Optimal. Leaf size=165 \[ -\frac{13 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{5 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{386 \sqrt{3 x^2+5 x+2}}{75 \sqrt{2 x+3}}-\frac{26 \sqrt{3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}+\frac{193 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{25 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10421, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {834, 843, 718, 424, 419} \[ -\frac{386 \sqrt{3 x^2+5 x+2}}{75 \sqrt{2 x+3}}-\frac{26 \sqrt{3 x^2+5 x+2}}{15 (2 x+3)^{3/2}}-\frac{13 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{193 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{25 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 834
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{5-x}{(3+2 x)^{5/2} \sqrt{2+5 x+3 x^2}} \, dx &=-\frac{26 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^{3/2}}-\frac{2}{15} \int \frac{-19+\frac{39 x}{2}}{(3+2 x)^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{26 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^{3/2}}-\frac{386 \sqrt{2+5 x+3 x^2}}{75 \sqrt{3+2 x}}+\frac{4}{75} \int \frac{\frac{771}{4}+\frac{579 x}{4}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{26 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^{3/2}}-\frac{386 \sqrt{2+5 x+3 x^2}}{75 \sqrt{3+2 x}}-\frac{13}{10} \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx+\frac{193}{50} \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{26 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^{3/2}}-\frac{386 \sqrt{2+5 x+3 x^2}}{75 \sqrt{3+2 x}}-\frac{\left (13 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 x^2}{3}}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{5 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (193 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 x^2}{3}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{25 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=-\frac{26 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^{3/2}}-\frac{386 \sqrt{2+5 x+3 x^2}}{75 \sqrt{3+2 x}}+\frac{193 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{25 \sqrt{3} \sqrt{2+5 x+3 x^2}}-\frac{13 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{5 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.320846, size = 183, normalized size = 1.11 \[ \frac{193 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{5/2} \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )-2 \left (77 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{5/2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+65 \left (3 x^2+5 x+2\right )\right )}{75 (2 x+3)^{3/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.022, size = 203, normalized size = 1.2 \begin{align*}{\frac{1}{750} \left ( 256\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}-386\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) x\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-20-30\,x}+384\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) -579\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) -23160\,{x}^{3}-77240\,{x}^{2}-79840\,x-25760 \right ) \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x - 5}{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}{\left (x - 5\right )}}{24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54}, x\right ) \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}{4 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 12 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 9 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{4 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 12 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 9 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x - 5}{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (2 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]