Optimal. Leaf size=42 \[ \frac{2 b (c+d x)^{9/2}}{9 d^2}-\frac{2 (c+d x)^{7/2} (b c-a d)}{7 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0148131, antiderivative size = 42, normalized size of antiderivative = 1., 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 = {43} \[ \frac{2 b (c+d x)^{9/2}}{9 d^2}-\frac{2 (c+d x)^{7/2} (b c-a d)}{7 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int (a+b x) (c+d x)^{5/2} \, dx &=\int \left (\frac{(-b c+a d) (c+d x)^{5/2}}{d}+\frac{b (c+d x)^{7/2}}{d}\right ) \, dx\\ &=-\frac{2 (b c-a d) (c+d x)^{7/2}}{7 d^2}+\frac{2 b (c+d x)^{9/2}}{9 d^2}\\ \end{align*}
Mathematica [A] time = 0.0224357, size = 30, normalized size = 0.71 \[ \frac{2 (c+d x)^{7/2} (9 a d-2 b c+7 b d x)}{63 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 27, normalized size = 0.6 \begin{align*}{\frac{14\,bdx+18\,ad-4\,bc}{63\,{d}^{2}} \left ( dx+c \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.949677, size = 45, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (7 \,{\left (d x + c\right )}^{\frac{9}{2}} b - 9 \,{\left (b c - a d\right )}{\left (d x + c\right )}^{\frac{7}{2}}\right )}}{63 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.75748, size = 205, normalized size = 4.88 \begin{align*} \frac{2 \,{\left (7 \, b d^{4} x^{4} - 2 \, b c^{4} + 9 \, a c^{3} d +{\left (19 \, b c d^{3} + 9 \, a d^{4}\right )} x^{3} + 3 \,{\left (5 \, b c^{2} d^{2} + 9 \, a c d^{3}\right )} x^{2} +{\left (b c^{3} d + 27 \, a c^{2} d^{2}\right )} x\right )} \sqrt{d x + c}}{63 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.17768, size = 194, normalized size = 4.62 \begin{align*} \begin{cases} \frac{2 a c^{3} \sqrt{c + d x}}{7 d} + \frac{6 a c^{2} x \sqrt{c + d x}}{7} + \frac{6 a c d x^{2} \sqrt{c + d x}}{7} + \frac{2 a d^{2} x^{3} \sqrt{c + d x}}{7} - \frac{4 b c^{4} \sqrt{c + d x}}{63 d^{2}} + \frac{2 b c^{3} x \sqrt{c + d x}}{63 d} + \frac{10 b c^{2} x^{2} \sqrt{c + d x}}{21} + \frac{38 b c d x^{3} \sqrt{c + d x}}{63} + \frac{2 b d^{2} x^{4} \sqrt{c + d x}}{9} & \text{for}\: d \neq 0 \\c^{\frac{5}{2}} \left (a x + \frac{b x^{2}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.07119, size = 263, normalized size = 6.26 \begin{align*} \frac{2 \,{\left (105 \,{\left (d x + c\right )}^{\frac{3}{2}} a c^{2} + 42 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a c + \frac{21 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} b c^{2}}{d} + 3 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2}\right )} a + \frac{6 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2}\right )} b c}{d} + \frac{{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} - 135 \,{\left (d x + c\right )}^{\frac{7}{2}} c + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3}\right )} b}{d}\right )}}{315 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]