Optimal. Leaf size=14 \[ \frac {(c+d x)^3}{3 d} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {626, 32} \begin {gather*} \frac {(c+d x)^3}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 626
Rubi steps
\begin {align*} \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^2} \, dx &=\int (c+d x)^2 \, dx\\ &=\frac {(c+d x)^3}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {(c+d x)^3}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.04, size = 14, normalized size = 1.00 \begin {gather*} \frac {(c+d x)^3}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 20, normalized size = 1.43 \begin {gather*} \frac {1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.17, size = 84, normalized size = 6.00 \begin {gather*} \frac {{\left (\frac {3 \, b^{2} c^{2}}{{\left (b x + a\right )}^{2}} + \frac {3 \, b c d}{b x + a} - \frac {6 \, a b c d}{{\left (b x + a\right )}^{2}} - \frac {3 \, a d^{2}}{b x + a} + \frac {3 \, a^{2} d^{2}}{{\left (b x + a\right )}^{2}} + d^{2}\right )} {\left (b x + a\right )}^{3}}{3 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 13, normalized size = 0.93 \begin {gather*} \frac {\left (d x +c \right )^{3}}{3 d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.09, size = 20, normalized size = 1.43 \begin {gather*} \frac {1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 20, normalized size = 1.43 \begin {gather*} c^2\,x+c\,d\,x^2+\frac {d^2\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.13, size = 19, normalized size = 1.36 \begin {gather*} c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________