Optimal. Leaf size=160 \[ -\frac {695}{42} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {104 \sqrt {2+3 x} (3+5 x)^{3/2}}{21 \sqrt {1-2 x}}+\frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {4621}{42} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {139}{42} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 159,
164, 114, 120} \begin {gather*} -\frac {139}{42} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {4621}{42} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {\sqrt {3 x+2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {104 \sqrt {3 x+2} (5 x+3)^{3/2}}{21 \sqrt {1-2 x}}-\frac {695}{42} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(3+5 x)^{3/2} \left (\frac {59}{2}+45 x\right )}{(1-2 x)^{3/2} \sqrt {2+3 x}} \, dx\\ &=-\frac {104 \sqrt {2+3 x} (3+5 x)^{3/2}}{21 \sqrt {1-2 x}}+\frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{21} \int \frac {\left (-\frac {4065}{2}-\frac {6255 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {695}{42} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {104 \sqrt {2+3 x} (3+5 x)^{3/2}}{21 \sqrt {1-2 x}}+\frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1}{189} \int \frac {\frac {263295}{4}+\frac {207945 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {695}{42} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {104 \sqrt {2+3 x} (3+5 x)^{3/2}}{21 \sqrt {1-2 x}}+\frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1529}{84} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {4621}{42} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {695}{42} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {104 \sqrt {2+3 x} (3+5 x)^{3/2}}{21 \sqrt {1-2 x}}+\frac {\sqrt {2+3 x} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {4621}{42} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {139}{42} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A]
time = 7.56, size = 120, normalized size = 0.75 \begin {gather*} -\frac {6 \sqrt {2+3 x} \sqrt {3+5 x} \left (1193-3408 x+350 x^2\right )+9242 \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 )-4655 \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 )}{252 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 229, normalized size = 1.43
method | result | size |
default | \(-\frac {\left (9174 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-18484 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4587 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+9242 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+31500 x^{4}-266820 x^{3}-268542 x^{2}+13314 x +42948\right ) \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{252 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(229\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {-\frac {7645}{14} x^{2}-\frac {29051}{42} x -\frac {1529}{7}}{\sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {29255 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1764 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {23105 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{882 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {25 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{12}+\frac {121 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{48 \left (-\frac {1}{2}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(244\) |
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 = 0.18, size = 45, normalized size = 0.28 \begin {gather*} -\frac {{\left (350 \, x^{2} - 3408 \, x + 1193\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{42 \, {\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 {\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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