\(\int \frac {(d+e x)^m}{(c d^2+2 c d e x+c e^2 x^2)^2} \, dx\) [1092]

Optimal result

Integrand size = 30, antiderivative size = 24 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=-\frac {(d+e x)^{-3+m}}{c^2 e (3-m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=-\frac {(d+e x)^{m-3}}{c^2 e (3-m)} \]

[In]

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

[Out]

-((d + e*x)^(-3 + m)/(c^2*e*(3 - 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*} \text {integral}& = \int \frac {(d+e x)^{-4+m}}{c^2} \, dx \\ & = \frac {\int (d+e x)^{-4+m} \, dx}{c^2} \\ & = -\frac {(d+e x)^{-3+m}}{c^2 e (3-m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {(d+e x)^{-3+m}}{c^2 e (-3+m)} \]

[In]

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

[Out]

(d + e*x)^(-3 + m)/(c^2*e*(-3 + m))

Maple [A] (verified)

Time = 3.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (21) = 42\).

Time = 0.39 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.17 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {{\left (e x + d\right )}^{m}}{c^{2} d^{3} e m - 3 \, c^{2} d^{3} e + {\left (c^{2} e^{4} m - 3 \, c^{2} e^{4}\right )} x^{3} + 3 \, {\left (c^{2} d e^{3} m - 3 \, c^{2} d e^{3}\right )} x^{2} + 3 \, {\left (c^{2} d^{2} e^{2} m - 3 \, c^{2} d^{2} e^{2}\right )} x} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (17) = 34\).

Time = 0.80 (sec) , antiderivative size = 136, normalized size of antiderivative = 5.67 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\begin {cases} \frac {x}{c^{2} d} & \text {for}\: e = 0 \wedge m = 3 \\\frac {d^{m} x}{c^{2} d^{4}} & \text {for}\: e = 0 \\\frac {\log {\left (\frac {d}{e} + x \right )}}{c^{2} e} & \text {for}\: m = 3 \\\frac {\left (d + e x\right )^{m}}{c^{2} d^{3} e m - 3 c^{2} d^{3} e + 3 c^{2} d^{2} e^{2} m x - 9 c^{2} d^{2} e^{2} x + 3 c^{2} d e^{3} m x^{2} - 9 c^{2} d e^{3} x^{2} + c^{2} e^{4} m x^{3} - 3 c^{2} e^{4} x^{3}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x/(c**2*d), Eq(e, 0) & Eq(m, 3)), (d**m*x/(c**2*d**4), Eq(e, 0)), (log(d/e + x)/(c**2*e), Eq(m, 3))
, ((d + e*x)**m/(c**2*d**3*e*m - 3*c**2*d**3*e + 3*c**2*d**2*e**2*m*x - 9*c**2*d**2*e**2*x + 3*c**2*d*e**3*m*x
**2 - 9*c**2*d*e**3*x**2 + c**2*e**4*m*x**3 - 3*c**2*e**4*x**3), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (21) = 42\).

Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {{\left (e x + d\right )}^{m}}{c^{2} e^{4} {\left (m - 3\right )} x^{3} + 3 \, c^{2} d e^{3} {\left (m - 3\right )} x^{2} + 3 \, c^{2} d^{2} e^{2} {\left (m - 3\right )} x + c^{2} d^{3} e {\left (m - 3\right )}} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{m}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.71 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.08 \[ \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx=\frac {{\left (d+e\,x\right )}^m}{c^2\,e^4\,\left (m-3\right )\,\left (x^3+\frac {d^3}{e^3}+\frac {3\,d\,x^2}{e}+\frac {3\,d^2\,x}{e^2}\right )} \]

[In]

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

[Out]

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