Optimal. Leaf size=129 \[ \frac{24353 \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )}{36 \sqrt{66} \sqrt{2 x-5}}-\frac{25}{36} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}-\frac{2135 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{108 \sqrt{5-2 x}} \]
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Rubi [A] time = 0.0627385, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {167, 24, 158, 114, 113, 121, 119} \[ -\frac{25}{36} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}+\frac{24353 \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{36 \sqrt{66} \sqrt{2 x-5}}-\frac{2135 \sqrt{11} \sqrt{2 x-5} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{108 \sqrt{5-2 x}} \]
Antiderivative was successfully verified.
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Rule 167
Rule 24
Rule 158
Rule 114
Rule 113
Rule 121
Rule 119
Rubi steps
\begin{align*} \int \frac{(7+5 x)^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx &=-\frac{25}{36} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}+\frac{1}{72} \int \frac{21021+74795 x+42700 x^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)} \, dx\\ &=-\frac{25}{36} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}+\frac{\int \frac{75075+213500 x}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{1800}\\ &=-\frac{25}{36} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}+\frac{2135}{36} \int \frac{\sqrt{-5+2 x}}{\sqrt{2-3 x} \sqrt{1+4 x}} \, dx+\frac{24353}{72} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx\\ &=-\frac{25}{36} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}+\frac{\left (24353 \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{22} \sqrt{-5+2 x}}+\frac{\left (2135 \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}{36 \sqrt{5-2 x}}\\ &=-\frac{25}{36} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}-\frac{2135 \sqrt{11} \sqrt{-5+2 x} E\left (\sin ^{-1}\left (\frac{2 \sqrt{2-3 x}}{\sqrt{11}}\right )|-\frac{1}{2}\right )}{108 \sqrt{5-2 x}}+\frac{24353 \sqrt{5-2 x} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{1+4 x}\right )|\frac{1}{3}\right )}{36 \sqrt{66} \sqrt{-5+2 x}}\\ \end{align*}
Mathematica [A] time = 0.238652, size = 115, normalized size = 0.89 \[ \frac{24353 \sqrt{66} \sqrt{5-2 x} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right ),\frac{1}{3}\right )+1650 \sqrt{2-3 x} \sqrt{4 x+1} (5-2 x)-23485 \sqrt{66} \sqrt{5-2 x} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{11}} \sqrt{4 x+1}\right )|\frac{1}{3}\right )}{2376 \sqrt{2 x-5}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 140, normalized size = 1.1 \begin{align*}{\frac{1}{57024\,{x}^{3}-166320\,{x}^{2}+49896\,x+23760}\sqrt{2-3\,x}\sqrt{2\,x-5}\sqrt{4\,x+1} \left ( 73059\,\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 ) -46970\,\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 ) -39600\,{x}^{3}+115500\,{x}^{2}-34650\,x-16500 \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 )}^{2}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}\,{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 (25 \, x^{2} + 70 \, x + 49\right )} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{24 \, x^{3} - 70 \, x^{2} + 21 \, x + 10}, 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{\left (5 x + 7\right )^{2}}{\sqrt{2 - 3 x} \sqrt{2 x - 5} \sqrt{4 x + 1}}\, 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 )}^{2}}{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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