Optimal. Leaf size=23 \[ -\frac {2}{3 b d (c+d (a+b x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {33, 32}
\begin {gather*} -\frac {2}{3 b d (d (a+b x)+c)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 33
Rubi steps
\begin {align*} \int \frac {1}{(c+d (a+b x))^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{(c+d x)^{5/2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac {2}{3 b d (c+d (a+b x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} -\frac {2}{3 b d (c+a d+b d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 3.19, size = 108, normalized size = 4.70 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {x}{c^{\frac {5}{2}}},b\text {==}0\text {\&\&}d\text {==}0\right \},\left \{\frac {x}{\left (a d+c\right )^{\frac {5}{2}}},b\text {==}0\right \},\left \{\frac {x}{c^{\frac {5}{2}}},d\text {==}0\right \}\right \},\frac {-2 \sqrt {a d+b d x+c}}{3 a^2 b d^3+6 a b^2 d^3 x+6 a b c d^2+3 b^3 d^3 x^2+6 b^2 c d^2 x+3 b c^2 d}\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 20, normalized size = 0.87
method | result | size |
gosper | \(-\frac {2}{3 \left (b d x +a d +c \right )^{\frac {3}{2}} b d}\) | \(20\) |
derivativedivides | \(-\frac {2}{3 \left (b d x +a d +c \right )^{\frac {3}{2}} b d}\) | \(20\) |
default | \(-\frac {2}{3 \left (b d x +a d +c \right )^{\frac {3}{2}} b d}\) | \(20\) |
trager | \(-\frac {2}{3 \left (b d x +a d +c \right )^{\frac {3}{2}} b d}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.25, size = 19, normalized size = 0.83 \begin {gather*} -\frac {2}{3 \, {\left ({\left (b x + a\right )} d + c\right )}^{\frac {3}{2}} b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (19) = 38\).
time = 0.31, size = 68, normalized size = 2.96 \begin {gather*} -\frac {2 \, \sqrt {b d x + a d + c}}{3 \, {\left (b^{3} d^{3} x^{2} + a^{2} b d^{3} + 2 \, a b c d^{2} + b c^{2} d + 2 \, {\left (a b^{2} d^{3} + b^{2} c d^{2}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.46, size = 102, normalized size = 4.43 \begin {gather*} \begin {cases} \frac {x}{c^{\frac {5}{2}}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x}{\left (a d + c\right )^{\frac {5}{2}}} & \text {for}\: b = 0 \\\frac {x}{c^{\frac {5}{2}}} & \text {for}\: d = 0 \\- \frac {2 \sqrt {a d + b d x + c}}{3 a^{2} b d^{3} + 6 a b^{2} d^{3} x + 6 a b c d^{2} + 3 b^{3} d^{3} x^{2} + 6 b^{2} c d^{2} x + 3 b c^{2} d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 33, normalized size = 1.43 \begin {gather*} -\frac {2}{3 b d \sqrt {a d+b d x+c} \left (a d+b d x+c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.18, size = 19, normalized size = 0.83 \begin {gather*} -\frac {2}{3\,b\,d\,{\left (c+d\,\left (a+b\,x\right )\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________