3.3056 \(\int (a+b x)^{-n} (c+d x) (e+f x)^{-3+n} \, dx\)

Optimal. Leaf size=125 \[ -\frac {(a+b x)^{1-n} (d e-c f) (e+f x)^{n-2}}{f (2-n) (b e-a f)}-\frac {(a+b x)^{1-n} (e+f x)^{n-1} (a d f (2-n)-b (c f+d (e-e n)))}{f (1-n) (2-n) (b e-a f)^2} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 123, 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} \[ \frac {(a+b x)^{1-n} (e+f x)^{n-1} (-a d f (2-n)+b c f+b d (e-e n))}{f (1-n) (2-n) (b e-a f)^2}-\frac {(a+b x)^{1-n} (d e-c f) (e+f x)^{n-2}}{f (2-n) (b e-a f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 84, normalized size = 0.67 \[ \frac {(a+b x)^{1-n} (e+f x)^{n-2} (a c f (n-1)-a d e+a d f (n-2) x+b c (f x-e (n-2))-b d e (n-1) x)}{(n-2) (n-1) (b e-a f)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.00, size = 326, normalized size = 2.61 \[ -\frac {{\left (a^{2} c e f - {\left (b^{2} d e f + {\left (b^{2} c - 2 \, a b d\right )} f^{2} - {\left (b^{2} d e f - a b d f^{2}\right )} n\right )} x^{3} - {\left (2 \, a b c - a^{2} d\right )} e^{2} - {\left (b^{2} d e^{2} - 2 \, a^{2} d f^{2} + {\left (3 \, b^{2} c - 2 \, a b d\right )} e f - {\left (b^{2} d e^{2} + b^{2} c e f - {\left (a b c + a^{2} d\right )} f^{2}\right )} n\right )} x^{2} + {\left (a b c e^{2} - a^{2} c e f\right )} n - {\left (2 \, b^{2} c e^{2} - a^{2} c f^{2} + {\left (2 \, a b c - 3 \, a^{2} d\right )} e f + {\left (a^{2} d e f + a^{2} c f^{2} - {\left (b^{2} c + a b d\right )} e^{2}\right )} n\right )} x\right )} {\left (f x + e\right )}^{n - 3}}{{\left (2 \, b^{2} e^{2} - 4 \, a b e f + 2 \, a^{2} f^{2} + {\left (b^{2} e^{2} - 2 \, a b e f + a^{2} f^{2}\right )} n^{2} - 3 \, {\left (b^{2} e^{2} - 2 \, a b e f + a^{2} f^{2}\right )} n\right )} {\left (b x + a\right )}^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.16, size = 1085, normalized size = 8.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 160, normalized size = 1.28 \[ \frac {\left (b x +a \right ) \left (a d f n x -b d e n x +a c f n -2 a d f x -b c e n +b c f x +b d e x -a c f -a d e +2 b c e \right ) \left (b x +a \right )^{-n} \left (f x +e \right )^{n -2}}{a^{2} f^{2} n^{2}-2 a b e f \,n^{2}+b^{2} e^{2} n^{2}-3 a^{2} f^{2} n +6 a b e f n -3 b^{2} e^{2} n +2 a^{2} f^{2}-4 a b e f +2 b^{2} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )} {\left (f x + e\right )}^{n - 3}}{{\left (b x + a\right )}^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.80, size = 360, normalized size = 2.88 \[ \frac {b\,f\,x^3\,{\left (e+f\,x\right )}^{n-3}\,\left (b\,c\,f-2\,a\,d\,f+b\,d\,e+a\,d\,f\,n-b\,d\,e\,n\right )}{{\left (a\,f-b\,e\right )}^2\,{\left (a+b\,x\right )}^n\,\left (n^2-3\,n+2\right )}-\frac {x^2\,{\left (e+f\,x\right )}^{n-3}\,\left (2\,a^2\,d\,f^2-b^2\,d\,e^2-3\,b^2\,c\,e\,f-a^2\,d\,f^2\,n+b^2\,d\,e^2\,n+2\,a\,b\,d\,e\,f-a\,b\,c\,f^2\,n+b^2\,c\,e\,f\,n\right )}{{\left (a\,f-b\,e\right )}^2\,{\left (a+b\,x\right )}^n\,\left (n^2-3\,n+2\right )}-\frac {a\,e\,{\left (e+f\,x\right )}^{n-3}\,\left (a\,c\,f+a\,d\,e-2\,b\,c\,e-a\,c\,f\,n+b\,c\,e\,n\right )}{{\left (a\,f-b\,e\right )}^2\,{\left (a+b\,x\right )}^n\,\left (n^2-3\,n+2\right )}-\frac {x\,{\left (e+f\,x\right )}^{n-3}\,\left (a^2\,c\,f^2-2\,b^2\,c\,e^2+3\,a^2\,d\,e\,f-a^2\,c\,f^2\,n+b^2\,c\,e^2\,n-2\,a\,b\,c\,e\,f+a\,b\,d\,e^2\,n-a^2\,d\,e\,f\,n\right )}{{\left (a\,f-b\,e\right )}^2\,{\left (a+b\,x\right )}^n\,\left (n^2-3\,n+2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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