3.3136 \(\int \frac {(a+b x)^m (c+d x)^{3-m}}{(e+f x)^3} \, dx\)

Optimal. Leaf size=453 \[ \frac {(a+b x)^m (d e-c f) (c+d x)^{-m} \left (-a^2 d^2 f^2 \left (m^2-5 m+6\right )+2 a b d f (3-m) (2 d e-c f m)-\left (b^2 \left (-c^2 f^2 (1-m) m-4 c d e f m+6 d^2 e^2\right )\right )\right ) \, _2F_1\left (1,-m;1-m;\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 f^4 m (b e-a f)^2}+\frac {d^3 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m (b (3 d e-c f (3-m))-a d f m) \, _2F_1\left (m,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{b f^4 m (m+1) (b c-a d)}-\frac {3 d^3 (a+b x)^{m+1} (d e-c f) (c+d x)^{-m}}{f^4 m (b c-a d)}+\frac {(a+b x)^{m+1} (d e-c f)^2 (c+d x)^{-m} (b (c f (1-m)+5 d e)-a d f (6-m))}{2 f^3 (e+f x) (b e-a f)^2}-\frac {(a+b x)^{m+1} (d e-c f)^3 (c+d x)^{-m}}{2 f^3 (e+f x)^2 (b e-a f)} \]

[Out]

-3*d^3*(-c*f+d*e)*(b*x+a)^(1+m)/(-a*d+b*c)/f^4/m/((d*x+c)^m)-1/2*(-c*f+d*e)^3*(b*x+a)^(1+m)/f^3/(-a*f+b*e)/((d
*x+c)^m)/(f*x+e)^2+1/2*(-c*f+d*e)^2*(b*(5*d*e+c*f*(1-m))-a*d*f*(6-m))*(b*x+a)^(1+m)/f^3/(-a*f+b*e)^2/((d*x+c)^
m)/(f*x+e)+1/2*(-c*f+d*e)*(2*a*b*d*f*(3-m)*(-c*f*m+2*d*e)-b^2*(6*d^2*e^2-4*c*d*e*f*m-c^2*f^2*(1-m)*m)-a^2*d^2*
f^2*(m^2-5*m+6))*(b*x+a)^m*hypergeom([1, -m],[1-m],(-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))/f^4/(-a*f+b*e)^2/m/(
(d*x+c)^m)+d^3*(b*(3*d*e-c*f*(3-m))-a*d*f*m)*(b*x+a)^(1+m)*(b*(d*x+c)/(-a*d+b*c))^m*hypergeom([m, 1+m],[2+m],-
d*(b*x+a)/(-a*d+b*c))/b/(-a*d+b*c)/f^4/m/(1+m)/((d*x+c)^m)

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Rubi [C]  time = 0.05, antiderivative size = 113, normalized size of antiderivative = 0.25, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {137, 136} \[ \frac {(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m F_1\left (m+1;m-3,3;m+2;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b (m+1) (b e-a f)^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 136

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*e - a*
f)^p*(a + b*x)^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f
))])/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rule 137

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^
FracPart[n]/((b/(b*c - a*d))^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*((b*c)/(b*c
- a*d) + (b*d*x)/(b*c - a*d))^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&
!IntegerQ[n] && IntegerQ[p] &&  !GtQ[b/(b*c - a*d), 0] &&  !SimplerQ[c + d*x, a + b*x]

Rubi steps

\begin {align*} \int \frac {(a+b x)^m (c+d x)^{3-m}}{(e+f x)^3} \, dx &=\frac {\left ((b c-a d)^3 (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m\right ) \int \frac {(a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{3-m}}{(e+f x)^3} \, dx}{b^3}\\ &=\frac {(b c-a d)^3 (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m F_1\left (1+m;-3+m,3;2+m;-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b (b e-a f)^3 (1+m)}\\ \end {align*}

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Mathematica [C]  time = 0.44, size = 111, normalized size = 0.25 \[ \frac {(b c-a d)^3 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m F_1\left (m+1;m-3,3;m+2;\frac {d (a+b x)}{a d-b c},\frac {f (a+b x)}{a f-b e}\right )}{b (m+1) (b e-a f)^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 1.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m + 3}}{f^{3} x^{3} + 3 \, e f^{2} x^{2} + 3 \, e^{2} f x + e^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-m +3}}{\left (f x +e \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{3-m}}{{\left (e+f\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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