3.8.80 \(\int (a e+c d x)^n (d+e x)^m (a d e+(c d^2+a e^2) x+c d e x^2)^{-m} \, dx\) [780]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 = {872} \begin {gather*} \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)} \end {gather*}

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 872

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] &&  !Intege
rQ[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*}

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Mathematica [A]
time = 0.04, size = 53, normalized size = 0.82 \begin {gather*} \frac {(a e+c d x)^{1+n} (d+e x)^m ((a e+c d x) (d+e x))^{-m}}{c d-c d m+c d n} \end {gather*}

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 = 0.17, size = 64, normalized size = 0.98

method	result	size
gosper	\(-\frac {\left (c d x +a e \right )^{1+n} \left (e x +d \right )^{m} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{-m}}{c d \left (-1+m -n \right )}\)	\(64\)
risch	\(-\frac {\left (e x +d \right )^{m} \left (c d x +a e \right )^{n} \left (c d x +a e \right ) {\mathrm e}^{\frac {m \left (i \pi \mathrm {csgn}\left (i \left (e x +d \right ) \left (c d x +a e \right )\right )^{3}-i \pi \mathrm {csgn}\left (i \left (e x +d \right ) \left (c d x +a e \right )\right )^{2} \mathrm {csgn}\left (i \left (e x +d \right )\right )-i \pi \mathrm {csgn}\left (i \left (e x +d \right ) \left (c d x +a e \right )\right )^{2} \mathrm {csgn}\left (i \left (c d x +a e \right )\right )+i \pi \,\mathrm {csgn}\left (i \left (e x +d \right ) \left (c d x +a e \right )\right ) \mathrm {csgn}\left (i \left (e x +d \right )\right ) \mathrm {csgn}\left (i \left (c d x +a e \right )\right )-2 \ln \left (e x +d \right )-2 \ln \left (c d x +a e \right )\right )}{2}}}{c d \left (-1+m -n \right )}\)	\(198\)

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,method=_RETURNVERBOSE)

[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 = 0.29, size = 52, normalized size = 0.80 \begin {gather*} -\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 {gather*}

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 = 4.09, size = 71, normalized size = 1.09 \begin {gather*} -\frac {{\left (c d x + a e\right )} {\left (c d x + a e\right )}^{n} {\left (x e + d\right )}^{m} e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right )\right )}}{c d m - c d n - c d} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 4.29, size = 114, normalized size = 1.75 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 3.54, size = 63, normalized size = 0.97 \begin {gather*} \frac {{\left (a\,e+c\,d\,x\right )}^{n+1}\,{\left (d+e\,x\right )}^m}{c\,d\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m\,\left (n-m+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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