3.3091 \(\int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx\)

Optimal. Leaf size=114 \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-2}}{d (m+2) (b c-a d)}-\frac{(a+b x)^{m+1} (c+d x)^{-m-1} (a d f (m+2)-b (c f (m+1)+d e))}{d (m+1) (m+2) (b c-a d)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0523963, antiderivative size = 112, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {79, 37} \[ \frac{(a+b x)^{m+1} (d e-c f) (c+d x)^{-m-2}}{d (m+2) (b c-a d)}+\frac{(a+b x)^{m+1} (c+d x)^{-m-1} (-a d f (m+2)+b c f (m+1)+b d e)}{d (m+1) (m+2) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^(-3 - m)*(e + f*x),x]

[Out]

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

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c,
d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimplerQ[p, 1]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx &=\frac{(d e-c f) (a+b x)^{1+m} (c+d x)^{-2-m}}{d (b c-a d) (2+m)}+\frac{(b d e+b c f (1+m)-a d f (2+m)) \int (a+b x)^m (c+d x)^{-2-m} \, dx}{d (b c-a d) (2+m)}\\ &=\frac{(d e-c f) (a+b x)^{1+m} (c+d x)^{-2-m}}{d (b c-a d) (2+m)}+\frac{(b d e+b c f (1+m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{-1-m}}{d (b c-a d)^2 (1+m) (2+m)}\\ \end{align*}

Mathematica [A]  time = 0.0628513, size = 82, normalized size = 0.72 \[ \frac{(a+b x)^{m+1} (c+d x)^{-m-2} (b (c e (m+2)+c f (m+1) x+d e x)-a (c f+d e (m+1)+d f (m+2) x))}{(m+1) (m+2) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m*(c + d*x)^(-3 - m)*(e + f*x),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 158, normalized size = 1.4 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+m} \left ( dx+c \right ) ^{-2-m} \left ( adfmx-bcfmx+adem+2\,adfx-bcem-bcfx-bdex+acf+ade-2\,bce \right ) }{{a}^{2}{d}^{2}{m}^{2}-2\,abcd{m}^{2}+{b}^{2}{c}^{2}{m}^{2}+3\,{a}^{2}{d}^{2}m-6\,abcdm+3\,{b}^{2}{c}^{2}m+2\,{a}^{2}{d}^{2}-4\,abcd+2\,{b}^{2}{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^(-3-m)*(f*x+e),x)

[Out]

-(b*x+a)^(1+m)*(d*x+c)^(-2-m)*(a*d*f*m*x-b*c*f*m*x+a*d*e*m+2*a*d*f*x-b*c*e*m-b*c*f*x-b*d*e*x+a*c*f+a*d*e-2*b*c
*e)/(a^2*d^2*m^2-2*a*b*c*d*m^2+b^2*c^2*m^2+3*a^2*d^2*m-6*a*b*c*d*m+3*b^2*c^2*m+2*a^2*d^2-4*a*b*c*d+2*b^2*c^2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-3-m)*(f*x+e),x, algorithm="maxima")

[Out]

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 3), x)

________________________________________________________________________________________

Fricas [B]  time = 1.74312, size = 671, normalized size = 5.89 \begin{align*} -\frac{{\left (a^{2} c^{2} f -{\left (b^{2} d^{2} e +{\left (b^{2} c d - a b d^{2}\right )} f m +{\left (b^{2} c d - 2 \, a b d^{2}\right )} f\right )} x^{3} -{\left (a b c^{2} - a^{2} c d\right )} e m -{\left (3 \, b^{2} c d e +{\left (b^{2} c^{2} - 2 \, a b c d - 2 \, a^{2} d^{2}\right )} f +{\left ({\left (b^{2} c d - a b d^{2}\right )} e +{\left (b^{2} c^{2} - a^{2} d^{2}\right )} f\right )} m\right )} x^{2} -{\left (2 \, a b c^{2} - a^{2} c d\right )} e +{\left (3 \, a^{2} c d f -{\left (2 \, b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2}\right )} e -{\left ({\left (b^{2} c^{2} - a^{2} d^{2}\right )} e +{\left (a b c^{2} - a^{2} c d\right )} f\right )} m\right )} x\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}}{2 \, b^{2} c^{2} - 4 \, a b c d + 2 \, a^{2} d^{2} +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m^{2} + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-3-m)*(f*x+e),x, algorithm="fricas")

[Out]

-(a^2*c^2*f - (b^2*d^2*e + (b^2*c*d - a*b*d^2)*f*m + (b^2*c*d - 2*a*b*d^2)*f)*x^3 - (a*b*c^2 - a^2*c*d)*e*m -
(3*b^2*c*d*e + (b^2*c^2 - 2*a*b*c*d - 2*a^2*d^2)*f + ((b^2*c*d - a*b*d^2)*e + (b^2*c^2 - a^2*d^2)*f)*m)*x^2 -
(2*a*b*c^2 - a^2*c*d)*e + (3*a^2*c*d*f - (2*b^2*c^2 + 2*a*b*c*d - a^2*d^2)*e - ((b^2*c^2 - a^2*d^2)*e + (a*b*c
^2 - a^2*c*d)*f)*m)*x)*(b*x + a)^m*(d*x + c)^(-m - 3)/(2*b^2*c^2 - 4*a*b*c*d + 2*a^2*d^2 + (b^2*c^2 - 2*a*b*c*
d + a^2*d^2)*m^2 + 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*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((b*x+a)**m*(d*x+c)**(-3-m)*(f*x+e),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-m - 3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-3-m)*(f*x+e),x, algorithm="giac")

[Out]

integrate((f*x + e)*(b*x + a)^m*(d*x + c)^(-m - 3), x)