3.3153 \(\int (a+b x)^m (c+d x)^{-1-\frac{d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac{(b c-a d) f (1+m)}{b (d e-c f)}} \, dx\)

Optimal. Leaf size=101 \[ \frac{b (a+b x)^{m+1} (c+d x)^{-\frac{d (m+1) (b e-a f)}{b (d e-c f)}} (e+f x)^{\frac{f (m+1) (b c-a d)}{b (d e-c f)}}}{(m+1) (b c-a d) (b e-a f)} \]

[Out]

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

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Rubi [A]  time = 0.033537, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 77, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.013, Rules used = {95} \[ \frac{b (a+b x)^{m+1} (c+d x)^{-\frac{d (m+1) (b e-a f)}{b (d e-c f)}} (e+f x)^{\frac{f (m+1) (b c-a d)}{b (d e-c f)}}}{(m+1) (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^m*(c + d*x)^(-1 - (d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f)))*(e + f*x)^(-1 + ((b*c - a*d)*f*(1 + m
))/(b*(d*e - c*f))),x]

[Out]

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

Rule 95

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.125438, size = 101, normalized size = 1. \[ \frac{b (a+b x)^{m+1} (c+d x)^{-\frac{d (m+1) (b e-a f)}{b (d e-c f)}} (e+f x)^{\frac{f (m+1) (b c-a d)}{b (d e-c f)}}}{(m+1) (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^m*(c + d*x)^(-1 - (d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f)))*(e + f*x)^(-1 + ((b*c - a*d)*f*
(1 + m))/(b*(d*e - c*f))),x]

[Out]

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

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Maple [A]  time = 0.004, size = 162, normalized size = 1.6 \begin{align*}{\frac{b \left ( bx+a \right ) ^{1+m}}{{a}^{2}dfm-abcfm-abdem+{b}^{2}cem+{a}^{2}df-abcf-abde+{b}^{2}ce} \left ( fx+e \right ) ^{1+{\frac{adfm-bcfm+adf-2\,bcf+bde}{b \left ( cf-de \right ) }}} \left ( dx+c \right ) ^{1-{\frac{adfm-bdem+adf+bcf-2\,bde}{b \left ( cf-de \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 2.30364, size = 311, normalized size = 3.08 \begin{align*} \frac{{\left (b^{2} x + a b\right )} e^{\left (\frac{a d f m \log \left (d x + c\right )}{b d e - b c f} - \frac{a d f m \log \left (f x + e\right )}{b d e - b c f} + \frac{a d f \log \left (d x + c\right )}{b d e - b c f} - \frac{d e m \log \left (d x + c\right )}{d e - c f} - \frac{a d f \log \left (f x + e\right )}{b d e - b c f} + \frac{c f m \log \left (f x + e\right )}{d e - c f} + m \log \left (b x + a\right ) - \frac{d e \log \left (d x + c\right )}{d e - c f} + \frac{c f \log \left (f x + e\right )}{d e - c f}\right )}}{b^{2} c e{\left (m + 1\right )} + a^{2} d f{\left (m + 1\right )} -{\left (d e{\left (m + 1\right )} + c f{\left (m + 1\right )}\right )} a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-1-d*(-a*f+b*e)*(1+m)/b/(-c*f+d*e))*(f*x+e)^(-1+(-a*d+b*c)*f*(1+m)/b/(-c*f+d*e)),
x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.9023, size = 458, normalized size = 4.53 \begin{align*} \frac{{\left (b^{2} d f x^{3} + a b c e +{\left (b^{2} d e +{\left (b^{2} c + a b d\right )} f\right )} x^{2} +{\left (a b c f +{\left (b^{2} c + a b d\right )} e\right )} x\right )}{\left (b x + a\right )}^{m}}{{\left ({\left (b^{2} c - a b d\right )} e -{\left (a b c - a^{2} d\right )} f +{\left ({\left (b^{2} c - a b d\right )} e -{\left (a b c - a^{2} d\right )} f\right )} m\right )}{\left (d x + c\right )}^{\frac{2 \, b d e -{\left (b c + a d\right )} f +{\left (b d e - a d f\right )} m}{b d e - b c f}}{\left (f x + e\right )}^{\frac{b d e -{\left (b c - a d\right )} f m -{\left (2 \, b c - a d\right )} f}{b d e - b c f}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-1-d*(-a*f+b*e)*(1+m)/b/(-c*f+d*e))*(f*x+e)^(-1+(-a*d+b*c)*f*(1+m)/b/(-c*f+d*e)),
x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{m}{\left (d x + c\right )}^{-\frac{{\left (b e - a f\right )} d{\left (m + 1\right )}}{{\left (d e - c f\right )} b} - 1}{\left (f x + e\right )}^{\frac{{\left (b c - a d\right )} f{\left (m + 1\right )}}{{\left (d e - c f\right )} b} - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(-1-d*(-a*f+b*e)*(1+m)/b/(-c*f+d*e))*(f*x+e)^(-1+(-a*d+b*c)*f*(1+m)/b/(-c*f+d*e)),
x, algorithm="giac")

[Out]

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