3.3050 \(\int (a+b x) (c+d x)^{-3+n} (e+f x)^{-n} \, dx\)
Optimal. Leaf size=123 \[ \frac{(b c-a d) (c+d x)^{n-2} (e+f x)^{1-n}}{d (2-n) (d e-c f)}+\frac{(c+d x)^{n-1} (e+f x)^{1-n} (a d f+b (c f (1-n)-d e (2-n)))}{d (1-n) (2-n) (d e-c f)^2} \]
[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) - d*e*(2
- n)))*(c + d*x)^(-1 + n)*(e + f*x)^(1 - n))/(d*(d*e - c*f)^2*(1 - n)*(2 - n))
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Rubi [A] time = 0.0580893, 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 = {79, 37} \[ \frac{(b c-a d) (c+d x)^{n-2} (e+f x)^{1-n}}{d (2-n) (d e-c f)}+\frac{(c+d x)^{n-1} (e+f x)^{1-n} (a d f+b c f (1-n)-b d e (2-n))}{d (1-n) (2-n) (d e-c f)^2} \]
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 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) (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*}
Mathematica [A] time = 0.0686276, size = 82, normalized size = 0.67 \[ \frac{(c+d x)^{n-2} (e+f x)^{1-n} (-a c f (n-2)+a d e (n-1)+a d f x-b c (e+f (n-1) x)+b d e (n-2) x)}{(n-2) (n-1) (d e-c f)^2} \]
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 = 0.005, size = 161, normalized size = 1.3 \begin{align*} -{\frac{ \left ( dx+c \right ) ^{-2+n} \left ( fx+e \right ) \left ( bcfnx-bdenx+acfn-aden-adfx-bcfx+2\,bdex-2\,acf+ade+bce \right ) }{ \left ({c}^{2}{f}^{2}{n}^{2}-2\,cdef{n}^{2}+{d}^{2}{e}^{2}{n}^{2}-3\,{c}^{2}{f}^{2}n+6\,cdefn-3\,{d}^{2}{e}^{2}n+2\,{c}^{2}{f}^{2}-4\,cdef+2\,{d}^{2}{e}^{2} \right ) \left ( fx+e \right ) ^{n}}} \end{align*}
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)
[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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}{\left (d x + c\right )}^{n - 3}}{{\left (f x + e\right )}^{n}}\,{d x} \end{align*}
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] time = 1.78985, size = 666, normalized size = 5.41 \begin{align*} \frac{{\left (2 \, a c^{2} e f -{\left (2 \, b d^{2} e f -{\left (b c d + a d^{2}\right )} f^{2} -{\left (b d^{2} e f - b c d f^{2}\right )} n\right )} x^{3} -{\left (b c^{2} + a c d\right )} e^{2} -{\left (2 \, b d^{2} e^{2} + 2 \, b c d e f -{\left (b c^{2} + 3 \, a c d\right )} f^{2} -{\left (b d^{2} e^{2} + a d^{2} e f -{\left (b c^{2} + a c d\right )} f^{2}\right )} n\right )} x^{2} +{\left (a c d e^{2} - a c^{2} e f\right )} n +{\left (2 \, a c d e f + 2 \, a c^{2} f^{2} -{\left (3 \, b c d + a d^{2}\right )} e^{2} -{\left (b c^{2} e f + a c^{2} f^{2} -{\left (b c d + a d^{2}\right )} e^{2}\right )} n\right )} x\right )}{\left (d x + c\right )}^{n - 3}}{{\left (2 \, d^{2} e^{2} - 4 \, c d e f + 2 \, c^{2} f^{2} +{\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} n^{2} - 3 \,{\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} n\right )}{\left (f x + e\right )}^{n}} \end{align*}
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]
(2*a*c^2*e*f - (2*b*d^2*e*f - (b*c*d + a*d^2)*f^2 - (b*d^2*e*f - b*c*d*f^2)*n)*x^3 - (b*c^2 + a*c*d)*e^2 - (2*
b*d^2*e^2 + 2*b*c*d*e*f - (b*c^2 + 3*a*c*d)*f^2 - (b*d^2*e^2 + a*d^2*e*f - (b*c^2 + a*c*d)*f^2)*n)*x^2 + (a*c*
d*e^2 - a*c^2*e*f)*n + (2*a*c*d*e*f + 2*a*c^2*f^2 - (3*b*c*d + a*d^2)*e^2 - (b*c^2*e*f + a*c^2*f^2 - (b*c*d +
a*d^2)*e^2)*n)*x)*(d*x + c)^(n - 3)/((2*d^2*e^2 - 4*c*d*e*f + 2*c^2*f^2 + (d^2*e^2 - 2*c*d*e*f + c^2*f^2)*n^2
- 3*(d^2*e^2 - 2*c*d*e*f + c^2*f^2)*n)*(f*x + e)^n)
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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((b*x+a)*(d*x+c)**(-3+n)/((f*x+e)**n),x)
[Out]
Timed out
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Giac [B] time = 1.92276, size = 1428, normalized size = 11.61 \begin{align*} \text{result too large to display} \end{align*}
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)