Optimal. Leaf size=195 \[ \frac{2 \sqrt{\frac{3}{143}} (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right ),-\frac{23}{39}\right )}{31 \sqrt{2 x-5} \sqrt{4 x+1}}+\frac{10 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5-2 x}{5 x+7}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{22}} \sqrt{4 x+1}}{\sqrt{5 x+7}}\right )|\frac{62}{39}\right )}{713 \sqrt{2 x-5} \sqrt{\frac{2-3 x}{5 x+7}}} \]
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Rubi [A] time = 0.182889, antiderivative size = 270, normalized size of antiderivative = 1.38, number of steps used = 8, number of rules used = 7, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.189, Rules used = {171, 170, 418, 176, 422, 492, 411} \[ -\frac{10 \sqrt{2 x-5} \sqrt{4 x+1}}{897 \sqrt{2-3 x} \sqrt{5 x+7}}+\frac{6 \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{31 \sqrt{253} \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{5 \sqrt{\frac{22}{31}} \sqrt{4 x+1} F\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{2 x-5}}{\sqrt{5 x+7}}\right )|\frac{39}{62}\right )}{1209 \sqrt{2-3 x} \sqrt{-\frac{4 x+1}{2-3 x}}}+\frac{10 \sqrt{\frac{22}{31}} \sqrt{4 x+1} E\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{2 x-5}}{\sqrt{5 x+7}}\right )|\frac{39}{62}\right )}{897 \sqrt{2-3 x} \sqrt{-\frac{4 x+1}{2-3 x}}} \]
Antiderivative was successfully verified.
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Rule 171
Rule 170
Rule 418
Rule 176
Rule 422
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{3/2}} \, dx &=\frac{3}{31} \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx+\frac{5}{31} \int \frac{\sqrt{2-3 x}}{\sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{3/2}} \, dx\\ &=\frac{\left (5 \sqrt{2} \sqrt{2-3 x} \sqrt{\frac{1+4 x}{7+5 x}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{31 x^2}{11}}}{\sqrt{1+\frac{23 x^2}{22}}} \, dx,x,\frac{\sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )}{1209 \sqrt{1+4 x} \sqrt{-\frac{2-3 x}{7+5 x}}}+\frac{\left (3 \sqrt{\frac{2}{253}} \sqrt{-\frac{-5+2 x}{2-3 x}} \sqrt{7+5 x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{2}} \sqrt{1+\frac{31 x^2}{23}}} \, dx,x,\frac{\sqrt{1+4 x}}{\sqrt{2-3 x}}\right )}{31 \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{2-3 x}}}\\ &=\frac{6 \sqrt{7+5 x} F\left (\tan ^{-1}\left (\frac{\sqrt{1+4 x}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{31 \sqrt{253} \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{5-2 x}}}+\frac{\left (5 \sqrt{2} \sqrt{2-3 x} \sqrt{\frac{1+4 x}{7+5 x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{23 x^2}{22}} \sqrt{1+\frac{31 x^2}{11}}} \, dx,x,\frac{\sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )}{1209 \sqrt{1+4 x} \sqrt{-\frac{2-3 x}{7+5 x}}}+\frac{\left (5 \sqrt{2} \sqrt{2-3 x} \sqrt{\frac{1+4 x}{7+5 x}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{23 x^2}{22}} \sqrt{1+\frac{31 x^2}{11}}} \, dx,x,\frac{\sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )}{429 \sqrt{1+4 x} \sqrt{-\frac{2-3 x}{7+5 x}}}\\ &=-\frac{10 \sqrt{-5+2 x} \sqrt{1+4 x}}{897 \sqrt{2-3 x} \sqrt{7+5 x}}+\frac{6 \sqrt{7+5 x} F\left (\tan ^{-1}\left (\frac{\sqrt{1+4 x}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{31 \sqrt{253} \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{5-2 x}}}-\frac{5 \sqrt{\frac{22}{31}} \sqrt{1+4 x} F\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )|\frac{39}{62}\right )}{1209 \sqrt{2-3 x} \sqrt{-\frac{1+4 x}{2-3 x}}}-\frac{\left (10 \sqrt{2} \sqrt{2-3 x} \sqrt{\frac{1+4 x}{7+5 x}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{23 x^2}{22}}}{\left (1+\frac{31 x^2}{11}\right )^{3/2}} \, dx,x,\frac{\sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )}{897 \sqrt{1+4 x} \sqrt{-\frac{2-3 x}{7+5 x}}}\\ &=-\frac{10 \sqrt{-5+2 x} \sqrt{1+4 x}}{897 \sqrt{2-3 x} \sqrt{7+5 x}}+\frac{10 \sqrt{\frac{22}{31}} \sqrt{1+4 x} E\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )|\frac{39}{62}\right )}{897 \sqrt{2-3 x} \sqrt{-\frac{1+4 x}{2-3 x}}}+\frac{6 \sqrt{7+5 x} F\left (\tan ^{-1}\left (\frac{\sqrt{1+4 x}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{31 \sqrt{253} \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{5-2 x}}}-\frac{5 \sqrt{\frac{22}{31}} \sqrt{1+4 x} F\left (\tan ^{-1}\left (\frac{\sqrt{\frac{31}{11}} \sqrt{-5+2 x}}{\sqrt{7+5 x}}\right )|\frac{39}{62}\right )}{1209 \sqrt{2-3 x} \sqrt{-\frac{1+4 x}{2-3 x}}}\\ \end{align*}
Mathematica [A] time = 1.57502, size = 237, normalized size = 1.22 \[ -\frac{2 \sqrt{2 x-5} \sqrt{4 x+1} \left (-23 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right ),\frac{39}{62}\right )+1705 \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )-55 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )\right )}{305877 \sqrt{2-3 x} \sqrt{5 x+7} \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 599, normalized size = 3.1 \begin{align*}{\frac{2}{36705240\,{x}^{4}-55669614\,{x}^{3}-117762645\,{x}^{2}+60257769\,x+21411390}\sqrt{2-3\,x}\sqrt{2\,x-5}\sqrt{4\,x+1}\sqrt{7+5\,x} \left ( 1104\,\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{2\,x-5}{4\,x+1}}}\sqrt{{\frac{-2+3\,x}{4\,x+1}}}{x}^{2}{\it EllipticF} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}},1/39\,\sqrt{31}\sqrt{78} \right ) +880\,\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{2\,x-5}{4\,x+1}}}\sqrt{{\frac{-2+3\,x}{4\,x+1}}}{x}^{2}{\it EllipticE} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}},1/39\,\sqrt{31}\sqrt{78} \right ) +552\,\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{2\,x-5}{4\,x+1}}}\sqrt{{\frac{-2+3\,x}{4\,x+1}}}x{\it EllipticF} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}},1/39\,\sqrt{31}\sqrt{78} \right ) +440\,\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{2\,x-5}{4\,x+1}}}\sqrt{{\frac{-2+3\,x}{4\,x+1}}}x{\it EllipticE} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}},1/39\,\sqrt{31}\sqrt{78} \right ) +69\,\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{2\,x-5}{4\,x+1}}}\sqrt{{\frac{-2+3\,x}{4\,x+1}}}{\it EllipticF} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}},1/39\,\sqrt{31}\sqrt{78} \right ) +55\,\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}}\sqrt{3}\sqrt{13}\sqrt{{\frac{2\,x-5}{4\,x+1}}}\sqrt{{\frac{-2+3\,x}{4\,x+1}}}{\it EllipticE} \left ( 1/31\,\sqrt{31}\sqrt{11}\sqrt{{\frac{7+5\,x}{4\,x+1}}},1/39\,\sqrt{31}\sqrt{78} \right ) +7590\,{x}^{2}-24035\,x+12650 \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{1}{{\left (5 \, x + 7\right )}^{\frac{3}{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{\sqrt{5 \, x + 7} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{600 \, x^{5} - 70 \, x^{4} - 3199 \, x^{3} - 1710 \, x^{2} + 1729 \, x + 490}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{{\left (5 \, x + 7\right )}^{\frac{3}{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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