Optimal. Leaf size=42 \[ \frac{2 (a+b x)^{5/2} (A b-a B)}{5 b^2}+\frac{2 B (a+b x)^{7/2}}{7 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0146335, 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} \[ \frac{2 (a+b x)^{5/2} (A b-a B)}{5 b^2}+\frac{2 B (a+b x)^{7/2}}{7 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int (a+b x)^{3/2} (A+B x) \, dx &=\int \left (\frac{(A b-a B) (a+b x)^{3/2}}{b}+\frac{B (a+b x)^{5/2}}{b}\right ) \, dx\\ &=\frac{2 (A b-a B) (a+b x)^{5/2}}{5 b^2}+\frac{2 B (a+b x)^{7/2}}{7 b^2}\\ \end{align*}
Mathematica [A] time = 0.0196335, size = 30, normalized size = 0.71 \[ \frac{2 (a+b x)^{5/2} (-2 a B+7 A b+5 b B x)}{35 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 27, normalized size = 0.6 \begin{align*}{\frac{10\,bBx+14\,Ab-4\,Ba}{35\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.26035, size = 45, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} B - 7 \,{\left (B a - A b\right )}{\left (b x + a\right )}^{\frac{5}{2}}\right )}}{35 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.31258, size = 155, normalized size = 3.69 \begin{align*} \frac{2 \,{\left (5 \, B b^{3} x^{3} - 2 \, B a^{3} + 7 \, A a^{2} b +{\left (8 \, B a b^{2} + 7 \, A b^{3}\right )} x^{2} +{\left (B a^{2} b + 14 \, A a b^{2}\right )} x\right )} \sqrt{b x + a}}{35 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.35107, size = 146, normalized size = 3.48 \begin{align*} \begin{cases} \frac{2 A a^{2} \sqrt{a + b x}}{5 b} + \frac{4 A a x \sqrt{a + b x}}{5} + \frac{2 A b x^{2} \sqrt{a + b x}}{5} - \frac{4 B a^{3} \sqrt{a + b x}}{35 b^{2}} + \frac{2 B a^{2} x \sqrt{a + b x}}{35 b} + \frac{16 B a x^{2} \sqrt{a + b x}}{35} + \frac{2 B b x^{3} \sqrt{a + b x}}{7} & \text{for}\: b \neq 0 \\a^{\frac{3}{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 = 1.16347, size = 140, normalized size = 3.33 \begin{align*} \frac{2 \,{\left (35 \,{\left (b x + a\right )}^{\frac{3}{2}} A a + 7 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} A + \frac{7 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} B a}{b} + \frac{{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}\right )} B}{b}\right )}}{105 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]