3.31.91 \(\int (a+b x)^m (c+d x)^{-3-m} (e+f x) \, dx\) [3091]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {80, 37} \begin {gather*} \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} \end {gather*}

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 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]

Rule 80

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]

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*}

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Mathematica [A]
time = 0.10, size = 82, normalized size = 0.72 \begin {gather*} \frac {(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)} \end {gather*}

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))

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Maple [A]
time = 0.10, size = 158, normalized size = 1.39

method	result	size
gosper	\(-\frac {\left (d x +c \right )^{-2-m} \left (b x +a \right )^{1+m} \left (a d f m x -b c f m x +a d e m +2 a d f x -b c e m -b c f x -b d e x +a c f +a d e -2 b c e \right )}{a^{2} d^{2} m^{2}-2 a b c d \,m^{2}+b^{2} c^{2} m^{2}+3 a^{2} d^{2} m -6 a b c d m +3 b^{2} c^{2} m +2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}\)	\(158\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (117) = 234\).
time = 1.64, size = 336, normalized size = 2.95 \begin {gather*} -\frac {{\left (a^{2} c^{2} f - {\left ({\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 ({\left (b^{2} c^{2} - a^{2} d^{2}\right )} f m + {\left (b^{2} c^{2} - 2 \, a b c d - 2 \, a^{2} d^{2}\right )} f\right )} x^{2} + {\left (3 \, a^{2} c d f - {\left (a b c^{2} - a^{2} c d\right )} f m\right )} x - {\left (b^{2} d^{2} x^{3} + 2 \, a b c^{2} - a^{2} c d + {\left (3 \, b^{2} c d + {\left (b^{2} c d - a b d^{2}\right )} m\right )} x^{2} + {\left (a b c^{2} - a^{2} c d\right )} m + {\left (2 \, b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} + {\left (b^{2} c^{2} - a^{2} d^{2}\right )} m\right )} x\right )} e\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 {gather*}

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

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]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 2.80, size = 360, normalized size = 3.16 \begin {gather*} \frac {b\,d\,x^3\,{\left (a+b\,x\right )}^m\,\left (b\,c\,f-2\,a\,d\,f+b\,d\,e-a\,d\,f\,m+b\,c\,f\,m\right )}{{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{m+3}\,\left (m^2+3\,m+2\right )}-\frac {x\,{\left (a+b\,x\right )}^m\,\left (a^2\,d^2\,e-2\,b^2\,c^2\,e+3\,a^2\,c\,d\,f+a^2\,d^2\,e\,m-b^2\,c^2\,e\,m-2\,a\,b\,c\,d\,e-a\,b\,c^2\,f\,m+a^2\,c\,d\,f\,m\right )}{{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{m+3}\,\left (m^2+3\,m+2\right )}-\frac {a\,c\,{\left (a+b\,x\right )}^m\,\left (a\,c\,f+a\,d\,e-2\,b\,c\,e+a\,d\,e\,m-b\,c\,e\,m\right )}{{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{m+3}\,\left (m^2+3\,m+2\right )}-\frac {x^2\,{\left (a+b\,x\right )}^m\,\left (2\,a^2\,d^2\,f-b^2\,c^2\,f-3\,b^2\,c\,d\,e+a^2\,d^2\,f\,m-b^2\,c^2\,f\,m+2\,a\,b\,c\,d\,f+a\,b\,d^2\,e\,m-b^2\,c\,d\,e\,m\right )}{{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{m+3}\,\left (m^2+3\,m+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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