Optimal. Leaf size=15 \[ -\frac {d}{4 b^2 (c+d x)^4} \]
________________________________________________________________________________________
Rubi [A] time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32} \begin {gather*} -\frac {d}{4 b^2 (c+d x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 32
Rubi steps
\begin {align*} \int \frac {1}{\left (\frac {b c}{d}+b x\right )^2 (c+d x)^3} \, dx &=\frac {d^2 \int \frac {1}{(c+d x)^5} \, dx}{b^2}\\ &=-\frac {d}{4 b^2 (c+d x)^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -\frac {d}{4 b^2 (c+d x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\frac {b c}{d}+b x\right )^2 (c+d x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.31, size = 59, normalized size = 3.93 \begin {gather*} -\frac {d}{4 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.98, size = 20, normalized size = 1.33 \begin {gather*} -\frac {b^{2}}{4 \, {\left (b x + \frac {b c}{d}\right )}^{4} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 14, normalized size = 0.93 \begin {gather*} -\frac {d}{4 \left (d x +c \right )^{4} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.35, size = 59, normalized size = 3.93 \begin {gather*} -\frac {d}{4 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 61, normalized size = 4.07 \begin {gather*} -\frac {d}{4\,\left (b^2\,c^4+4\,b^2\,c^3\,d\,x+6\,b^2\,c^2\,d^2\,x^2+4\,b^2\,c\,d^3\,x^3+b^2\,d^4\,x^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.36, size = 68, normalized size = 4.53 \begin {gather*} - \frac {d^{2}}{4 b^{2} c^{4} d + 16 b^{2} c^{3} d^{2} x + 24 b^{2} c^{2} d^{3} x^{2} + 16 b^{2} c d^{4} x^{3} + 4 b^{2} d^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________