Optimal. Leaf size=127 \[ \frac {11 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}-\frac {74 \sqrt {2+3 x} \sqrt {3+5 x}}{147 \sqrt {1-2 x}}-\frac {37}{49} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {13 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}} \]
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Rubi [A]
time = 0.03, antiderivative size = 127, 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 = {100, 157, 164,
114, 120} \begin {gather*} -\frac {13 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}}-\frac {37}{49} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {74 \sqrt {3 x+2} \sqrt {5 x+3}}{147 \sqrt {1-2 x}}+\frac {11 \sqrt {3 x+2} \sqrt {5 x+3}}{21 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx &=\frac {11 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}-\frac {1}{21} \int \frac {\frac {227}{2}+180 x}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}-\frac {74 \sqrt {2+3 x} \sqrt {3+5 x}}{147 \sqrt {1-2 x}}+\frac {2 \int \frac {\frac {7755}{4}+\frac {6105 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1617}\\ &=\frac {11 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}-\frac {74 \sqrt {2+3 x} \sqrt {3+5 x}}{147 \sqrt {1-2 x}}+\frac {13}{98} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {37}{49} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {11 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}-\frac {74 \sqrt {2+3 x} \sqrt {3+5 x}}{147 \sqrt {1-2 x}}-\frac {37}{49} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {13 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 = 6.32, size = 115, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {2+3 x} \sqrt {3+5 x} (3+148 x)-74 \sqrt {2-4 x} (-1+2 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+35 \sqrt {2-4 x} (-1+2 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{294 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(223\) vs.
\(2(93)=186\).
time = 0.10, size = 224, normalized size = 1.76
method | result | size |
default | \(-\frac {\left (78 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-148 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-39 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+74 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-4440 x^{3}-5714 x^{2}-1890 x -36\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}}{294 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(224\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {-\frac {370}{49} x^{2}-\frac {1406}{147} x -\frac {148}{49}}{\sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {235 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{2058 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {185 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1029 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {11 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{84 \left (-\frac {1}{2}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(225\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.24, size = 40, normalized size = 0.31 \begin {gather*} \frac {{\left (148 \, x + 3\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{147 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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