3.32.20 \(\int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^5} \, dx\) [3120]

Optimal. Leaf size=311 \[ -\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{4 (b e-a f) (d e-c f) (e+f x)^4}-\frac {f (b (5 d e-c f (3-m))-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{12 (b e-a f)^2 (d e-c f)^2 (e+f x)^3}-\frac {(b c-a d)^2 \left (2 a b d f (4 d e-c f (2-m)) (1+m)-a^2 d^2 f^2 \left (2+3 m+m^2\right )-b^2 \left (12 d^2 e^2-8 c d e f (2-m)+c^2 f^2 \left (6-5 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (3,1+m;2+m;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{12 (b e-a f)^5 (d e-c f)^2 (1+m)} \]

[Out]

-1/4*f*(b*x+a)^(1+m)*(d*x+c)^(2-m)/(-a*f+b*e)/(-c*f+d*e)/(f*x+e)^4-1/12*f*(b*(5*d*e-c*f*(3-m))-a*d*f*(2+m))*(b
*x+a)^(1+m)*(d*x+c)^(2-m)/(-a*f+b*e)^2/(-c*f+d*e)^2/(f*x+e)^3-1/12*(-a*d+b*c)^2*(2*a*b*d*f*(4*d*e-c*f*(2-m))*(
1+m)-a^2*d^2*f^2*(m^2+3*m+2)-b^2*(12*d^2*e^2-8*c*d*e*f*(2-m)+c^2*f^2*(m^2-5*m+6)))*(b*x+a)^(1+m)*(d*x+c)^(-1-m
)*hypergeom([3, 1+m],[2+m],(-c*f+d*e)*(b*x+a)/(-a*f+b*e)/(d*x+c))/(-a*f+b*e)^5/(-c*f+d*e)^2/(1+m)

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Rubi [A]
time = 0.24, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {105, 156, 12, 133} \begin {gather*} -\frac {(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m-1} \left (-a^2 d^2 f^2 \left (m^2+3 m+2\right )+2 a b d f (m+1) (4 d e-c f (2-m))-\left (b^2 \left (c^2 f^2 \left (m^2-5 m+6\right )-8 c d e f (2-m)+12 d^2 e^2\right )\right )\right ) \, _2F_1\left (3,m+1;m+2;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{12 (m+1) (b e-a f)^5 (d e-c f)^2}-\frac {f (a+b x)^{m+1} (c+d x)^{2-m} (-a d f (m+2)-b c f (3-m)+5 b d e)}{12 (e+f x)^3 (b e-a f)^2 (d e-c f)^2}-\frac {f (a+b x)^{m+1} (c+d x)^{2-m}}{4 (e+f x)^4 (b e-a f) (d e-c f)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^m*(c + d*x)^(1 - m))/(e + f*x)^5,x]

[Out]

-1/4*(f*(a + b*x)^(1 + m)*(c + d*x)^(2 - m))/((b*e - a*f)*(d*e - c*f)*(e + f*x)^4) - (f*(5*b*d*e - b*c*f*(3 -
m) - a*d*f*(2 + m))*(a + b*x)^(1 + m)*(c + d*x)^(2 - m))/(12*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^3) - ((b*c
- a*d)^2*(2*a*b*d*f*(4*d*e - c*f*(2 - m))*(1 + m) - a^2*d^2*f^2*(2 + 3*m + m^2) - b^2*(12*d^2*e^2 - 8*c*d*e*f*
(2 - m) + c^2*f^2*(6 - 5*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*Hypergeometric2F1[3, 1 + m, 2 + m, ((
d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/(12*(b*e - a*f)^5*(d*e - c*f)^2*(1 + m))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 105

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] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 133

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*c - a
*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2,
(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

Rule 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(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] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^5} \, dx &=-\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{4 (b e-a f) (d e-c f) (e+f x)^4}-\frac {\int \frac {(a+b x)^m (c+d x)^{1-m} (-b (4 d e-c f (3-m))+a d f (2+m)+b d f x)}{(e+f x)^4} \, dx}{4 (b e-a f) (d e-c f)}\\ &=-\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{4 (b e-a f) (d e-c f) (e+f x)^4}-\frac {f (5 b d e-b c f (3-m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{12 (b e-a f)^2 (d e-c f)^2 (e+f x)^3}+\frac {\int \frac {\left (-2 a b d f (4 d e-c f (2-m)) (1+m)+a^2 d^2 f^2 \left (2+3 m+m^2\right )+b^2 \left (12 d^2 e^2-8 c d e f (2-m)+c^2 f^2 \left (6-5 m+m^2\right )\right )\right ) (a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx}{12 (b e-a f)^2 (d e-c f)^2}\\ &=-\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{4 (b e-a f) (d e-c f) (e+f x)^4}-\frac {f (5 b d e-b c f (3-m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{12 (b e-a f)^2 (d e-c f)^2 (e+f x)^3}-\frac {\left (2 a b d f (4 d e-c f (2-m)) (1+m)-a^2 d^2 f^2 \left (2+3 m+m^2\right )-b^2 \left (12 d^2 e^2-8 c d e f (2-m)+c^2 f^2 \left (6-5 m+m^2\right )\right )\right ) \int \frac {(a+b x)^m (c+d x)^{1-m}}{(e+f x)^3} \, dx}{12 (b e-a f)^2 (d e-c f)^2}\\ &=-\frac {f (a+b x)^{1+m} (c+d x)^{2-m}}{4 (b e-a f) (d e-c f) (e+f x)^4}-\frac {f (5 b d e-b c f (3-m)-a d f (2+m)) (a+b x)^{1+m} (c+d x)^{2-m}}{12 (b e-a f)^2 (d e-c f)^2 (e+f x)^3}-\frac {(b c-a d)^2 \left (2 a b d f (4 d e-c f (2-m)) (1+m)-a^2 d^2 f^2 \left (2+3 m+m^2\right )-b^2 \left (12 d^2 e^2-8 c d e f (2-m)+c^2 f^2 \left (6-5 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m} \, _2F_1\left (3,1+m;2+m;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{12 (b e-a f)^5 (d e-c f)^2 (1+m)}\\ \end {align*}

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Mathematica [A]
time = 0.71, size = 271, normalized size = 0.87 \begin {gather*} \frac {(a+b x)^{1+m} (c+d x)^{-1-m} \left (-\frac {3 f (c+d x)^3}{(e+f x)^4}-\frac {f (5 b d e+b c f (-3+m)-a d f (2+m)) (c+d x)^3}{(b e-a f) (d e-c f) (e+f x)^3}+\frac {(b c-a d)^2 \left (-2 a b d f (4 d e+c f (-2+m)) (1+m)+a^2 d^2 f^2 \left (2+3 m+m^2\right )+b^2 \left (12 d^2 e^2+8 c d e f (-2+m)+c^2 f^2 \left (6-5 m+m^2\right )\right )\right ) \, _2F_1\left (3,1+m;2+m;\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^4 (d e-c f) (1+m)}\right )}{12 (b e-a f) (d e-c f)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^m*(c + d*x)^(1 - m))/(e + f*x)^5,x]

[Out]

((a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*((-3*f*(c + d*x)^3)/(e + f*x)^4 - (f*(5*b*d*e + b*c*f*(-3 + m) - a*d*f*(
2 + m))*(c + d*x)^3)/((b*e - a*f)*(d*e - c*f)*(e + f*x)^3) + ((b*c - a*d)^2*(-2*a*b*d*f*(4*d*e + c*f*(-2 + m))
*(1 + m) + a^2*d^2*f^2*(2 + 3*m + m^2) + b^2*(12*d^2*e^2 + 8*c*d*e*f*(-2 + m) + c^2*f^2*(6 - 5*m + m^2)))*Hype
rgeometric2F1[3, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)^4*(d*e - c*f)*(1
+ m))))/(12*(b*e - a*f)*(d*e - c*f))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{1-m}}{\left (f x +e \right )^{5}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^m*(d*x+c)^(1-m)/(f*x+e)^5,x)

[Out]

int((b*x+a)^m*(d*x+c)^(1-m)/(f*x+e)^5,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(1-m)/(f*x+e)^5,x, algorithm="maxima")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^5, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(1-m)/(f*x+e)^5,x, algorithm="fricas")

[Out]

integral((b*x + a)^m*(d*x + c)^(-m + 1)/(f^5*x^5 + 5*f^4*x^4*e + 10*f^3*x^3*e^2 + 10*f^2*x^2*e^3 + 5*f*x*e^4 +
e^5), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**m*(d*x+c)**(1-m)/(f*x+e)**5,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^m*(d*x+c)^(1-m)/(f*x+e)^5,x, algorithm="giac")

[Out]

integrate((b*x + a)^m*(d*x + c)^(-m + 1)/(f*x + e)^5, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^{1-m}}{{\left (e+f\,x\right )}^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^m*(c + d*x)^(1 - m))/(e + f*x)^5,x)

[Out]

int(((a + b*x)^m*(c + d*x)^(1 - m))/(e + f*x)^5, x)

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