3.2105 \(\int (d+e x)^m (a d e+(c d^2+a e^2) x+c d e x^2)^{-m} \, dx\)

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]

(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 = 0.02, 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 = {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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.88, size = 57, normalized size = 1.06 \[ -\frac {{\left (c d x + a e\right )} {\left (e x + d\right )}^{m}}{{\left (c d m - c d\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \]

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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giac [A]  time = 0.26, size = 87, normalized size = 1.61 \[ -\frac {{\left (x e + d\right )}^{m} c d x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right )\right )} + {\left (x e + d\right )}^{m} a e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 1\right )}}{c d m - c d} \]

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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maple [A]  time = 0.04, size = 57, normalized size = 1.06 \[ -\frac {\left (c d x +a e \right ) \left (e x +d \right )^{m} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{-m}}{\left (m -1\right ) c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c*d*x+a*e)/c/d/(m-1)*(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 = 1.16, size = 33, normalized size = 0.61 \[ -\frac {c d x + a e}{{\left (c d x + a e\right )}^{m} c d {\left (m - 1\right )}} \]

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

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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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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

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