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

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))

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Rubi [A]  time = 0.231958, 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 \[ \frac{g (d+e x)^m \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d e (2-m)}-\frac{(d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^2 d^2 e (1-m) (2-m)} \]

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))

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Rubi in Sympy [A]  time = 59.7488, size = 136, normalized size = 0.91 \[ \frac{g \left (d + e x\right )^{m} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{- m + 1}}{c d e \left (- m + 2\right )} - \frac{\left (d + e x\right )^{m - 1} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{- m + 1} \left (a e^{2} g - c d^{2} g m + c d^{2} g + c d e f m - 2 c d e f\right )}{c^{2} d^{2} e \left (- m + 1\right ) \left (- m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

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

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Mathematica [A]  time = 0.0912918, 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)))

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Maple [A]  time = 0.007, size = 89, normalized size = 0.6 \[ -{\frac{ \left ( ex+d \right ) ^{m} \left ( cdgmx+cdfm-xcdg+aeg-2\,cdf \right ) \left ( cdx+ae \right ) }{ \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{m}{c}^{2}{d}^{2} \left ({m}^{2}-3\,m+2 \right ) }} \]

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)

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Maxima [A]  time = 0.742372, size = 127, normalized size = 0.85 \[ -\frac{{\left (c d x + a e\right )}{\left (c d x + a e\right )}^{-m} f}{c d{\left (m - 1\right )}} - \frac{{\left (c^{2} d^{2}{\left (m - 1\right )} x^{2} + a c d e m x + a^{2} e^{2}\right )}{\left (c d x + a e\right )}^{-m} g}{{\left (m^{2} - 3 \, m + 2\right )} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)*(e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^m,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.285293, size = 196, normalized size = 1.31 \[ -\frac{{\left (a c d e f m - 2 \, a c d e f + a^{2} e^{2} g +{\left (c^{2} d^{2} g m - c^{2} d^{2} g\right )} x^{2} -{\left (2 \, c^{2} d^{2} f -{\left (c^{2} d^{2} f + a c d e g\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{{\left (c^{2} d^{2} m^{2} - 3 \, c^{2} d^{2} m + 2 \, c^{2} d^{2}\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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

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GIAC/XCAS [A]  time = 0.277623, size = 298, normalized size = 1.99 \[ -\frac{c^{2} d^{2} g m x^{2} e^{\left (-m{\rm ln}\left (c d x + a e\right )\right )} + c^{2} d^{2} f m x e^{\left (-m{\rm ln}\left (c d x + a e\right )\right )} - c^{2} d^{2} g x^{2} e^{\left (-m{\rm ln}\left (c d x + a e\right )\right )} + a c d g m x e^{\left (-m{\rm ln}\left (c d x + a e\right ) + 1\right )} - 2 \, c^{2} d^{2} f x e^{\left (-m{\rm ln}\left (c d x + a e\right )\right )} + a c d f m e^{\left (-m{\rm ln}\left (c d x + a e\right ) + 1\right )} - 2 \, a c d f e^{\left (-m{\rm ln}\left (c d x + a e\right ) + 1\right )} + a^{2} g e^{\left (-m{\rm ln}\left (c d x + a e\right ) + 2\right )}}{c^{2} d^{2} m^{2} - 3 \, c^{2} d^{2} m + 2 \, c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)*(e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^m,x, algorithm="giac")

[Out]

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