Optimal. Leaf size=63 \[ -\frac{2 (A b-2 a B)}{b^3 \sqrt{a+b x}}+\frac{2 a (A b-a B)}{3 b^3 (a+b x)^{3/2}}+\frac{2 B \sqrt{a+b x}}{b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0250434, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ -\frac{2 (A b-2 a B)}{b^3 \sqrt{a+b x}}+\frac{2 a (A b-a B)}{3 b^3 (a+b x)^{3/2}}+\frac{2 B \sqrt{a+b x}}{b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin{align*} \int \frac{x (A+B x)}{(a+b x)^{5/2}} \, dx &=\int \left (\frac{a (-A b+a B)}{b^2 (a+b x)^{5/2}}+\frac{A b-2 a B}{b^2 (a+b x)^{3/2}}+\frac{B}{b^2 \sqrt{a+b x}}\right ) \, dx\\ &=\frac{2 a (A b-a B)}{3 b^3 (a+b x)^{3/2}}-\frac{2 (A b-2 a B)}{b^3 \sqrt{a+b x}}+\frac{2 B \sqrt{a+b x}}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0309279, size = 46, normalized size = 0.73 \[ \frac{16 a^2 B-4 a b (A-6 B x)+6 b^2 x (B x-A)}{3 b^3 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.002, size = 47, normalized size = 0.8 \begin{align*} -{\frac{-6\,{b}^{2}B{x}^{2}+6\,{b}^{2}Ax-24\,abBx+4\,Aba-16\,B{a}^{2}}{3\,{b}^{3}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.10789, size = 78, normalized size = 1.24 \begin{align*} \frac{2 \,{\left (\frac{3 \, \sqrt{b x + a} B}{b} - \frac{B a^{2} - A a b - 3 \,{\left (2 \, B a - A b\right )}{\left (b x + a\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b}\right )}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.22115, size = 147, normalized size = 2.33 \begin{align*} \frac{2 \,{\left (3 \, B b^{2} x^{2} + 8 \, B a^{2} - 2 \, A a b + 3 \,{\left (4 \, B a b - A b^{2}\right )} x\right )} \sqrt{b x + a}}{3 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.65409, size = 211, normalized size = 3.35 \begin{align*} \begin{cases} - \frac{4 A a b}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} - \frac{6 A b^{2} x}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} + \frac{16 B a^{2}}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} + \frac{24 B a b x}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} + \frac{6 B b^{2} x^{2}}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{2}}{2} + \frac{B x^{3}}{3}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.23299, size = 74, normalized size = 1.17 \begin{align*} \frac{2 \, \sqrt{b x + a} B}{b^{3}} + \frac{2 \,{\left (6 \,{\left (b x + a\right )} B a - B a^{2} - 3 \,{\left (b x + a\right )} A b + A a b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]