Optimal. Leaf size=87 \[ \frac{3 a^2 A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 \sqrt{b}}+\frac{1}{4} A x \left (a+b x^2\right )^{3/2}+\frac{3}{8} a A x \sqrt{a+b x^2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0261586, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {641, 195, 217, 206} \[ \frac{3 a^2 A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 \sqrt{b}}+\frac{1}{4} A x \left (a+b x^2\right )^{3/2}+\frac{3}{8} a A x \sqrt{a+b x^2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 641
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int (A+B x) \left (a+b x^2\right )^{3/2} \, dx &=\frac{B \left (a+b x^2\right )^{5/2}}{5 b}+A \int \left (a+b x^2\right )^{3/2} \, dx\\ &=\frac{1}{4} A x \left (a+b x^2\right )^{3/2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b}+\frac{1}{4} (3 a A) \int \sqrt{a+b x^2} \, dx\\ &=\frac{3}{8} a A x \sqrt{a+b x^2}+\frac{1}{4} A x \left (a+b x^2\right )^{3/2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b}+\frac{1}{8} \left (3 a^2 A\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx\\ &=\frac{3}{8} a A x \sqrt{a+b x^2}+\frac{1}{4} A x \left (a+b x^2\right )^{3/2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b}+\frac{1}{8} \left (3 a^2 A\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )\\ &=\frac{3}{8} a A x \sqrt{a+b x^2}+\frac{1}{4} A x \left (a+b x^2\right )^{3/2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b}+\frac{3 a^2 A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0696007, size = 88, normalized size = 1.01 \[ \frac{\sqrt{a+b x^2} \left (8 a^2 B+a b x (25 A+16 B x)+2 b^2 x^3 (5 A+4 B x)\right )+15 a^2 A \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{40 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 69, normalized size = 0.8 \begin{align*}{\frac{B}{5\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Ax}{4} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,aAx}{8}\sqrt{b{x}^{2}+a}}+{\frac{3\,A{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.63008, size = 425, normalized size = 4.89 \begin{align*} \left [\frac{15 \, A a^{2} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \,{\left (8 \, B b^{2} x^{4} + 10 \, A b^{2} x^{3} + 16 \, B a b x^{2} + 25 \, A a b x + 8 \, B a^{2}\right )} \sqrt{b x^{2} + a}}{80 \, b}, -\frac{15 \, A a^{2} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (8 \, B b^{2} x^{4} + 10 \, A b^{2} x^{3} + 16 \, B a b x^{2} + 25 \, A a b x + 8 \, B a^{2}\right )} \sqrt{b x^{2} + a}}{40 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 6.30866, size = 219, normalized size = 2.52 \begin{align*} \frac{A a^{\frac{3}{2}} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{A a^{\frac{3}{2}} x}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A \sqrt{a} b x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 \sqrt{b}} + \frac{A b^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + B a \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) + B b \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18071, size = 103, normalized size = 1.18 \begin{align*} -\frac{3 \, A a^{2} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, \sqrt{b}} + \frac{1}{40} \, \sqrt{b x^{2} + a}{\left (\frac{8 \, B a^{2}}{b} +{\left (25 \, A a + 2 \,{\left (8 \, B a +{\left (4 \, B b x + 5 \, A b\right )} x\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]