Optimal. Leaf size=22 \[ \frac{2 (a c+b c x)^{9/2}}{9 b c^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0043641, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {21, 32} \[ \frac{2 (a c+b c x)^{9/2}}{9 b c^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 32
Rubi steps
\begin{align*} \int \frac{(a+b x)^5}{(a c+b c x)^{3/2}} \, dx &=\frac{\int (a c+b c x)^{7/2} \, dx}{c^5}\\ &=\frac{2 (a c+b c x)^{9/2}}{9 b c^6}\\ \end{align*}
Mathematica [A] time = 0.0148942, size = 25, normalized size = 1.14 \[ \frac{2 (a+b x)^6}{9 b (c (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 23, normalized size = 1.1 \begin{align*}{\frac{2\, \left ( bx+a \right ) ^{6}}{9\,b} \left ( bcx+ac \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.977158, size = 24, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (b c x + a c\right )}^{\frac{9}{2}}}{9 \, b c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.07865, size = 120, normalized size = 5.45 \begin{align*} \frac{2 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt{b c x + a c}}{9 \, b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.3438, size = 73, normalized size = 3.32 \begin{align*} \begin{cases} \frac{2 b^{\frac{7}{2}} \left (\frac{a}{b} + x\right )^{\frac{9}{2}}}{9 c^{\frac{3}{2}}} & \text{for}\: \left |{\frac{a}{b} + x}\right | > 1 \vee \left |{\frac{a}{b} + x}\right | < 1 \\\frac{b^{\frac{7}{2}}{G_{2, 2}^{1, 1}\left (\begin{matrix} 1 & \frac{11}{2} \\\frac{9}{2} & 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )}}{c^{\frac{3}{2}}} + \frac{b^{\frac{7}{2}}{G_{2, 2}^{0, 2}\left (\begin{matrix} \frac{11}{2}, 1 & \\ & \frac{9}{2}, 0 \end{matrix} \middle |{\frac{a}{b} + x} \right )}}{c^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.06055, size = 359, normalized size = 16.32 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{b c x + a c} a^{4} - \frac{420 \,{\left (3 \, \sqrt{b c x + a c} a c -{\left (b c x + a c\right )}^{\frac{3}{2}}\right )} a^{3}}{c} + \frac{126 \,{\left (15 \, \sqrt{b c x + a c} a^{2} c^{2} - 10 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a c + 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}}\right )} a^{2}}{c^{2}} - \frac{36 \,{\left (35 \, \sqrt{b c x + a c} a^{3} c^{3} - 35 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{2} c^{2} + 21 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a c - 5 \,{\left (b c x + a c\right )}^{\frac{7}{2}}\right )} a}{c^{3}} + \frac{315 \, \sqrt{b c x + a c} a^{4} c^{4} - 420 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a^{3} c^{3} + 378 \,{\left (b c x + a c\right )}^{\frac{5}{2}} a^{2} c^{2} - 180 \,{\left (b c x + a c\right )}^{\frac{7}{2}} a c + 35 \,{\left (b c x + a c\right )}^{\frac{9}{2}}}{c^{4}}\right )}}{315 \, b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]