Optimal. Leaf size=23 \[ -\frac {1}{3 b d \left (a+b (c+d x)^3\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {372, 261} \[ -\frac {1}{3 b d \left (a+b (c+d x)^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 372
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{\left (a+b (c+d x)^3\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {1}{3 b d \left (a+b (c+d x)^3\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \[ -\frac {1}{3 b d \left (a+b (c+d x)^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.55, size = 52, normalized size = 2.26 \[ -\frac {1}{3 \, {\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + {\left (b^{2} c^{3} + a b\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.19, size = 21, normalized size = 0.91 \[ -\frac {1}{3 \, {\left ({\left (d x + c\right )}^{3} b + a\right )} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.00, size = 44, normalized size = 1.91 \[ -\frac {1}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.63, size = 21, normalized size = 0.91 \[ -\frac {1}{3 \, {\left ({\left (d x + c\right )}^{3} b + a\right )} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.21, size = 43, normalized size = 1.87 \[ -\frac {1}{3\,b\,d\,\left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.96, size = 58, normalized size = 2.52 \[ - \frac {1}{3 a b d + 3 b^{2} c^{3} d + 9 b^{2} c^{2} d^{2} x + 9 b^{2} c d^{3} x^{2} + 3 b^{2} d^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________