3.3125 \(\int (a+b x)^m (c+d x)^{-1-\frac{d (b e-a f) (1+m)}{b (d e-c f)}} (e+f x)^{-1+\frac{(b c-a d) f (1+m)}{b (d e-c f)}} \, dx\)

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 = 0.134797, 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 \[ \frac{b (a+b x)^{m+1} (c+d x)^{-\frac{d (m+1) (b e-a f)}{b (d e-c f)}} (e+f x)^{\frac{f (m+1) (b c-a d)}{b (d e-c f)}}}{(m+1) (b c-a d) (b e-a f)} \]

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))))

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Rubi in Sympy [A]  time = 18.4627, size = 76, normalized size = 0.75 \[ \frac{b \left (a + b x\right )^{m + 1} \left (c + d x\right )^{- \frac{d \left (m + 1\right ) \left (a f - b e\right )}{b \left (c f - d e\right )}} \left (e + f x\right )^{\frac{f \left (m + 1\right ) \left (a d - b c\right )}{b \left (c f - d e\right )}}}{\left (m + 1\right ) \left (a d - b c\right ) \left (a f - b e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

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

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Mathematica [C]  time = 18.3187, size = 1732, normalized size = 17.15 \[ -\frac{f (a f-b e) (c+d x)^{-\frac{d (b e-a f) (m+1)}{b (d e-c f)}} (e+f x)^{\frac{(b c-a d) f (m+1)}{b (d e-c f)}} F_1\left (m+1;\frac{d (b e-a f) (m+1)}{b (d e-c f)},\frac{(b c-a d) f (m+1)}{b (c f-d e)}+1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right ) (a+b x)^{m+1}}{(b e-a f) (c f-d e) (m+1) \left (\frac{f \left (\frac{(b c-a d) f (m+1)}{b (c f-d e)}+1\right ) (a+b x) F_1\left (m+1;\frac{d (b e-a f) (m+1)}{b (d e-c f)},\frac{(b c-a d) f (m+1)}{b (c f-d e)}+1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{m+1}+\frac{f (-b d e-a d f (m+1)+b c f (m+2)) (a+b x) F_1\left (m+1;\frac{d (b e-a f) (m+1)}{b (d e-c f)},\frac{(b c-a d) f (m+1)}{b (c f-d e)}+1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (d e-c f) (m+1)}+b (e+f x) F_1\left (m+1;\frac{d (b e-a f) (m+1)}{b (d e-c f)},\frac{(b c-a d) f (m+1)}{b (c f-d e)}+1;m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+\frac{(a+b x) (e+f x) \left (\frac{d^2 (m+1) F_1\left (m+2;\frac{d (b e-a f) (m+1)}{b (d e-c f)}+1,\frac{(b c-a d) f (m+1)}{b (c f-d e)}+1;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right ) (b e-a f)^2}{b c-a d}+f (b d e+a d f (m+1)-b c f (m+2)) F_1\left (m+2;\frac{d (b e-a f) (m+1)}{b (d e-c f)},\frac{(b c-a d) f (m+1)}{b (c f-d e)}+2;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )}{(b e-a f) (c f-d e) (m+2)}\right )}-\frac{d (a d-b c) (c+d x)^{-\frac{d (b e-a f) (m+1)}{b (d e-c f)}} (e+f x)^{\frac{(b c-a d) f (m+1)}{b (d e-c f)}} F_1\left (m+1;\frac{d (b e-a f) (m+1)}{b (d e-c f)}+1,\frac{(b c-a d) f (m+1)}{b (c f-d e)};m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right ) (a+b x)^{m+1}}{(b c-a d) (d e-c f) (m+1) \left (\frac{d \left (\frac{d (b e-a f) (m+1)}{b (d e-c f)}+1\right ) (a+b x) F_1\left (m+1;\frac{d (b e-a f) (m+1)}{b (d e-c f)}+1,\frac{(b c-a d) f (m+1)}{b (c f-d e)};m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{m+1}-\frac{d (-b c f-a d (m+1) f+b d e (m+2)) (a+b x) F_1\left (m+1;\frac{d (b e-a f) (m+1)}{b (d e-c f)}+1,\frac{(b c-a d) f (m+1)}{b (c f-d e)};m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b (d e-c f) (m+1)}+b (c+d x) F_1\left (m+1;\frac{d (b e-a f) (m+1)}{b (d e-c f)}+1,\frac{(b c-a d) f (m+1)}{b (c f-d e)};m+2;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )+\frac{(a+b x) (c+d x) \left (\frac{(b c-a d)^2 f^2 (m+1) F_1\left (m+2;\frac{d (b e-a f) (m+1)}{b (d e-c f)}+1,\frac{(b c-a d) f (m+1)}{b (c f-d e)}+1;m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )}{b e-a f}-d (-b c f-a d (m+1) f+b d e (m+2)) F_1\left (m+2;\frac{d (b e-a f) (m+1)}{b (d e-c f)}+2,\frac{(b c-a d) f (m+1)}{b (c f-d e)};m+3;\frac{d (a+b x)}{a d-b c},\frac{f (a+b x)}{a f-b e}\right )\right )}{(b c-a d) (d e-c f) (m+2)}\right )} \]

Warning: Unable to verify antiderivative.

[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]

-((f*(-(b*e) + a*f)*(a + b*x)^(1 + m)*(e + f*x)^(((b*c - a*d)*f*(1 + m))/(b*(d*e
- c*f)))*AppellF1[1 + m, (d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f)), 1 + ((b*c - a
*d)*f*(1 + m))/(b*(-(d*e) + c*f)), 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a +
b*x))/(-(b*e) + a*f)])/((b*e - a*f)*(-(d*e) + c*f)*(1 + m)*(c + d*x)^((d*(b*e -
a*f)*(1 + m))/(b*(d*e - c*f)))*((f*(1 + ((b*c - a*d)*f*(1 + m))/(b*(-(d*e) + c*f
)))*(a + b*x)*AppellF1[1 + m, (d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f)), 1 + ((b*c
- a*d)*f*(1 + m))/(b*(-(d*e) + c*f)), 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(
a + b*x))/(-(b*e) + a*f)])/(1 + m) + (f*(-(b*d*e) - a*d*f*(1 + m) + b*c*f*(2 + m
))*(a + b*x)*AppellF1[1 + m, (d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f)), 1 + ((b*c
- a*d)*f*(1 + m))/(b*(-(d*e) + c*f)), 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a
+ b*x))/(-(b*e) + a*f)])/(b*(d*e - c*f)*(1 + m)) + b*(e + f*x)*AppellF1[1 + m,
(d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f)), 1 + ((b*c - a*d)*f*(1 + m))/(b*(-(d*e)
+ c*f)), 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)] + ((
a + b*x)*(e + f*x)*(f*(b*d*e + a*d*f*(1 + m) - b*c*f*(2 + m))*AppellF1[2 + m, (d
*(b*e - a*f)*(1 + m))/(b*(d*e - c*f)), 2 + ((b*c - a*d)*f*(1 + m))/(b*(-(d*e) +
c*f)), 3 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)] + (d^2
*(b*e - a*f)^2*(1 + m)*AppellF1[2 + m, 1 + (d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f
)), 1 + ((b*c - a*d)*f*(1 + m))/(b*(-(d*e) + c*f)), 3 + m, (d*(a + b*x))/(-(b*c)
+ a*d), (f*(a + b*x))/(-(b*e) + a*f)])/(b*c - a*d)))/((b*e - a*f)*(-(d*e) + c*f
)*(2 + m))))) - (d*(-(b*c) + a*d)*(a + b*x)^(1 + m)*(e + f*x)^(((b*c - a*d)*f*(1
+ m))/(b*(d*e - c*f)))*AppellF1[1 + m, 1 + (d*(b*e - a*f)*(1 + m))/(b*(d*e - c*
f)), ((b*c - a*d)*f*(1 + m))/(b*(-(d*e) + c*f)), 2 + m, (d*(a + b*x))/(-(b*c) +
a*d), (f*(a + b*x))/(-(b*e) + a*f)])/((b*c - a*d)*(d*e - c*f)*(1 + m)*(c + d*x)^
((d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f)))*((d*(1 + (d*(b*e - a*f)*(1 + m))/(b*(d
*e - c*f)))*(a + b*x)*AppellF1[1 + m, 1 + (d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f)
), ((b*c - a*d)*f*(1 + m))/(b*(-(d*e) + c*f)), 2 + m, (d*(a + b*x))/(-(b*c) + a*
d), (f*(a + b*x))/(-(b*e) + a*f)])/(1 + m) - (d*(-(b*c*f) - a*d*f*(1 + m) + b*d*
e*(2 + m))*(a + b*x)*AppellF1[1 + m, 1 + (d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f))
, ((b*c - a*d)*f*(1 + m))/(b*(-(d*e) + c*f)), 2 + m, (d*(a + b*x))/(-(b*c) + a*d
), (f*(a + b*x))/(-(b*e) + a*f)])/(b*(d*e - c*f)*(1 + m)) + b*(c + d*x)*AppellF1
[1 + m, 1 + (d*(b*e - a*f)*(1 + m))/(b*(d*e - c*f)), ((b*c - a*d)*f*(1 + m))/(b*
(-(d*e) + c*f)), 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*
f)] + ((a + b*x)*(c + d*x)*(((b*c - a*d)^2*f^2*(1 + m)*AppellF1[2 + m, 1 + (d*(b
*e - a*f)*(1 + m))/(b*(d*e - c*f)), 1 + ((b*c - a*d)*f*(1 + m))/(b*(-(d*e) + c*f
)), 3 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)])/(b*e - a
*f) - d*(-(b*c*f) - a*d*f*(1 + m) + b*d*e*(2 + m))*AppellF1[2 + m, 2 + (d*(b*e -
a*f)*(1 + m))/(b*(d*e - c*f)), ((b*c - a*d)*f*(1 + m))/(b*(-(d*e) + c*f)), 3 +
m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)]))/((b*c - a*d)*(d
*e - c*f)*(2 + m))))

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Maple [A]  time = 0.008, size = 162, normalized size = 1.6 \[{\frac{b \left ( bx+a \right ) ^{1+m}}{{a}^{2}dfm-abcfm-abdem+{b}^{2}cem+{a}^{2}df-abcf-abde+{b}^{2}ce} \left ( fx+e \right ) ^{1+{\frac{adfm-bcfm+adf-2\,bcf+bde}{b \left ( cf-de \right ) }}} \left ( dx+c \right ) ^{1-{\frac{adfm-bdem+adf+bcf-2\,bde}{b \left ( cf-de \right ) }}}} \]

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 [A]  time = 1.61304, size = 311, normalized size = 3.08 \[ \frac{{\left (b^{2} x + a b\right )} e^{\left (\frac{a d f m \log \left (d x + c\right )}{b d e - b c f} - \frac{a d f m \log \left (f x + e\right )}{b d e - b c f} + \frac{a d f \log \left (d x + c\right )}{b d e - b c f} - \frac{d e m \log \left (d x + c\right )}{d e - c f} - \frac{a d f \log \left (f x + e\right )}{b d e - b c f} + \frac{c f m \log \left (f x + e\right )}{d e - c f} + m \log \left (b x + a\right ) - \frac{d e \log \left (d x + c\right )}{d e - c f} + \frac{c f \log \left (f x + e\right )}{d e - c f}\right )}}{b^{2} c e{\left (m + 1\right )} + a^{2} d f{\left (m + 1\right )} -{\left (d e{\left (m + 1\right )} + c f{\left (m + 1\right )}\right )} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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 [A]  time = 0.38499, size = 304, normalized size = 3.01 \[ \frac{{\left (b^{2} d f x^{3} + a b c e +{\left (b^{2} d e +{\left (b^{2} c + a b d\right )} f\right )} x^{2} +{\left (a b c f +{\left (b^{2} c + a b d\right )} e\right )} x\right )}{\left (b x + a\right )}^{m}}{{\left ({\left (b^{2} c - a b d\right )} e -{\left (a b c - a^{2} d\right )} f +{\left ({\left (b^{2} c - a b d\right )} e -{\left (a b c - a^{2} d\right )} f\right )} m\right )}{\left (d x + c\right )}^{\frac{2 \, b d e -{\left (b c + a d\right )} f +{\left (b d e - a d f\right )} m}{b d e - b c f}}{\left (f x + e\right )}^{\frac{b d e -{\left (b c - a d\right )} f m -{\left (2 \, b c - a d\right )} f}{b d e - b c f}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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. \[ \text{Timed out} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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)