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

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 203, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.26, 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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fricas [F]  time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{b x + a}, x\right ) \]

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]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{b x + a}\,{d x} \]

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)

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maple [F]  time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (D x^{3}+C \,x^{2}+B x +A \right ) \left (d x +c \right )^{n}}{b x +a}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{b x + a}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c+d\,x\right )}^n\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{a+b\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c + d x\right )^{n} \left (A + B x + C x^{2} + D x^{3}\right )}{a + b x}\, dx \]

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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