Optimal. Leaf size=253 \[ \frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {3+5 x}}{823543 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}+\frac {98642 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543}-\frac {65672 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543} \]
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Rubi [A]
time = 0.07, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {100, 155, 157,
164, 114, 120} \begin {gather*} -\frac {65672 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543}+\frac {98642 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543}+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {5 x+3}}{823543 \sqrt {3 x+2}}-\frac {33778 \sqrt {1-2 x} \sqrt {5 x+3}}{117649 (3 x+2)^{3/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {5 x+3}}{16807 (3 x+2)^{5/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {5 x+3}}{2401 (3 x+2)^{7/2}}+\frac {220 \sqrt {5 x+3}}{49 \sqrt {1-2 x} (3 x+2)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx &=\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {1}{21} \int \frac {\left (-\frac {345}{2}-315 x\right ) \sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {1}{147} \int \frac {-\frac {34305}{2}-\frac {58275 x}{2}}{\sqrt {1-2 x} (2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {2 \int \frac {-\frac {397335}{4}-\frac {340875 x}{2}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{7203}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {4 \int \frac {-\frac {1461615}{4}-\frac {2572425 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{252105}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {8 \int \frac {-\frac {4326885}{8}-\frac {3800025 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{5294205}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {3+5 x}}{823543 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {16 \int \frac {-\frac {1468575}{8}+\frac {11097225 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{37059435}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {3+5 x}}{823543 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}-\frac {98642 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{823543}+\frac {361196 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{823543}\\ &=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {3+5 x}}{823543 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}+\frac {98642 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543}-\frac {65672 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{823543}\\ \end {align*}
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Mathematica [A]
time = 9.04, size = 113, normalized size = 0.45 \begin {gather*} \frac {2 \left (\frac {\sqrt {3+5 x} \left (866085+6524789 x+10746933 x^2-7681599 x^3-28748088 x^4-15980004 x^5\right )}{(1-2 x)^{3/2} (2+3 x)^{7/2}}+\sqrt {2} \left (-49321 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+591115 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{2470629} \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. \(490\) vs.
\(2(185)=370\).
time = 0.11, size = 491, normalized size = 1.94
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{50421 \left (-\frac {1}{2}+x \right )^{2}}-\frac {1364 \left (-30 x^{2}-38 x -12\right )}{50421 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {194 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{453789 \left (\frac {2}{3}+x \right )^{3}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{194481 \left (\frac {2}{3}+x \right )^{4}}-\frac {10602 \left (-30 x^{2}-3 x +9\right )}{823543 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {1814 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{352947 \left (\frac {2}{3}+x \right )^{2}}+\frac {65270 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{17294403 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {493210 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{17294403 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(325\) |
default | \(-\frac {2 \sqrt {1-2 x}\, \left (2663334 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+29256876 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+3995001 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+43885314 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+887778 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+9752292 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-986420 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-10835880 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+79900020 x^{6}-394568 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-4334352 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+191680452 x^{5}+124652259 x^{4}-30689868 x^{3}-64864744 x^{2}-23904792 x -2598255\right )}{2470629 \left (2+3 x \right )^{\frac {7}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\) | \(491\) |
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 = 0.25, size = 80, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (15980004 \, x^{5} + 28748088 \, x^{4} + 7681599 \, x^{3} - 10746933 \, x^{2} - 6524789 \, x - 866085\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2470629 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\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 = -0.00 \begin {gather*} \int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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