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

Optimal. Leaf size=246 \[ -\frac{2 (d+e x)^{m-1} (c d f-a e g) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^3 d^3 e (1-m) (2-m) (3-m)}+\frac{2 g (d+e x)^m (c d f-a e g) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c^2 d^2 e (2-m) (3-m)}+\frac{(f+g x)^2 (d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (3-m)} \]

[Out]

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

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Rubi [A]  time = 0.204594, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068, Rules used = {870, 794, 648} \[ -\frac{2 (d+e x)^{m-1} (c d f-a e g) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m} \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )}{c^3 d^3 e (1-m) (2-m) (3-m)}+\frac{2 g (d+e x)^m (c d f-a e g) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c^2 d^2 e (2-m) (3-m)}+\frac{(f+g x)^2 (d+e x)^{m-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{1-m}}{c d (3-m)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 870

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] - Dist[(n*(c*e*f + c*d*g
- b*e*g))/(c*e*(m - n - 1)), Int[(d + e*x)^m*(f + g*x)^(n - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, 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] &&  !Integ
erQ[p] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (IntegerQ[2*p] || IntegerQ[n])

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)^2 \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} (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d (3-m)}+\frac{(2 (c d f-a e g)) \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}{c d (3-m)}\\ &=\frac{2 g (c d f-a e 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^2 d^2 e (2-m) (3-m)}+\frac{(d+e x)^{-1+m} (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d (3-m)}-\frac{\left (2 (c d f-a e g) \left (a e^2 g+c d (d g (1-m)-e f (2-m))\right )\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^2 d^2 e (2-m) (3-m)}\\ &=-\frac{2 (c d f-a e g) \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^3 d^3 e (1-m) (2-m) (3-m)}+\frac{2 g (c d f-a e 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^2 d^2 e (2-m) (3-m)}+\frac{(d+e x)^{-1+m} (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1-m}}{c d (3-m)}\\ \end{align*}

Mathematica [A]  time = 0.11471, size = 131, normalized size = 0.53 \[ -\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} \left (2 a^2 e^2 g^2+2 a c d e g (f (m-3)+g (m-1) x)+c^2 d^2 \left (f^2 \left (m^2-5 m+6\right )+2 f g \left (m^2-4 m+3\right ) x+g^2 \left (m^2-3 m+2\right ) x^2\right )\right )}{c^3 d^3 (m-3) (m-2) (m-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^m*(f + g*x)^2)/(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)*(2*a^2*e^2*g^2 + 2*a*c*d*e*g*(f*(-3 + m) + g*(-1 + m)*
x) + c^2*d^2*(f^2*(6 - 5*m + m^2) + 2*f*g*(3 - 4*m + m^2)*x + g^2*(2 - 3*m + m^2)*x^2)))/(c^3*d^3*(-3 + m)*(-2
+ m)*(-1 + m)))

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Maple [A]  time = 0.055, size = 235, normalized size = 1. \begin{align*} -{\frac{ \left ({c}^{2}{d}^{2}{g}^{2}{m}^{2}{x}^{2}+2\,{c}^{2}{d}^{2}fg{m}^{2}x-3\,{c}^{2}{d}^{2}{g}^{2}m{x}^{2}+2\,acde{g}^{2}mx+{c}^{2}{d}^{2}{f}^{2}{m}^{2}-8\,{c}^{2}{d}^{2}fgmx+2\,{g}^{2}{x}^{2}{c}^{2}{d}^{2}+2\,acdefgm-2\,acde{g}^{2}x-5\,{c}^{2}{d}^{2}{f}^{2}m+6\,{c}^{2}{d}^{2}fgx+2\,{a}^{2}{e}^{2}{g}^{2}-6\,acdefg+6\,{c}^{2}{d}^{2}{f}^{2} \right ) \left ( cdx+ae \right ) \left ( ex+d \right ) ^{m}}{ \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{m}{c}^{3}{d}^{3} \left ({m}^{3}-6\,{m}^{2}+11\,m-6 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c*d*x+a*e)*(c^2*d^2*g^2*m^2*x^2+2*c^2*d^2*f*g*m^2*x-3*c^2*d^2*g^2*m*x^2+2*a*c*d*e*g^2*m*x+c^2*d^2*f^2*m^2-8*
c^2*d^2*f*g*m*x+2*c^2*d^2*g^2*x^2+2*a*c*d*e*f*g*m-2*a*c*d*e*g^2*x-5*c^2*d^2*f^2*m+6*c^2*d^2*f*g*x+2*a^2*e^2*g^
2-6*a*c*d*e*f*g+6*c^2*d^2*f^2)*(e*x+d)^m/((c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^m)/c^3/d^3/(m^3-6*m^2+11*m-6)

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Maxima [A]  time = 1.10489, size = 261, normalized size = 1.06 \begin{align*} -\frac{{\left (c d x + a e\right )} f^{2}}{{\left (c d x + a e\right )}^{m} c d{\left (m - 1\right )}} - \frac{2 \,{\left (c^{2} d^{2}{\left (m - 1\right )} x^{2} + a c d e m x + a^{2} e^{2}\right )} f g}{{\left (m^{2} - 3 \, m + 2\right )}{\left (c d x + a e\right )}^{m} c^{2} d^{2}} - \frac{{\left ({\left (m^{2} - 3 \, m + 2\right )} c^{3} d^{3} x^{3} +{\left (m^{2} - m\right )} a c^{2} d^{2} e x^{2} + 2 \, a^{2} c d e^{2} m x + 2 \, a^{3} e^{3}\right )} g^{2}}{{\left (m^{3} - 6 \, m^{2} + 11 \, m - 6\right )}{\left (c d x + a e\right )}^{m} c^{3} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)^2/((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^2/((c*d*x + a*e)^m*c*d*(m - 1)) - 2*(c^2*d^2*(m - 1)*x^2 + a*c*d*e*m*x + a^2*e^2)*f*g/((m^2 -
3*m + 2)*(c*d*x + a*e)^m*c^2*d^2) - ((m^2 - 3*m + 2)*c^3*d^3*x^3 + (m^2 - m)*a*c^2*d^2*e*x^2 + 2*a^2*c*d*e^2*
m*x + 2*a^3*e^3)*g^2/((m^3 - 6*m^2 + 11*m - 6)*(c*d*x + a*e)^m*c^3*d^3)

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Fricas [A]  time = 1.34981, size = 695, normalized size = 2.83 \begin{align*} -\frac{{\left (a c^{2} d^{2} e f^{2} m^{2} + 6 \, a c^{2} d^{2} e f^{2} - 6 \, a^{2} c d e^{2} f g + 2 \, a^{3} e^{3} g^{2} +{\left (c^{3} d^{3} g^{2} m^{2} - 3 \, c^{3} d^{3} g^{2} m + 2 \, c^{3} d^{3} g^{2}\right )} x^{3} +{\left (6 \, c^{3} d^{3} f g +{\left (2 \, c^{3} d^{3} f g + a c^{2} d^{2} e g^{2}\right )} m^{2} -{\left (8 \, c^{3} d^{3} f g + a c^{2} d^{2} e g^{2}\right )} m\right )} x^{2} -{\left (5 \, a c^{2} d^{2} e f^{2} - 2 \, a^{2} c d e^{2} f g\right )} m +{\left (6 \, c^{3} d^{3} f^{2} +{\left (c^{3} d^{3} f^{2} + 2 \, a c^{2} d^{2} e f g\right )} m^{2} -{\left (5 \, c^{3} d^{3} f^{2} + 6 \, a c^{2} d^{2} e f g - 2 \, a^{2} c d e^{2} g^{2}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{{\left (c^{3} d^{3} m^{3} - 6 \, c^{3} d^{3} m^{2} + 11 \, c^{3} d^{3} m - 6 \, c^{3} d^{3}\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)^2/((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^m),x, algorithm="fricas")

[Out]

-(a*c^2*d^2*e*f^2*m^2 + 6*a*c^2*d^2*e*f^2 - 6*a^2*c*d*e^2*f*g + 2*a^3*e^3*g^2 + (c^3*d^3*g^2*m^2 - 3*c^3*d^3*g
^2*m + 2*c^3*d^3*g^2)*x^3 + (6*c^3*d^3*f*g + (2*c^3*d^3*f*g + a*c^2*d^2*e*g^2)*m^2 - (8*c^3*d^3*f*g + a*c^2*d^
2*e*g^2)*m)*x^2 - (5*a*c^2*d^2*e*f^2 - 2*a^2*c*d*e^2*f*g)*m + (6*c^3*d^3*f^2 + (c^3*d^3*f^2 + 2*a*c^2*d^2*e*f*
g)*m^2 - (5*c^3*d^3*f^2 + 6*a*c^2*d^2*e*f*g - 2*a^2*c*d*e^2*g^2)*m)*x)*(e*x + d)^m/((c^3*d^3*m^3 - 6*c^3*d^3*m
^2 + 11*c^3*d^3*m - 6*c^3*d^3)*(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. \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)**2/((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 [B]  time = 1.32125, size = 1324, normalized size = 5.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(g*x+f)^2/((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^3*d^3*g^2*m^2*x^3*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) + 2*(x*e + d)^m*c^3*d^3*f*g*m^2*x^2
*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) - 3*(x*e + d)^m*c^3*d^3*g^2*m*x^3*e^(-m*log(c*d*x + a*e) - m*log(x*e
+ d)) + (x*e + d)^m*a*c^2*d^2*g^2*m^2*x^2*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) + (x*e + d)^m*c^3*d^3*
f^2*m^2*x*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) - 8*(x*e + d)^m*c^3*d^3*f*g*m*x^2*e^(-m*log(c*d*x + a*e) -
m*log(x*e + d)) + 2*(x*e + d)^m*c^3*d^3*g^2*x^3*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) + 2*(x*e + d)^m*a*c^2
*d^2*f*g*m^2*x*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) - (x*e + d)^m*a*c^2*d^2*g^2*m*x^2*e^(-m*log(c*d*x
+ a*e) - m*log(x*e + d) + 1) - 5*(x*e + d)^m*c^3*d^3*f^2*m*x*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) + 6*(x*e
+ d)^m*c^3*d^3*f*g*x^2*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) + (x*e + d)^m*a*c^2*d^2*f^2*m^2*e^(-m*log(c*d
*x + a*e) - m*log(x*e + d) + 1) - 6*(x*e + d)^m*a*c^2*d^2*f*g*m*x*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1)
+ 6*(x*e + d)^m*c^3*d^3*f^2*x*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)) + 2*(x*e + d)^m*a^2*c*d*g^2*m*x*e^(-m*
log(c*d*x + a*e) - m*log(x*e + d) + 2) - 5*(x*e + d)^m*a*c^2*d^2*f^2*m*e^(-m*log(c*d*x + a*e) - m*log(x*e + d)
+ 1) + 2*(x*e + d)^m*a^2*c*d*f*g*m*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 2) + 6*(x*e + d)^m*a*c^2*d^2*f^2
*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 1) - 6*(x*e + d)^m*a^2*c*d*f*g*e^(-m*log(c*d*x + a*e) - m*log(x*e +
d) + 2) + 2*(x*e + d)^m*a^3*g^2*e^(-m*log(c*d*x + a*e) - m*log(x*e + d) + 3))/(c^3*d^3*m^3 - 6*c^3*d^3*m^2 +
11*c^3*d^3*m - 6*c^3*d^3)