3.31.50 \(\int (a+b x) (c+d x)^{-3+n} (e+f x)^{-n} \, dx\) [3050]

Optimal. Leaf size=123 \[ \frac {(b c-a d) (c+d x)^{-2+n} (e+f x)^{1-n}}{d (d e-c f) (2-n)}+\frac {(a d f+b (c f (1-n)-d e (2-n))) (c+d x)^{-1+n} (e+f x)^{1-n}}{d (d e-c f)^2 (1-n) (2-n)} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 122, normalized size of antiderivative = 0.99, 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 {(b c-a d) (c+d x)^{n-2} (e+f x)^{1-n}}{d (2-n) (d e-c f)}+\frac {(c+d x)^{n-1} (e+f x)^{1-n} (a d f+b c f (1-n)-b d e (2-n))}{d (1-n) (2-n) (d e-c f)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.11, size = 82, normalized size = 0.67 \begin {gather*} \frac {(c+d x)^{-2+n} (e+f x)^{1-n} (-a c f (-2+n)+a d e (-1+n)+a d f x+b d e (-2+n) x-b c (e+f (-1+n) x))}{(d e-c f)^2 (-2+n) (-1+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.10, size = 161, normalized size = 1.31

method	result	size
gosper	\(-\frac {\left (d x +c \right )^{-2+n} \left (f x +e \right ) \left (b c f n x -b d e n x +a c f n -a d e n -a d f x -b c f x +2 b d e x -2 a c f +a d e +b c e \right ) \left (f x +e \right )^{-n}}{c^{2} f^{2} n^{2}-2 c d e f \,n^{2}+d^{2} e^{2} n^{2}-3 c^{2} f^{2} n +6 c d e f n -3 d^{2} e^{2} n +2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}}\)	\(161\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(d*x+c)^(-3+n)/((f*x+e)^n),x,method=_RETURNVERBOSE)

[Out]

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

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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)*(d*x+c)^(-3+n)/((f*x+e)^n),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (119) = 238\).
time = 2.10, size = 320, normalized size = 2.60 \begin {gather*} -\frac {{\left ({\left (b c d f^{2} n - {\left (b c d + a d^{2}\right )} f^{2}\right )} x^{3} + {\left ({\left (b c^{2} + a c d\right )} f^{2} n - {\left (b c^{2} + 3 \, a c d\right )} f^{2}\right )} x^{2} + {\left (a c^{2} f^{2} n - 2 \, a c^{2} f^{2}\right )} x - {\left (a c d n - b c^{2} - a c d + {\left (b d^{2} n - 2 \, b d^{2}\right )} x^{2} - {\left (3 \, b c d + a d^{2} - {\left (b c d + a d^{2}\right )} n\right )} x\right )} e^{2} + {\left (a c^{2} f n - 2 \, a c^{2} f - {\left (b d^{2} f n - 2 \, b d^{2} f\right )} x^{3} - {\left (a d^{2} f n - 2 \, b c d f\right )} x^{2} + {\left (b c^{2} f n - 2 \, a c d f\right )} x\right )} e\right )} {\left (d x + c\right )}^{n - 3}}{{\left (c^{2} f^{2} n^{2} - 3 \, c^{2} f^{2} n + 2 \, c^{2} f^{2} + {\left (d^{2} n^{2} - 3 \, d^{2} n + 2 \, d^{2}\right )} e^{2} - 2 \, {\left (c d f n^{2} - 3 \, c d f n + 2 \, c d f\right )} e\right )} {\left (f x + e\right )}^{n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)**(-3+n)/((f*x+e)**n),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3006 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1058 vs. \(2 (119) = 238\).
time = 0.51, size = 1058, normalized size = 8.60 \begin {gather*} -\frac {\frac {b c d f^{2} n x^{3} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right )\right )}}{{\left (f x + e\right )}^{n}} - \frac {b d^{2} f n x^{3} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 1\right )}}{{\left (f x + e\right )}^{n}} + \frac {b c^{2} f^{2} n x^{2} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right )\right )}}{{\left (f x + e\right )}^{n}} + \frac {a c d f^{2} n x^{2} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right )\right )}}{{\left (f x + e\right )}^{n}} - \frac {b c d f^{2} x^{3} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right )\right )}}{{\left (f x + e\right )}^{n}} - \frac {a d^{2} f^{2} x^{3} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right )\right )}}{{\left (f x + e\right )}^{n}} - \frac {a d^{2} f n x^{2} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 1\right )}}{{\left (f x + e\right )}^{n}} + \frac {2 \, b d^{2} f x^{3} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 1\right )}}{{\left (f x + e\right )}^{n}} + \frac {a c^{2} f^{2} n x e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right )\right )}}{{\left (f x + e\right )}^{n}} - \frac {b c^{2} f^{2} x^{2} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right )\right )}}{{\left (f x + e\right )}^{n}} - \frac {3 \, a c d f^{2} x^{2} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right )\right )}}{{\left (f x + e\right )}^{n}} - \frac {b d^{2} n x^{2} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 2\right )}}{{\left (f x + e\right )}^{n}} + \frac {b c^{2} f n x e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 1\right )}}{{\left (f x + e\right )}^{n}} + \frac {2 \, b c d f x^{2} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 1\right )}}{{\left (f x + e\right )}^{n}} - \frac {2 \, a c^{2} f^{2} x e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right )\right )}}{{\left (f x + e\right )}^{n}} - \frac {b c d n x e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 2\right )}}{{\left (f x + e\right )}^{n}} - \frac {a d^{2} n x e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 2\right )}}{{\left (f x + e\right )}^{n}} + \frac {2 \, b d^{2} x^{2} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 2\right )}}{{\left (f x + e\right )}^{n}} + \frac {a c^{2} f n e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 1\right )}}{{\left (f x + e\right )}^{n}} - \frac {2 \, a c d f x e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 1\right )}}{{\left (f x + e\right )}^{n}} - \frac {a c d n e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 2\right )}}{{\left (f x + e\right )}^{n}} + \frac {3 \, b c d x e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 2\right )}}{{\left (f x + e\right )}^{n}} + \frac {a d^{2} x e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 2\right )}}{{\left (f x + e\right )}^{n}} - \frac {2 \, a c^{2} f e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 1\right )}}{{\left (f x + e\right )}^{n}} + \frac {b c^{2} e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 2\right )}}{{\left (f x + e\right )}^{n}} + \frac {a c d e^{\left (n \log \left (d x + c\right ) - 3 \, \log \left (d x + c\right ) + 2\right )}}{{\left (f x + e\right )}^{n}}}{c^{2} f^{2} n^{2} - 2 \, c d f n^{2} e - 3 \, c^{2} f^{2} n + d^{2} n^{2} e^{2} + 6 \, c d f n e + 2 \, c^{2} f^{2} - 3 \, d^{2} n e^{2} - 4 \, c d f e + 2 \, d^{2} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 2.85, size = 358, normalized size = 2.91 \begin {gather*} \frac {x\,{\left (c+d\,x\right )}^{n-3}\,\left (2\,a\,c^2\,f^2-a\,d^2\,e^2-3\,b\,c\,d\,e^2-a\,c^2\,f^2\,n+a\,d^2\,e^2\,n+2\,a\,c\,d\,e\,f+b\,c\,d\,e^2\,n-b\,c^2\,e\,f\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^2\,\left (n^2-3\,n+2\right )}+\frac {x^2\,{\left (c+d\,x\right )}^{n-3}\,\left (b\,c^2\,f^2-2\,b\,d^2\,e^2+3\,a\,c\,d\,f^2-b\,c^2\,f^2\,n+b\,d^2\,e^2\,n-2\,b\,c\,d\,e\,f-a\,c\,d\,f^2\,n+a\,d^2\,e\,f\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^2\,\left (n^2-3\,n+2\right )}-\frac {c\,e\,{\left (c+d\,x\right )}^{n-3}\,\left (a\,d\,e-2\,a\,c\,f+b\,c\,e+a\,c\,f\,n-a\,d\,e\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^2\,\left (n^2-3\,n+2\right )}+\frac {d\,f\,x^3\,{\left (c+d\,x\right )}^{n-3}\,\left (a\,d\,f+b\,c\,f-2\,b\,d\,e-b\,c\,f\,n+b\,d\,e\,n\right )}{{\left (e+f\,x\right )}^n\,{\left (c\,f-d\,e\right )}^2\,\left (n^2-3\,n+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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