Optimal. Leaf size=160 \[ -\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {1076}{675} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {8}{15} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {31588 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3375}-\frac {12758 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3375} \]
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Rubi [A]
time = 0.03, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 159, 164,
114, 120} \begin {gather*} -\frac {12758 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3375}+\frac {31588 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3375}-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{3 \sqrt {3 x+2}}-\frac {8}{15} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {1076}{675} \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}+\frac {2}{3} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {8}{15} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4}{225} \int \frac {\left (-\frac {2895}{4}-\frac {4035 x}{2}\right ) \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {1076}{675} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {8}{15} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {8 \int \frac {-\frac {73785}{8}-\frac {118455 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{10125}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {1076}{675} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {8}{15} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {31588 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3375}+\frac {70169 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3375}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {1076}{675} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {8}{15} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {31588 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3375}-\frac {12758 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3375}\\ \end {align*}
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Mathematica [A]
time = 5.80, size = 102, normalized size = 0.64 \begin {gather*} \frac {\frac {30 \sqrt {1-2 x} \sqrt {3+5 x} \left (-1661-534 x+180 x^2\right )}{\sqrt {2+3 x}}-31588 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+242095 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{10125} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 143, normalized size = 0.89
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (210507 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+31588 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-54000 x^{4}+154800 x^{3}+530520 x^{2}+1770 x -149490\right )}{10125 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(143\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {8 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45}-\frac {436 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{675}-\frac {4919 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{14175 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {31588 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{14175 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {98 \left (-30 x^{2}-3 x +9\right )}{81 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(240\) |
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 = 0.15, size = 33, normalized size = 0.21 \begin {gather*} \frac {2 \, {\left (180 \, x^{2} - 534 \, x - 1661\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{675 \, \sqrt {3 \, x + 2}} \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 (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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