2 + c^3*d^5*(m - 5)*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (21) = 42\).
time = 2.00, size = 139, normalized size = 5.79 \begin {gather*} \frac {{\left (x e + d\right )}^{m}}{{\left (c^{3} m - 5 \, c^{3}\right )} x^{5} e^{6} + 5 \, {\left (c^{3} d m - 5 \, c^{3} d\right )} x^{4} e^{5} + 10 \, {\left (c^{3} d^{2} m - 5 \, c^{3} d^{2}\right )} x^{3} e^{4} + 10 \, {\left (c^{3} d^{3} m - 5 \, c^{3} d^{3}\right )} x^{2} e^{3} + 5 \, {\left (c^{3} d^{4} m - 5 \, c^{3} d^{4}\right )} x e^{2} + {\left (c^{3} d^{5} m - 5 \, c^{3} d^{5}\right )} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="fricas")

[Out]

(x*e + d)^m/((c^3*m - 5*c^3)*x^5*e^6 + 5*(c^3*d*m - 5*c^3*d)*x^4*e^5 + 10*(c^3*d^2*m - 5*c^3*d^2)*x^3*e^4 + 10
*(c^3*d^3*m - 5*c^3*d^3)*x^2*e^3 + 5*(c^3*d^4*m - 5*c^3*d^4)*x*e^2 + (c^3*d^5*m - 5*c^3*d^5)*e)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (17) = 34\).
time = 1.80, size = 201, normalized size = 8.38 \begin {gather*} \begin {cases} \frac {x}{c^{3} d} & \text {for}\: e = 0 \wedge m = 5 \\\frac {d^{m} x}{c^{3} d^{6}} & \text {for}\: e = 0 \\\frac {\log {\left (\frac {d}{e} + x \right )}}{c^{3} e} & \text {for}\: m = 5 \\\frac {\left (d + e x\right )^{m}}{c^{3} d^{5} e m - 5 c^{3} d^{5} e + 5 c^{3} d^{4} e^{2} m x - 25 c^{3} d^{4} e^{2} x + 10 c^{3} d^{3} e^{3} m x^{2} - 50 c^{3} d^{3} e^{3} x^{2} + 10 c^{3} d^{2} e^{4} m x^{3} - 50 c^{3} d^{2} e^{4} x^{3} + 5 c^{3} d e^{5} m x^{4} - 25 c^{3} d e^{5} x^{4} + c^{3} e^{6} m x^{5} - 5 c^{3} e^{6} x^{5}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m/(c*e**2*x**2+2*c*d*e*x+c*d**2)**3,x)

[Out]

Piecewise((x/(c**3*d), Eq(e, 0) & Eq(m, 5)), (d**m*x/(c**3*d**6), Eq(e, 0)), (log(d/e + x)/(c**3*e), Eq(m, 5))
, ((d + e*x)**m/(c**3*d**5*e*m - 5*c**3*d**5*e + 5*c**3*d**4*e**2*m*x - 25*c**3*d**4*e**2*x + 10*c**3*d**3*e**
3*m*x**2 - 50*c**3*d**3*e**3*x**2 + 10*c**3*d**2*e**4*m*x**3 - 50*c**3*d**2*e**4*x**3 + 5*c**3*d*e**5*m*x**4 -
25*c**3*d*e**5*x**4 + c**3*e**6*m*x**5 - 5*c**3*e**6*x**5), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="giac")

[Out]

integrate((x*e + d)^m/(c*x^2*e^2 + 2*c*d*x*e + c*d^2)^3, x)

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Mupad [B]
time = 0.55, size = 72, normalized size = 3.00 \begin {gather*} \frac {{\left (d+e\,x\right )}^m}{c^3\,e^6\,\left (m-5\right )\,\left (x^5+\frac {d^5}{e^5}+\frac {5\,d\,x^4}{e}+\frac {5\,d^4\,x}{e^4}+\frac {10\,d^2\,x^3}{e^2}+\frac {10\,d^3\,x^2}{e^3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^m/(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^3,x)

[Out]

(d + e*x)^m/(c^3*e^6*(m - 5)*(x^5 + d^5/e^5 + (5*d*x^4)/e + (5*d^4*x)/e^4 + (10*d^2*x^3)/e^2 + (10*d^3*x^2)/e^
3))

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