Optimal. Leaf size=96 \[ -\frac{2 \sqrt{x} \sqrt{b x+c x^2} (4 b B-5 A c)}{15 c^2}+\frac{4 b \sqrt{b x+c x^2} (4 b B-5 A c)}{15 c^3 \sqrt{x}}+\frac{2 B x^{3/2} \sqrt{b x+c x^2}}{5 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0787564, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {794, 656, 648} \[ -\frac{2 \sqrt{x} \sqrt{b x+c x^2} (4 b B-5 A c)}{15 c^2}+\frac{4 b \sqrt{b x+c x^2} (4 b B-5 A c)}{15 c^3 \sqrt{x}}+\frac{2 B x^{3/2} \sqrt{b x+c x^2}}{5 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 794
Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{x^{3/2} (A+B x)}{\sqrt{b x+c x^2}} \, dx &=\frac{2 B x^{3/2} \sqrt{b x+c x^2}}{5 c}+\frac{\left (2 \left (\frac{3}{2} (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right )\right ) \int \frac{x^{3/2}}{\sqrt{b x+c x^2}} \, dx}{5 c}\\ &=-\frac{2 (4 b B-5 A c) \sqrt{x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 B x^{3/2} \sqrt{b x+c x^2}}{5 c}+\frac{(2 b (4 b B-5 A c)) \int \frac{\sqrt{x}}{\sqrt{b x+c x^2}} \, dx}{15 c^2}\\ &=\frac{4 b (4 b B-5 A c) \sqrt{b x+c x^2}}{15 c^3 \sqrt{x}}-\frac{2 (4 b B-5 A c) \sqrt{x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 B x^{3/2} \sqrt{b x+c x^2}}{5 c}\\ \end{align*}
Mathematica [A] time = 0.040393, size = 55, normalized size = 0.57 \[ \frac{2 \sqrt{x (b+c x)} \left (-2 b c (5 A+2 B x)+c^2 x (5 A+3 B x)+8 b^2 B\right )}{15 c^3 \sqrt{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 59, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -3\,B{c}^{2}{x}^{2}-5\,A{c}^{2}x+4\,Bbcx+10\,Abc-8\,{b}^{2}B \right ) }{15\,{c}^{3}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05934, size = 101, normalized size = 1.05 \begin{align*} \frac{2 \,{\left (c^{2} x^{2} - b c x - 2 \, b^{2}\right )} A}{3 \, \sqrt{c x + b} c^{2}} + \frac{2 \,{\left (3 \, c^{3} x^{3} - b c^{2} x^{2} + 4 \, b^{2} c x + 8 \, b^{3}\right )} B}{15 \, \sqrt{c x + b} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.64604, size = 131, normalized size = 1.36 \begin{align*} \frac{2 \,{\left (3 \, B c^{2} x^{2} + 8 \, B b^{2} - 10 \, A b c -{\left (4 \, B b c - 5 \, A c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{15 \, c^{3} \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2}} \left (A + B x\right )}{\sqrt{x \left (b + c x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17006, size = 112, normalized size = 1.17 \begin{align*} \frac{2 \,{\left (3 \,{\left (c x + b\right )}^{\frac{5}{2}} B - 10 \,{\left (c x + b\right )}^{\frac{3}{2}} B b + 15 \, \sqrt{c x + b} B b^{2} + 5 \,{\left (c x + b\right )}^{\frac{3}{2}} A c - 15 \, \sqrt{c x + b} A b c\right )}}{15 \, c^{3}} - \frac{4 \,{\left (4 \, B b^{\frac{5}{2}} - 5 \, A b^{\frac{3}{2}} c\right )}}{15 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]