Optimal. Leaf size=42 \[ -\frac {2 (b c-a d) (c+d x)^{5/2}}{5 d^2}+\frac {2 b (c+d x)^{7/2}}{7 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45}
\begin {gather*} \frac {2 b (c+d x)^{7/2}}{7 d^2}-\frac {2 (c+d x)^{5/2} (b c-a d)}{5 d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rubi steps
\begin {align*} \int (a+b x) (c+d x)^{3/2} \, dx &=\int \left (\frac {(-b c+a d) (c+d x)^{3/2}}{d}+\frac {b (c+d x)^{5/2}}{d}\right ) \, dx\\ &=-\frac {2 (b c-a d) (c+d x)^{5/2}}{5 d^2}+\frac {2 b (c+d x)^{7/2}}{7 d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 30, normalized size = 0.71 \begin {gather*} \frac {2 (c+d x)^{5/2} (-2 b c+7 a d+5 b d x)}{35 d^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 = 2.33, size = 78, normalized size = 1.86 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (-2 b c^3+c^2 d \left (7 a+b x\right )+d^2 x \left (14 a c+7 a d x+8 b c x+5 b d x^2\right )\right ) \sqrt {c+d x}}{35 d^2},d\text {!=}0\right \}\right \},c^{\frac {3}{2}} \left (a x+\frac {b x^2}{2}\right )\right ] \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.13, size = 34, normalized size = 0.81
method | result | size |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (5 b d x +7 a d -2 b c \right )}{35 d^{2}}\) | \(27\) |
derivativedivides | \(\frac {\frac {2 b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right ) \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{2}}\) | \(34\) |
default | \(\frac {\frac {2 b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right ) \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{2}}\) | \(34\) |
trager | \(\frac {2 \left (5 b \,d^{3} x^{3}+7 a \,d^{3} x^{2}+8 b c \,d^{2} x^{2}+14 a c \,d^{2} x +b \,c^{2} d x +7 a \,c^{2} d -2 b \,c^{3}\right ) \sqrt {d x +c}}{35 d^{2}}\) | \(70\) |
risch | \(\frac {2 \left (5 b \,d^{3} x^{3}+7 a \,d^{3} x^{2}+8 b c \,d^{2} x^{2}+14 a c \,d^{2} x +b \,c^{2} d x +7 a \,c^{2} d -2 b \,c^{3}\right ) \sqrt {d x +c}}{35 d^{2}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 33, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} b - 7 \, {\left (b c - a d\right )} {\left (d x + c\right )}^{\frac {5}{2}}\right )}}{35 \, d^{2}} \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. 69 vs.
\(2 (34) = 68\).
time = 0.30, size = 69, normalized size = 1.64 \begin {gather*} \frac {2 \, {\left (5 \, b d^{3} x^{3} - 2 \, b c^{3} + 7 \, a c^{2} d + {\left (8 \, b c d^{2} + 7 \, a d^{3}\right )} x^{2} + {\left (b c^{2} d + 14 \, a c d^{2}\right )} x\right )} \sqrt {d x + c}}{35 \, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.17, size = 146, normalized size = 3.48 \begin {gather*} \begin {cases} \frac {2 a c^{2} \sqrt {c + d x}}{5 d} + \frac {4 a c x \sqrt {c + d x}}{5} + \frac {2 a d x^{2} \sqrt {c + d x}}{5} - \frac {4 b c^{3} \sqrt {c + d x}}{35 d^{2}} + \frac {2 b c^{2} x \sqrt {c + d x}}{35 d} + \frac {16 b c x^{2} \sqrt {c + d x}}{35} + \frac {2 b d x^{3} \sqrt {c + d x}}{7} & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (a x + \frac {b x^{2}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 192 vs.
\(2 (34) = 68\).
time = 0.00, size = 300, normalized size = 7.14 \begin {gather*} \frac {\frac {2 b d^{2} \left (\frac {1}{7} \sqrt {c+d x} \left (c+d x\right )^{3}-\frac {3}{5} \sqrt {c+d x} \left (c+d x\right )^{2} c+\sqrt {c+d x} \left (c+d x\right ) c^{2}-\sqrt {c+d x} c^{3}\right )}{d^{3}}+\frac {2 a d^{2} \left (\frac {1}{5} \sqrt {c+d x} \left (c+d x\right )^{2}-\frac {2}{3} \sqrt {c+d x} \left (c+d x\right ) c+\sqrt {c+d x} c^{2}\right )}{d^{2}}+\frac {4 b c d \left (\frac {1}{5} \sqrt {c+d x} \left (c+d x\right )^{2}-\frac {2}{3} \sqrt {c+d x} \left (c+d x\right ) c+\sqrt {c+d x} c^{2}\right )}{d^{2}}+4 a c \left (\frac {1}{3} \sqrt {c+d x} \left (c+d x\right )-c \sqrt {c+d x}\right )+\frac {2 b c^{2} \left (\frac {1}{3} \sqrt {c+d x} \left (c+d x\right )-c \sqrt {c+d x}\right )}{d}+2 a c^{2} \sqrt {c+d x}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.21, size = 29, normalized size = 0.69 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{5/2}\,\left (7\,a\,d-7\,b\,c+5\,b\,\left (c+d\,x\right )\right )}{35\,d^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________