Optimal. Leaf size=216 \[ \frac{a^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b^2 (n+1) (b c-a d) (b e-a f)}-\frac{(a d+b c) (e+f x)^{n+1}}{b^2 d^2 f (n+1)}-\frac{c^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^2 (n+1) (b c-a d) (d e-c f)}-\frac{e (e+f x)^{n+1}}{b d f^2 (n+1)}+\frac{(e+f x)^{n+2}}{b d f^2 (n+2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15416, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {180, 43, 68} \[ \frac{a^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b^2 (n+1) (b c-a d) (b e-a f)}-\frac{(a d+b c) (e+f x)^{n+1}}{b^2 d^2 f (n+1)}-\frac{c^3 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^2 (n+1) (b c-a d) (d e-c f)}-\frac{e (e+f x)^{n+1}}{b d f^2 (n+1)}+\frac{(e+f x)^{n+2}}{b d f^2 (n+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 180
Rule 43
Rule 68
Rubi steps
\begin{align*} \int \frac{x^3 (e+f x)^n}{(a+b x) (c+d x)} \, dx &=\int \left (\frac{(-b c-a d) (e+f x)^n}{b^2 d^2}+\frac{x (e+f x)^n}{b d}-\frac{a^3 (e+f x)^n}{b^2 (b c-a d) (a+b x)}-\frac{c^3 (e+f x)^n}{d^2 (-b c+a d) (c+d x)}\right ) \, dx\\ &=-\frac{(b c+a d) (e+f x)^{1+n}}{b^2 d^2 f (1+n)}+\frac{\int x (e+f x)^n \, dx}{b d}-\frac{a^3 \int \frac{(e+f x)^n}{a+b x} \, dx}{b^2 (b c-a d)}+\frac{c^3 \int \frac{(e+f x)^n}{c+d x} \, dx}{d^2 (b c-a d)}\\ &=-\frac{(b c+a d) (e+f x)^{1+n}}{b^2 d^2 f (1+n)}+\frac{a^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{b (e+f x)}{b e-a f}\right )}{b^2 (b c-a d) (b e-a f) (1+n)}-\frac{c^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{d (e+f x)}{d e-c f}\right )}{d^2 (b c-a d) (d e-c f) (1+n)}+\frac{\int \left (-\frac{e (e+f x)^n}{f}+\frac{(e+f x)^{1+n}}{f}\right ) \, dx}{b d}\\ &=-\frac{e (e+f x)^{1+n}}{b d f^2 (1+n)}-\frac{(b c+a d) (e+f x)^{1+n}}{b^2 d^2 f (1+n)}+\frac{(e+f x)^{2+n}}{b d f^2 (2+n)}+\frac{a^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{b (e+f x)}{b e-a f}\right )}{b^2 (b c-a d) (b e-a f) (1+n)}-\frac{c^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{d (e+f x)}{d e-c f}\right )}{d^2 (b c-a d) (d e-c f) (1+n)}\\ \end{align*}
Mathematica [A] time = 0.496155, size = 174, normalized size = 0.81 \[ \frac{(e+f x)^{n+1} \left (\frac{a^3 \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b e-a f}+\frac{(b c-a d) (c f-d e) (a d f (n+2)+b c f (n+2)+b d (e-f (n+1) x))-b^2 c^3 f^2 (n+2) \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^2 f^2 (n+2) (d e-c f)}\right )}{b^2 (n+1) (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.077, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{n}{x}^{3}}{ \left ( bx+a \right ) \left ( dx+c \right ) }}\, 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 (f x + e\right )}^{n} x^{3}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (f x + e\right )}^{n} x^{3}}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \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{x^{3} \left (e + f x\right )^{n}}{\left (a + b x\right ) \left (c + d x\right )}\, 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 (f x + e\right )}^{n} x^{3}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]