Optimal. Leaf size=54 \[ \frac {b^2 (a+b x)^{1+m} \, _2F_1\left (3,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^3 (1+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {70}
\begin {gather*} \frac {b^2 (a+b x)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac {d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 70
Rubi steps
\begin {align*} \int \frac {(a+b x)^m}{(c+d x)^3} \, dx &=\frac {b^2 (a+b x)^{1+m} \, _2F_1\left (3,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^3 (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 54, normalized size = 1.00 \begin {gather*} \frac {b^2 (a+b x)^{1+m} \, _2F_1\left (3,1+m;2+m;-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^3 (1+m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{m}}{\left (d x +c \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x\right )^{m}}{\left (c + d x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^m}{{\left (c+d\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________