Optimal. Leaf size=125 \[ -\frac {(a+b x)^{p+1} (B d-A e) (d+e x)^{-p-1}}{e (p+1) (b d-a e)}-\frac {B (a+b x)^p (d+e x)^{-p} \left (-\frac {e (a+b x)}{b d-a e}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac {b (d+e x)}{b d-a e}\right )}{e^2 p} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 70, 69} \[ -\frac {(a+b x)^{p+1} (B d-A e) (d+e x)^{-p-1}}{e (p+1) (b d-a e)}-\frac {B (a+b x)^p (d+e x)^{-p} \left (-\frac {e (a+b x)}{b d-a e}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac {b (d+e x)}{b d-a e}\right )}{e^2 p} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 70
Rule 79
Rubi steps
\begin {align*} \int (a+b x)^p (A+B x) (d+e x)^{-2-p} \, dx &=-\frac {(B d-A e) (a+b x)^{1+p} (d+e x)^{-1-p}}{e (b d-a e) (1+p)}+\frac {B \int (a+b x)^p (d+e x)^{-1-p} \, dx}{e}\\ &=-\frac {(B d-A e) (a+b x)^{1+p} (d+e x)^{-1-p}}{e (b d-a e) (1+p)}+\frac {\left (B (a+b x)^p \left (\frac {e (a+b x)}{-b d+a e}\right )^{-p}\right ) \int (d+e x)^{-1-p} \left (-\frac {a e}{b d-a e}-\frac {b e x}{b d-a e}\right )^p \, dx}{e}\\ &=-\frac {(B d-A e) (a+b x)^{1+p} (d+e x)^{-1-p}}{e (b d-a e) (1+p)}-\frac {B (a+b x)^p \left (-\frac {e (a+b x)}{b d-a e}\right )^{-p} (d+e x)^{-p} \, _2F_1\left (-p,-p;1-p;\frac {b (d+e x)}{b d-a e}\right )}{e^2 p}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 114, normalized size = 0.91 \[ \frac {(a+b x)^p (d+e x)^{-p} \left (\frac {e (a+b x) (A e-B d)}{(p+1) (d+e x) (b d-a e)}-\frac {B \left (\frac {e (a+b x)}{a e-b d}\right )^{-p} \, _2F_1\left (-p,-p;1-p;\frac {b (d+e x)}{b d-a e}\right )}{p}\right )}{e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B x + A\right )} {\left (b x + a\right )}^{p} {\left (e x + d\right )}^{-p - 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B x + A\right )} {\left (b x + a\right )}^{p} {\left (e x + d\right )}^{-p - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \left (B x +A \right ) \left (b x +a \right )^{p} \left (e x +d \right )^{-p -2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B x + A\right )} {\left (b x + a\right )}^{p} {\left (e x + d\right )}^{-p - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^p}{{\left (d+e\,x\right )}^{p+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________