3.30 \(\int \frac{(c+d x)^n (A+B x+C x^2+D x^3)}{(a+b x)^2} \, dx\)

Optimal. Leaf size=220 \[ \frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right ) \left (-a^2 b (3 c D+C d (n+2))+a^3 d D (n+3)+a b^2 (B d (n+1)+2 c C)-b^3 (A d n+B c)\right )}{b^3 (n+1) (b c-a d)^2}-\frac{(c+d x)^{n+1} \left (A-\frac{a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}+\frac{(c+d x)^{n+1} (-2 a d D-b c D+b C d)}{b^3 d^2 (n+1)}+\frac{D (c+d x)^{n+2}}{b^2 d^2 (n+2)} \]

[Out]

((b*C*d - b*c*D - 2*a*d*D)*(c + d*x)^(1 + n))/(b^3*d^2*(1 + n)) - ((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*(c +
d*x)^(1 + n))/((b*c - a*d)*(a + b*x)) + (D*(c + d*x)^(2 + n))/(b^2*d^2*(2 + n)) + ((a^3*d*D*(3 + n) - b^3*(B*c
+ A*d*n) + a*b^2*(2*c*C + B*d*(1 + n)) - a^2*b*(3*c*D + C*d*(2 + n)))*(c + d*x)^(1 + n)*Hypergeometric2F1[1,
1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/(b^3*(b*c - a*d)^2*(1 + n))

________________________________________________________________________________________

Rubi [A]  time = 0.523727, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1621, 951, 80, 68} \[ \frac{(c+d x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right ) \left (-a^2 b (3 c D+C d (n+2))+a^3 d D (n+3)+a b^2 (B d (n+1)+2 c C)-b^3 (A d n+B c)\right )}{b^3 (n+1) (b c-a d)^2}-\frac{(c+d x)^{n+1} \left (A-\frac{a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{(a+b x) (b c-a d)}+\frac{(c+d x)^{n+1} (-2 a d D-b c D+b C d)}{b^3 d^2 (n+1)}+\frac{D (c+d x)^{n+2}}{b^2 d^2 (n+2)} \]

Antiderivative was successfully verified.

[In]

Int[((c + d*x)^n*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^2,x]

[Out]

((b*C*d - b*c*D - 2*a*d*D)*(c + d*x)^(1 + n))/(b^3*d^2*(1 + n)) - ((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*(c +
d*x)^(1 + n))/((b*c - a*d)*(a + b*x)) + (D*(c + d*x)^(2 + n))/(b^2*d^2*(2 + n)) + ((a^3*d*D*(3 + n) - b^3*(B*c
+ A*d*n) + a*b^2*(2*c*C + B*d*(1 + n)) - a^2*b*(3*c*D + C*d*(2 + n)))*(c + d*x)^(1 + n)*Hypergeometric2F1[1,
1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/(b^3*(b*c - a*d)^2*(1 + n))

Rule 1621

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> With[{Qx = PolynomialQuotient[Px,
a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/((m + 1)*
(b*c - a*d)), x] + Dist[1/((m + 1)*(b*c - a*d)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*ExpandToSum[(m + 1)*(b*c -
a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && ILtQ[m, -1] && GtQ[Expo
n[Px, x], 2]

Rule 951

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])

Rule 80

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

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

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

Mathematica [A]  time = 0.19947, size = 180, normalized size = 0.82 \[ \frac{(c+d x)^{n+1} \left (\frac{d \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \, _2F_1\left (2,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)^2}-\frac{\left (3 a^2 D-2 a b C+b^2 B\right ) \, _2F_1\left (1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)}+\frac{-2 a d D-b c D+b C d}{d^2 (n+1)}+\frac{b D (c+d x)}{d^2 (n+2)}\right )}{b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((c + d*x)^n*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^2,x]

[Out]

((c + d*x)^(1 + n)*((b*C*d - b*c*D - 2*a*d*D)/(d^2*(1 + n)) + (b*D*(c + d*x))/(d^2*(2 + n)) - ((b^2*B - 2*a*b*
C + 3*a^2*D)*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)*(1 + n)) + (d*(A*b^3
- a*(b^2*B - a*b*C + a^2*D))*Hypergeometric2F1[2, 1 + n, 2 + n, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^2*(1
+ n))))/b^3

________________________________________________________________________________________

Maple [F]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{n} \left ( D{x}^{3}+C{x}^{2}+Bx+A \right ) }{ \left ( bx+a \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^n*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,x)

[Out]

int((d*x+c)^n*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (D x^{3} + C x^{2} + B x + A\right )}{\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^n*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,x, algorithm="maxima")

[Out]

integrate((D*x^3 + C*x^2 + B*x + A)*(d*x + c)^n/(b*x + a)^2, x)

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^n*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c + d x\right )^{n} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (a + b x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**n*(D*x**3+C*x**2+B*x+A)/(b*x+a)**2,x)

[Out]

Integral((c + d*x)**n*(A + B*x + C*x**2 + D*x**3)/(a + b*x)**2, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (D x^{3} + C x^{2} + B x + A\right )}{\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^n*(D*x^3+C*x^2+B*x+A)/(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((D*x^3 + C*x^2 + B*x + A)*(d*x + c)^n/(b*x + a)^2, x)