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

Optimal. Leaf size=65 \[ \frac{(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} (a e+c d x)^n}{c d (-m+n+1)} \]

[Out]

((a*e + c*d*x)^n*(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 + n))

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Rubi [A]  time = 0.0442912, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.021, Rules used = {858} \[ \frac{(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} (a e+c d x)^n}{c d (-m+n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*e + c*d*x)^n*(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 + n))

Rule 858

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

Rubi steps

\begin{align*} \int (a e+c d x)^n (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{(a e+c d x)^n (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+n)}\\ \end{align*}

Mathematica [A]  time = 0.0297271, size = 53, normalized size = 0.82 \[ \frac{(d+e x)^m ((d+e x) (a e+c d x))^{-m} (a e+c d x)^{n+1}}{-c d m+c d n+c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.049, size = 64, normalized size = 1. \begin{align*} -{\frac{ \left ( cdx+ae \right ) ^{1+n} \left ( ex+d \right ) ^{m}}{cd \left ( -1+m-n \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((c*d*x+a*e)^n*(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)^(1+n)/c/d/(-1+m-n)*(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 = 1.47024, size = 66, normalized size = 1.02 \begin{align*} -\frac{{\left (c d x + a e\right )} e^{\left (-m \log \left (c d x + a e\right ) + n \log \left (c d x + a e\right )\right )}}{c d{\left (m - n - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 1.36945, size = 144, normalized size = 2.22 \begin{align*} -\frac{{\left (c d x + a e\right )}{\left (c d x + a e\right )}^{n}{\left (e x + d\right )}^{m} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (e x + d\right )\right )}}{c d m - c d n - c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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((c*d*x+a*e)**n*(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 = 1.18307, size = 154, normalized size = 2.37 \begin{align*} -\frac{{\left (c d x + a e\right )}^{n}{\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 (c d x + a e\right )}^{n}{\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 n - c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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