3.32.37 \(\int \frac {(a+b x)^{1-n} (c+d x)^{1+n}}{b c+a d+2 b d x} \, dx\) [3137]

Optimal. Leaf size=245 \[ \frac {(b c-a d) (3-2 n) (a+b x)^{2-n} (c+d x)^{-1+n}}{8 b^3 (1-n)}+\frac {d (a+b x)^{3-n} (c+d x)^{-1+n}}{4 b^3}+\frac {(b c-a d)^2 (a+b x)^{1-n} (c+d x)^{-1+n} \, _2F_1\left (1,-1+n;n;-\frac {b (c+d x)}{d (a+b x)}\right )}{8 b^3 d (1-n)}-\frac {(b c-a d)^2 \left (1-2 n^2\right ) (a+b x)^{-n} \left (-\frac {d (a+b x)}{b c-a d}\right )^n (c+d x)^n \, _2F_1\left (-1+n,n;1+n;\frac {b (c+d x)}{b c-a d}\right )}{8 b^2 d^2 (1-n) n} \]

[Out]

1/8*(-a*d+b*c)*(3-2*n)*(b*x+a)^(2-n)*(d*x+c)^(-1+n)/b^3/(1-n)+1/4*d*(b*x+a)^(3-n)*(d*x+c)^(-1+n)/b^3+1/8*(-a*d
+b*c)^2*(b*x+a)^(1-n)*(d*x+c)^(-1+n)*hypergeom([1, -1+n],[n],-b*(d*x+c)/d/(b*x+a))/b^3/d/(1-n)-1/8*(-a*d+b*c)^
2*(-2*n^2+1)*(-d*(b*x+a)/(-a*d+b*c))^n*(d*x+c)^n*hypergeom([n, -1+n],[1+n],b*(d*x+c)/(-a*d+b*c))/b^2/d^2/(1-n)
/n/((b*x+a)^n)

________________________________________________________________________________________

Rubi [A]
time = 0.20, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {135, 133, 965, 80, 72, 71} \begin {gather*} \frac {(b c-a d)^2 (a+b x)^{1-n} (c+d x)^{n-1} \, _2F_1\left (1,n-1;n;-\frac {b (c+d x)}{d (a+b x)}\right )}{8 b^3 d (1-n)}+\frac {(3-2 n) (b c-a d) (a+b x)^{2-n} (c+d x)^{n-1}}{8 b^3 (1-n)}+\frac {d (a+b x)^{3-n} (c+d x)^{n-1}}{4 b^3}-\frac {\left (1-2 n^2\right ) (b c-a d)^2 (a+b x)^{-n} (c+d x)^n \left (-\frac {d (a+b x)}{b c-a d}\right )^n \, _2F_1\left (n-1,n;n+1;\frac {b (c+d x)}{b c-a d}\right )}{8 b^2 d^2 (1-n) n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(1 - n)*(c + d*x)^(1 + n))/(b*c + a*d + 2*b*d*x),x]

[Out]

((b*c - a*d)*(3 - 2*n)*(a + b*x)^(2 - n)*(c + d*x)^(-1 + n))/(8*b^3*(1 - n)) + (d*(a + b*x)^(3 - n)*(c + d*x)^
(-1 + n))/(4*b^3) + ((b*c - a*d)^2*(a + b*x)^(1 - n)*(c + d*x)^(-1 + n)*Hypergeometric2F1[1, -1 + n, n, -((b*(
c + d*x))/(d*(a + b*x)))])/(8*b^3*d*(1 - n)) - ((b*c - a*d)^2*(1 - 2*n^2)*(-((d*(a + b*x))/(b*c - a*d)))^n*(c
+ d*x)^n*Hypergeometric2F1[-1 + n, n, 1 + n, (b*(c + d*x))/(b*c - a*d)])/(8*b^2*d^2*(1 - n)*n*(a + b*x)^n)

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 72

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), 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*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c -
a*d)), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 80

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 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 135

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[(c*f - d*e)^
(m + n + 1)/f^(m + n + 1), Int[(a + b*x)^m/((c + d*x)^(m + 1)*(e + f*x)), x], x] + Dist[1/f^(m + n + 1), Int[(
(a + b*x)^m/(c + d*x)^(m + 1))*ExpandToSum[(f^(m + n + 1)*(c + d*x)^(m + n + 1) - (c*f - d*e)^(m + n + 1))/(e
+ f*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[m + n + 1, 0] && (LtQ[m, 0] || SumSimplerQ[m,
1] ||  !(LtQ[n, 0] || SumSimplerQ[n, 1]))

Rule 965

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x)^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Dist[1/(g*e^(2*p)*(m +
n + 2*p + 1)), Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x + c*x^2)^p - c^p*
(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x
] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && NeQ[m + n + 2*
p + 1, 0] && (IntegerQ[n] ||  !IntegerQ[m])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.56, size = 257, normalized size = 1.05 \begin {gather*} \frac {(-b c+a d) (a+b x)^{-n} (c+d x)^n \left ((b c-a d) \, _2F_1\left (1,-n;1-n;-\frac {d (a+b x)}{b (c+d x)}\right )+\frac {\left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (4 d n (1+n) (a+b x) \, _2F_1\left (-1-n,1-n;2-n;\frac {d (a+b x)}{-b c+a d}\right )-(-1+n) \left ((b c-a d) (1+n) \, _2F_1\left (-n,-n;1-n;\frac {d (a+b x)}{-b c+a d}\right )-2 b n (c+d x) \left (-\frac {b d (a+b x) (c+d x)}{(b c-a d)^2}\right )^n \, _2F_1\left (n,1+n;2+n;\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{-1+n^2}\right )}{8 b^2 d^2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(1 - n)*(c + d*x)^(1 + n))/(b*c + a*d + 2*b*d*x),x]

[Out]

((-(b*c) + a*d)*(c + d*x)^n*((b*c - a*d)*Hypergeometric2F1[1, -n, 1 - n, -((d*(a + b*x))/(b*(c + d*x)))] + (4*
d*n*(1 + n)*(a + b*x)*Hypergeometric2F1[-1 - n, 1 - n, 2 - n, (d*(a + b*x))/(-(b*c) + a*d)] - (-1 + n)*((b*c -
a*d)*(1 + n)*Hypergeometric2F1[-n, -n, 1 - n, (d*(a + b*x))/(-(b*c) + a*d)] - 2*b*n*(c + d*x)*(-((b*d*(a + b*
x)*(c + d*x))/(b*c - a*d)^2))^n*Hypergeometric2F1[n, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)]))/((-1 + n^2)*((
b*(c + d*x))/(b*c - a*d))^n)))/(8*b^2*d^2*n*(a + b*x)^n)

________________________________________________________________________________________

Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{1-n} \left (d x +c \right )^{1+n}}{2 b d x +a d +b c}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1-n)*(d*x+c)^(1+n)/(2*b*d*x+a*d+b*c),x)

[Out]

int((b*x+a)^(1-n)*(d*x+c)^(1+n)/(2*b*d*x+a*d+b*c),x)

________________________________________________________________________________________

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)^(1-n)*(d*x+c)^(1+n)/(2*b*d*x+a*d+b*c),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(-n + 1)*(d*x + c)^(n + 1)/(2*b*d*x + b*c + a*d), x)

________________________________________________________________________________________

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)^(1-n)*(d*x+c)^(1+n)/(2*b*d*x+a*d+b*c),x, algorithm="fricas")

[Out]

integral((b*x + a)^(-n + 1)*(d*x + c)^(n + 1)/(2*b*d*x + b*c + a*d), x)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(1-n)*(d*x+c)**(1+n)/(2*b*d*x+a*d+b*c),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

________________________________________________________________________________________

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)^(1-n)*(d*x+c)^(1+n)/(2*b*d*x+a*d+b*c),x, algorithm="giac")

[Out]

integrate((b*x + a)^(-n + 1)*(d*x + c)^(n + 1)/(2*b*d*x + b*c + a*d), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{1-n}\,{\left (c+d\,x\right )}^{n+1}}{a\,d+b\,c+2\,b\,d\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^(1 - n)*(c + d*x)^(n + 1))/(a*d + b*c + 2*b*d*x),x)

[Out]

int(((a + b*x)^(1 - n)*(c + d*x)^(n + 1))/(a*d + b*c + 2*b*d*x), x)

________________________________________________________________________________________