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

Optimal. Leaf size=24 \[ -\frac {(d+e x)^{m-1}}{c e (1-m)} \]

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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-1}}{c e (1-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),x]

[Out]

-((d + e*x)^(-1 + m)/(c*e*(1 - 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}{c d^2+2 c d e x+c e^2 x^2} \, dx &=\int \frac {(d+e x)^{-2+m}}{c} \, dx\\ &=\frac {\int (d+e x)^{-2+m} \, dx}{c}\\ &=-\frac {(d+e x)^{-1+m}}{c e (1-m)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.88 \begin {gather*} \frac {(d+e x)^{m-1}}{c e (m-1)} \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),x]

[Out]

(d + e*x)^(-1 + m)/(c*e*(-1 + m))

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IntegrateAlgebraic [F]  time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^m}{c d^2+2 c d e x+c e^2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 36, normalized size = 1.50 \begin {gather*} \frac {{\left (e x + d\right )}^{m}}{c d e m - c d e + {\left (c e^{2} m - c e^{2}\right )} x} \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),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + d\right )}^{m}}{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}\,{d x} \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),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 22, normalized size = 0.92 \begin {gather*} \frac {\left (e x +d \right )^{m -1}}{\left (m -1\right ) c e} \end {gather*}

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),x)

[Out]

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

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maxima [A]  time = 1.45, size = 27, normalized size = 1.12 \begin {gather*} \frac {{\left (e x + d\right )}^{m}}{c e^{2} {\left (m - 1\right )} x + c d e {\left (m - 1\right )}} \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),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.47, size = 28, normalized size = 1.17 \begin {gather*} \frac {{\left (d+e\,x\right )}^m}{c\,e^2\,\left (x+\frac {d}{e}\right )\,\left (m-1\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),x)

[Out]

(d + e*x)^m/(c*e^2*(x + d/e)*(m - 1))

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sympy [A]  time = 1.06, size = 63, normalized size = 2.62 \begin {gather*} \begin {cases} \text {NaN} & \text {for}\: d = 0 \wedge e = 0 \wedge m = 1 \\0^{m} \tilde {\infty } x & \text {for}\: d = - e x \\\frac {d^{m} x}{c d^{2}} & \text {for}\: e = 0 \\\frac {\log {\left (\frac {d}{e} + x \right )}}{c e} & \text {for}\: m = 1 \\\frac {\left (d + e x\right )^{m}}{c d e m - c d e + c e^{2} m x - c e^{2} x} & \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),x)

[Out]

Piecewise((nan, Eq(d, 0) & Eq(e, 0) & Eq(m, 1)), (0**m*zoo*x, Eq(d, -e*x)), (d**m*x/(c*d**2), Eq(e, 0)), (log(
d/e + x)/(c*e), Eq(m, 1)), ((d + e*x)**m/(c*d*e*m - c*d*e + c*e**2*m*x - c*e**2*x), True))

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