3.22.5 \(\int (d+e x)^m (a d e+(c d^2+a e^2) x+c d e x^2)^{-m} \, dx\) [2105]

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

[Out]

(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)

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Rubi [A]
time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, 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 = {662} \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}}{c d (1-m)} \end {gather*}

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 662

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*}

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Mathematica [A]
time = 0.03, size = 42, normalized size = 0.78 \begin {gather*} -\frac {(d+e x)^{-1+m} ((a e+c d x) (d+e x))^{1-m}}{c d (-1+m)} \end {gather*}

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 = 0.78, size = 57, normalized size = 1.06

method	result	size
gosper	\(-\frac {\left (c d x +a e \right ) \left (e x +d \right )^{m} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{-m}}{c d \left (-1+m \right )}\)	\(57\)
norman	\(\left (-\frac {x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{-1+m}-\frac {a e \,{\mathrm e}^{m \ln \left (e x +d \right )}}{c d \left (-1+m \right )}\right ) {\mathrm e}^{-m \ln \left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}\)	\(75\)
risch	\(-\frac {\left (c d x +a e \right ) \left (e x +d \right )^{m} {\mathrm e}^{-\frac {m \left (-i \mathrm {csgn}\left (i \left (e x +d \right ) \left (c d x +a e \right )\right )^{3} \pi +i \mathrm {csgn}\left (i \left (e x +d \right ) \left (c d x +a e \right )\right )^{2} \mathrm {csgn}\left (i \left (e x +d \right )\right ) \pi +i \mathrm {csgn}\left (i \left (e x +d \right ) \left (c d x +a e \right )\right )^{2} \mathrm {csgn}\left (i \left (c d x +a e \right )\right ) \pi -i \mathrm {csgn}\left (i \left (e x +d \right ) \left (c d x +a e \right )\right ) \mathrm {csgn}\left (i \left (e x +d \right )\right ) \mathrm {csgn}\left (i \left (c d x +a e \right )\right ) \pi +2 \ln \left (e x +d \right )+2 \ln \left (c d x +a e \right )\right )}{2}}}{\left (-1+m \right ) d c}\)	\(185\)

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,method=_RETURNVERBOSE)

[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 = 0.30, size = 35, normalized size = 0.65 \begin {gather*} -\frac {c d x + a e}{{\left (c d x + a e\right )}^{m} c d {\left (m - 1\right )}} \end {gather*}

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 = 2.69, size = 59, normalized size = 1.09 \begin {gather*} -\frac {{\left (c d x + a e\right )} {\left (x e + d\right )}^{m}}{{\left (c d m - c d\right )} {\left (c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e\right )}^{m}} \end {gather*}

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> Invalid NaN comparison

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Giac [A]
time = 1.68, size = 87, normalized size = 1.61 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 0.80, size = 57, normalized size = 1.06 \begin {gather*} -\frac {\left (a\,e+c\,d\,x\right )\,{\left (d+e\,x\right )}^m}{c\,d\,\left (m-1\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^m} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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