Optimal. Leaf size=162 \[ -\frac{4543 \sqrt{\frac{11}{6}} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{36 \sqrt{2 x-5}}+\frac{1}{4} \sqrt{2-3 x} \sqrt{2 x-5} (4 x+1)^{3/2}+\frac{95}{18} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}+\frac{1397 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{27 \sqrt{5-2 x}} \]
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Rubi [A] time = 0.0624533, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {154, 158, 114, 113, 121, 119} \[ \frac{1}{4} \sqrt{2-3 x} \sqrt{2 x-5} (4 x+1)^{3/2}+\frac{95}{18} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}-\frac{4543 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{36 \sqrt{2 x-5}}+\frac{1397 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{27 \sqrt{5-2 x}} \]
Antiderivative was successfully verified.
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Rule 154
Rule 158
Rule 114
Rule 113
Rule 121
Rule 119
Rubi steps
\begin{align*} \int \frac{\sqrt{2-3 x} \sqrt{1+4 x} (7+5 x)}{\sqrt{-5+2 x}} \, dx &=\frac{1}{4} \sqrt{2-3 x} \sqrt{-5+2 x} (1+4 x)^{3/2}+\frac{1}{20} \int \frac{\left (\frac{1065}{2}-950 x\right ) \sqrt{1+4 x}}{\sqrt{2-3 x} \sqrt{-5+2 x}} \, dx\\ &=\frac{95}{18} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}+\frac{1}{4} \sqrt{2-3 x} \sqrt{-5+2 x} (1+4 x)^{3/2}-\frac{1}{180} \int \frac{-\frac{29535}{2}+55880 x}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx\\ &=\frac{95}{18} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}+\frac{1}{4} \sqrt{2-3 x} \sqrt{-5+2 x} (1+4 x)^{3/2}-\frac{1397}{9} \int \frac{\sqrt{-5+2 x}}{\sqrt{2-3 x} \sqrt{1+4 x}} \, dx-\frac{49973}{72} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx\\ &=\frac{95}{18} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}+\frac{1}{4} \sqrt{2-3 x} \sqrt{-5+2 x} (1+4 x)^{3/2}-\frac{\left (4543 \sqrt{\frac{11}{2}} \sqrt{5-2 x}\right ) \int \frac{1}{\sqrt{2-3 x} \sqrt{\frac{10}{11}-\frac{4 x}{11}} \sqrt{1+4 x}} \, dx}{36 \sqrt{-5+2 x}}-\frac{\left (1397 \sqrt{-5+2 x}\right ) \int \frac{\sqrt{\frac{15}{11}-\frac{6 x}{11}}}{\sqrt{2-3 x} \sqrt{\frac{3}{11}+\frac{12 x}{11}}} \, dx}{9 \sqrt{5-2 x}}\\ &=\frac{95}{18} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}+\frac{1}{4} \sqrt{2-3 x} \sqrt{-5+2 x} (1+4 x)^{3/2}+\frac{1397 \sqrt{11} \sqrt{-5+2 x} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{27 \sqrt{5-2 x}}-\frac{4543 \sqrt{\frac{11}{6}} \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{1+4 x}\right )|\frac{1}{3}\right )}{36 \sqrt{-5+2 x}}\\ \end{align*}
Mathematica [A] time = 0.192639, size = 120, normalized size = 0.74 \[ \frac{-4543 \sqrt{66} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )+6 \sqrt{2-3 x} \sqrt{4 x+1} \left (72 x^2+218 x-995\right )+5588 \sqrt{66} \sqrt{5-2 x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{216 \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 145, normalized size = 0.9 \begin{align*} -{\frac{1}{5184\,{x}^{3}-15120\,{x}^{2}+4536\,x+2160}\sqrt{2-3\,x}\sqrt{2\,x-5}\sqrt{4\,x+1} \left ( 13629\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticF} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) -11176\,\sqrt{11}\sqrt{2-3\,x}\sqrt{5-2\,x}\sqrt{4\,x+1}{\it EllipticE} \left ( 2/11\,\sqrt{22-33\,x},i/2\sqrt{2} \right ) -5184\,{x}^{4}-13536\,{x}^{3}+79044\,{x}^{2}-27234\,x-11940 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 7\right )} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (5 \, x + 7\right )} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 - 3 x} \sqrt{4 x + 1} \left (5 x + 7\right )}{\sqrt{2 x - 5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 7\right )} \sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{\sqrt{2 \, x - 5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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