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

Optimal. Leaf size=203 \[ -\frac{(c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \, _2F_1\left (1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{b^3 (n+1) (b c-a d)}+\frac{(c+d x)^{n+1} \left (a^2 d^2 D-a b d (C d-c D)+b^2 \left (-\left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{b^3 d^3 (n+1)}+\frac{(c+d x)^{n+2} (-a d D-2 b c D+b C d)}{b^2 d^3 (n+2)}+\frac{D (c+d x)^{n+3}}{b d^3 (n+3)} \]

[Out]

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

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Rubi [A]  time = 0.179561, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1620, 68} \[ -\frac{(c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \, _2F_1\left (1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{b^3 (n+1) (b c-a d)}+\frac{(c+d x)^{n+1} \left (a^2 d^2 D-a b d (C d-c D)+b^2 \left (-\left (-B d^2+c^2 (-D)+c C d\right )\right )\right )}{b^3 d^3 (n+1)}+\frac{(c+d x)^{n+2} (-a d D-2 b c D+b C d)}{b^2 d^3 (n+2)}+\frac{D (c+d x)^{n+3}}{b d^3 (n+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

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

Mathematica [A]  time = 0.245645, size = 181, normalized size = 0.89 \[ \frac{(c+d x)^{n+1} \left (-\frac{\left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right ) \, _2F_1\left (1,n+1;n+2;\frac{b (c+d x)}{b c-a d}\right )}{(n+1) (b c-a d)}+\frac{a^2 d^2 D+a b d (c D-C d)+b^2 \left (B d^2+c^2 D-c C d\right )}{d^3 (n+1)}+\frac{b (c+d x) (-a d D-2 b c D+b C d)}{d^3 (n+2)}+\frac{b^2 D (c+d x)^2}{d^3 (n+3)}\right )}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{n} \left ( D{x}^{3}+C{x}^{2}+Bx+A \right ) }{bx+a}}\, 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),x)

[Out]

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

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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}}{b x + a}\,{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),x, algorithm="maxima")

[Out]

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

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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),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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 )}{a + b x}\, 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),x)

[Out]

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

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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}}{b x + a}\,{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),x, algorithm="giac")

[Out]

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