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

Optimal. Leaf size=150 \[ \frac{g (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d e (2-m)}-\frac{(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^2 d^2 e (1-m) (2-m)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0816185, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {794, 648} \[ \frac{g (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d e (2-m)}-\frac{(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^2 d^2 e (1-m) (2-m)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 794

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

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

Mathematica [A]  time = 0.0626045, size = 67, normalized size = 0.45 \[ -\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} (a e g+c d (f (m-2)+g (m-1) x))}{c^2 d^2 (m-2) (m-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^m*(f + g*x))/(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)*(a*e*g + c*d*(f*(-2 + m) + g*(-1 + m)*x)))/(c^2*d^2*(-
2 + m)*(-1 + m)))

________________________________________________________________________________________

Maple [A]  time = 0.048, size = 89, normalized size = 0.6 \begin{align*} -{\frac{ \left ( ex+d \right ) ^{m} \left ( cdgmx+cdfm-xcdg+aeg-2\,cdf \right ) \left ( cdx+ae \right ) }{ \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{m}{c}^{2}{d}^{2} \left ({m}^{2}-3\,m+2 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(e*x+d)^m*(c*d*g*m*x+c*d*f*m-c*d*g*x+a*e*g-2*c*d*f)*(c*d*x+a*e)/((c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^m)/c^2/d^2
/(m^2-3*m+2)

________________________________________________________________________________________

Maxima [A]  time = 1.06504, size = 127, normalized size = 0.85 \begin{align*} -\frac{{\left (c d x + a e\right )} f}{{\left (c d x + a e\right )}^{m} c d{\left (m - 1\right )}} - \frac{{\left (c^{2} d^{2}{\left (m - 1\right )} x^{2} + a c d e m x + a^{2} e^{2}\right )} g}{{\left (m^{2} - 3 \, m + 2\right )}{\left (c d x + a e\right )}^{m} c^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)/((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)*f/((c*d*x + a*e)^m*c*d*(m - 1)) - (c^2*d^2*(m - 1)*x^2 + a*c*d*e*m*x + a^2*e^2)*g/((m^2 - 3*m +
2)*(c*d*x + a*e)^m*c^2*d^2)

________________________________________________________________________________________

Fricas [A]  time = 1.36285, size = 292, normalized size = 1.95 \begin{align*} -\frac{{\left (a c d e f m - 2 \, a c d e f + a^{2} e^{2} g +{\left (c^{2} d^{2} g m - c^{2} d^{2} g\right )} x^{2} -{\left (2 \, c^{2} d^{2} f -{\left (c^{2} d^{2} f + a c d e g\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{{\left (c^{2} d^{2} m^{2} - 3 \, c^{2} d^{2} m + 2 \, c^{2} d^{2}\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*(g*x+f)/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x, algorithm="fricas")

[Out]

-(a*c*d*e*f*m - 2*a*c*d*e*f + a^2*e^2*g + (c^2*d^2*g*m - c^2*d^2*g)*x^2 - (2*c^2*d^2*f - (c^2*d^2*f + a*c*d*e*
g)*m)*x)*(e*x + d)^m/((c^2*d^2*m^2 - 3*c^2*d^2*m + 2*c^2*d^2)*(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^m)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.27108, size = 498, normalized size = 3.32 \begin{align*} -\frac{{\left (x e + d\right )}^{m} c^{2} d^{2} g m x^{2} 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} c^{2} d^{2} f m 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} c^{2} d^{2} g x^{2} 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 c d g m x e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 1\right )} - 2 \,{\left (x e + d\right )}^{m} c^{2} d^{2} f 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 c d f m e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 1\right )} - 2 \,{\left (x e + d\right )}^{m} a c d f e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 1\right )} +{\left (x e + d\right )}^{m} a^{2} g e^{\left (-m \log \left (c d x + a e\right ) - m \log \left (x e + d\right ) + 2\right )}}{c^{2} d^{2} m^{2} - 3 \, c^{2} d^{2} m + 2 \, c^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)/((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^2*d^2*g*m*x^2*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) + (x*e + d)^m*c^2*d^2*f*m*x*e^(-m*log(c
*d*x + a*e) - m*log(x*e + d)) - (x*e + d)^m*c^2*d^2*g*x^2*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) + (x*e + d)
^m*a*c*d*g*m*x*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) - 2*(x*e + d)^m*c^2*d^2*f*x*e^(-m*log(c*d*x + a*e)
- m*log(x*e + d)) + (x*e + d)^m*a*c*d*f*m*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) - 2*(x*e + d)^m*a*c*d*
f*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) + (x*e + d)^m*a^2*g*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 2
))/(c^2*d^2*m^2 - 3*c^2*d^2*m + 2*c^2*d^2)