3.771 \(\int (d+e x)^m (a d e+(c d^2+a e^2) x+c d e x^2)^{-m} \, dx\)

Optimal. Leaf size=54 \[ \frac{(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (1-m)} \]

[Out]

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

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Rubi [A]  time = 0.0149937, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.027, Rules used = {648} \[ \frac{(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (1-m)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0214816, size = 42, normalized size = 0.78 \[ -\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m}}{c d (m-1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 57, normalized size = 1.1 \begin{align*} -{\frac{ \left ( cdx+ae \right ) \left ( ex+d \right ) ^{m}}{cd \left ( -1+m \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{m}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01867, size = 45, normalized size = 0.83 \begin{align*} -\frac{c d x + a e}{{\left (c d x + a e\right )}^{m} c d{\left (m - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.40221, size = 116, normalized size = 2.15 \begin{align*} -\frac{{\left (c d x + a e\right )}{\left (e x + d\right )}^{m}}{{\left (c d m - c d\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.31714, size = 117, normalized size = 2.17 \begin{align*} -\frac{{\left (x e + d\right )}^{m} c d x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right )\right )} +{\left (x e + d\right )}^{m} a e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 1\right )}}{c d m - c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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