Optimal. Leaf size=190 \[ \frac {(a+b x)^{n+1} (2 a d-b c n) \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a^2 c^3 (n+1)}-\frac {d^2 (a+b x)^{n+1} (2 a d-b c (2-n)) \, _2F_1\left (1,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{c^3 (n+1) (b c-a d)^2}-\frac {d (b c-2 a d) (a+b x)^{n+1}}{a c^2 (c+d x) (b c-a d)}-\frac {(a+b x)^{n+1}}{a c x (c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {103, 151, 156, 65, 68} \[ \frac {(a+b x)^{n+1} (2 a d-b c n) \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )}{a^2 c^3 (n+1)}-\frac {d^2 (a+b x)^{n+1} (2 a d-b c (2-n)) \, _2F_1\left (1,n+1;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{c^3 (n+1) (b c-a d)^2}-\frac {d (b c-2 a d) (a+b x)^{n+1}}{a c^2 (c+d x) (b c-a d)}-\frac {(a+b x)^{n+1}}{a c x (c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 68
Rule 103
Rule 151
Rule 156
Rubi steps
\begin {align*} \int \frac {(a+b x)^n}{x^2 (c+d x)^2} \, dx &=-\frac {(a+b x)^{1+n}}{a c x (c+d x)}-\frac {\int \frac {(a+b x)^n (2 a d-b c n+b d (1-n) x)}{x (c+d x)^2} \, dx}{a c}\\ &=-\frac {d (b c-2 a d) (a+b x)^{1+n}}{a c^2 (b c-a d) (c+d x)}-\frac {(a+b x)^{1+n}}{a c x (c+d x)}+\frac {\int \frac {(a+b x)^n (-(b c-a d) (2 a d-b c n)+b d (b c-2 a d) n x)}{x (c+d x)} \, dx}{a c^2 (b c-a d)}\\ &=-\frac {d (b c-2 a d) (a+b x)^{1+n}}{a c^2 (b c-a d) (c+d x)}-\frac {(a+b x)^{1+n}}{a c x (c+d x)}-\frac {\left (d^2 (2 a d-b c (2-n))\right ) \int \frac {(a+b x)^n}{c+d x} \, dx}{c^3 (b c-a d)}-\frac {(2 a d-b c n) \int \frac {(a+b x)^n}{x} \, dx}{a c^3}\\ &=-\frac {d (b c-2 a d) (a+b x)^{1+n}}{a c^2 (b c-a d) (c+d x)}-\frac {(a+b x)^{1+n}}{a c x (c+d x)}-\frac {d^2 (2 a d-b c (2-n)) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{c^3 (b c-a d)^2 (1+n)}+\frac {(2 a d-b c n) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a^2 c^3 (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 176, normalized size = 0.93 \[ -\frac {(a+b x)^{n+1} \left (-x (c+d x) \left ((b c-a d)^2 (2 a d-b c n) \, _2F_1\left (1,n+1;n+2;\frac {b x}{a}+1\right )-a^2 d^2 (2 a d+b c (n-2)) \, _2F_1\left (1,n+1;n+2;\frac {d (a+b x)}{a d-b c}\right )\right )+a c^2 (n+1) (b c-a d)^2+a c d (n+1) x (a d-b c) (2 a d-b c)\right )}{a^2 c^3 (n+1) x (c+d x) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{n}}{d^{2} x^{4} + 2 \, c d x^{3} + c^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{n}}{\left (d x +c \right )^{2} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n}}{{\left (d x + c\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x\right )}^n}{x^2\,{\left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x\right )^{n}}{x^{2} \left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________