Optimal. Leaf size=112 \[ \frac {(d+e x)^{m+1} (a B e (m+1)-b (A e m+B d)) \, _2F_1\left (1,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{b (m+1) (b d-a e)^2}-\frac {(A b-a B) (d+e x)^{m+1}}{b (a+b x) (b d-a e)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {27, 78, 68} \[ \frac {(d+e x)^{m+1} (a B e (m+1)-b (A e m+B d)) \, _2F_1\left (1,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{b (m+1) (b d-a e)^2}-\frac {(A b-a B) (d+e x)^{m+1}}{b (a+b x) (b d-a e)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 68
Rule 78
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^m}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(A+B x) (d+e x)^m}{(a+b x)^2} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{1+m}}{b (b d-a e) (a+b x)}-\frac {(a B e (1+m)-b (B d+A e m)) \int \frac {(d+e x)^m}{a+b x} \, dx}{b (b d-a e)}\\ &=-\frac {(A b-a B) (d+e x)^{1+m}}{b (b d-a e) (a+b x)}+\frac {(a B e (1+m)-b (B d+A e m)) (d+e x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b (d+e x)}{b d-a e}\right )}{b (b d-a e)^2 (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 99, normalized size = 0.88 \[ \frac {(d+e x)^{m+1} \left (\frac {(a B e (m+1)-b (A e m+B d)) \, _2F_1\left (1,m+1;m+2;\frac {b (d+e x)}{b d-a e}\right )}{m+1}-\frac {(A b-a B) (b d-a e)}{a+b x}\right )}{b (b d-a e)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (B x +A \right ) \left (e x +d \right )^{m}}{b^{2} x^{2}+2 a b x +a^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{m}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^m}{a^2+2\,a\,b\,x+b^2\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B x\right ) \left (d + e x\right )^{m}}{\left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________