Optimal. Leaf size=117 \[ \frac {(a+b x)^{n+1} (c+d x)^{p+1} (a d (p+1)+b c (n+1)) \, _2F_1\left (1,n+p+2;p+2;\frac {b (c+d x)}{b c-a d}\right )}{b d (p+1) (n+p+2) (b c-a d)}+\frac {(a+b x)^{n+1} (c+d x)^{p+1}}{b d (n+p+2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 129, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {80, 70, 69} \[ \frac {(a+b x)^{n+1} (c+d x)^{p+1}}{b d (n+p+2)}-\frac {(a+b x)^{n+1} (c+d x)^p (a d (p+1)+b c (n+1)) \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \, _2F_1\left (n+1,-p;n+2;-\frac {d (a+b x)}{b c-a d}\right )}{b^2 d (n+1) (n+p+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 70
Rule 80
Rubi steps
\begin {align*} \int x (a+b x)^n (c+d x)^p \, dx &=\frac {(a+b x)^{1+n} (c+d x)^{1+p}}{b d (2+n+p)}-\frac {(b c (1+n)+a d (1+p)) \int (a+b x)^n (c+d x)^p \, dx}{b d (2+n+p)}\\ &=\frac {(a+b x)^{1+n} (c+d x)^{1+p}}{b d (2+n+p)}-\frac {\left ((b c (1+n)+a d (1+p)) (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p}\right ) \int (a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^p \, dx}{b d (2+n+p)}\\ &=\frac {(a+b x)^{1+n} (c+d x)^{1+p}}{b d (2+n+p)}-\frac {(b c (1+n)+a d (1+p)) (a+b x)^{1+n} (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \, _2F_1\left (1+n,-p;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{b^2 d (1+n) (2+n+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 105, normalized size = 0.90 \[ \frac {(a+b x)^{n+1} (c+d x)^p \left (b (c+d x)-\frac {(a d (p+1)+b c (n+1)) \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \, _2F_1\left (n+1,-p;n+2;\frac {d (a+b x)}{a d-b c}\right )}{n+1}\right )}{b^2 d (n+p+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{n} {\left (d x + c\right )}^{p} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int x \left (b x +a \right )^{n} \left (d x +c \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^p \,d x \]
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]
________________________________________________________________________________________