Optimal. Leaf size=125 \[ -\frac {62 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{49 \sqrt {33}}-\frac {12 \sqrt {1-2 x} \sqrt {5 x+3}}{49 \sqrt {3 x+2}}+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}+\frac {4}{49} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 152, 158, 113, 119} \[ -\frac {12 \sqrt {1-2 x} \sqrt {5 x+3}}{49 \sqrt {3 x+2}}+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} \sqrt {3 x+2}}-\frac {62 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}}+\frac {4}{49} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 99
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {2}{7} \int \frac {-4-\frac {15 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {12 \sqrt {1-2 x} \sqrt {3+5 x}}{49 \sqrt {2+3 x}}-\frac {4}{49} \int \frac {\frac {5}{4}+15 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {12 \sqrt {1-2 x} \sqrt {3+5 x}}{49 \sqrt {2+3 x}}-\frac {12}{49} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx+\frac {31}{49} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} \sqrt {2+3 x}}-\frac {12 \sqrt {1-2 x} \sqrt {3+5 x}}{49 \sqrt {2+3 x}}+\frac {4}{49} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {62 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 122, normalized size = 0.98 \[ \frac {35 \sqrt {2-4 x} (3 x+2) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+2 \sqrt {3 x+2} \sqrt {5 x+3} (12 x+1)-4 \sqrt {2-4 x} (3 x+2) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{49 \sqrt {1-2 x} (3 x+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 135, normalized size = 1.08 \[ -\frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {3 x +2}\, \left (120 x^{2}+82 x -4 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+35 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+6\right )}{49 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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