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]
________________________________________________________________________________________
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]
[Out]
Rule 1620
Rule 68
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]