3.2747 \(\int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=280 \[ \frac{2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}+\frac{62 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}}{2925}+\frac{3698 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}{482625}+\frac{142391 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{7239375}-\frac{569519 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{28153125}-\frac{400516993 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{2533781250}-\frac{13267820528 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{11402015625}-\frac{13267820528 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5182734375 \sqrt{33}}-\frac{1764163292393 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{20730937500 \sqrt{33}} \]

[Out]

(-13267820528*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/11402015625 - (40051699
3*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/2533781250 - (569519*Sqrt[1 - 2*x
]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/28153125 + (142391*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2
)*(3 + 5*x)^(5/2))/7239375 + (3698*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2)
)/482625 + (62*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/2925 + (2*(1 - 2
*x)^(5/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/75 - (1764163292393*EllipticE[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(20730937500*Sqrt[33]) - (13267820528*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5182734375*Sqrt[33])

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Rubi [A]  time = 0.644864, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}+\frac{62 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}}{2925}+\frac{3698 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}{482625}+\frac{142391 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{7239375}-\frac{569519 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{28153125}-\frac{400516993 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{2533781250}-\frac{13267820528 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{11402015625}-\frac{13267820528 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5182734375 \sqrt{33}}-\frac{1764163292393 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{20730937500 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2),x]

[Out]

(-13267820528*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/11402015625 - (40051699
3*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/2533781250 - (569519*Sqrt[1 - 2*x
]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/28153125 + (142391*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2
)*(3 + 5*x)^(5/2))/7239375 + (3698*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2)
)/482625 + (62*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/2925 + (2*(1 - 2
*x)^(5/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/75 - (1764163292393*EllipticE[ArcSin[
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(20730937500*Sqrt[33]) - (13267820528*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5182734375*Sqrt[33])

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Rubi in Sympy [A]  time = 62.4838, size = 258, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{45} - \frac{181 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{1755} + \frac{1594 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{10725} + \frac{298244 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{2606175} - \frac{5068747 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{91216125} - \frac{1089070189 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{4560806250} - \frac{25385346787 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{22804031250} - \frac{1764163292393 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{684120937500} - \frac{13267820528 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{181395703125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(2+3*x)**(5/2)*(3+5*x)**(3/2),x)

[Out]

2*(-2*x + 1)**(5/2)*(3*x + 2)**(7/2)*(5*x + 3)**(3/2)/45 - 181*(-2*x + 1)**(5/2)
*(3*x + 2)**(7/2)*sqrt(5*x + 3)/1755 + 1594*(-2*x + 1)**(3/2)*(3*x + 2)**(7/2)*s
qrt(5*x + 3)/10725 + 298244*sqrt(-2*x + 1)*(3*x + 2)**(7/2)*sqrt(5*x + 3)/260617
5 - 5068747*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/91216125 - 1089070189*
sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/4560806250 - 25385346787*sqrt(-2*x
+ 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/22804031250 - 1764163292393*sqrt(33)*elliptic_
e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/684120937500 - 13267820528*sqrt(35)*el
liptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/181395703125

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Mathematica [A]  time = 0.349957, size = 119, normalized size = 0.42 \[ \frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (547296750000 x^6+621672975000 x^5-336683182500 x^4-528977216250 x^3+48836706750 x^2+173484591165 x+12155574323\right )+\sqrt{2} \left (1764163292393 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-888487137545 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{684120937500} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2),x]

[Out]

(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(12155574323 + 173484591165*x + 48
836706750*x^2 - 528977216250*x^3 - 336683182500*x^4 + 621672975000*x^5 + 5472967
50000*x^6) + Sqrt[2]*(1764163292393*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]],
-33/2] - 888487137545*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/68412
0937500

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Maple [C]  time = 0.017, size = 194, normalized size = 0.7 \[{\frac{1}{20523628125000\,{x}^{3}[phone]\,{x}^{2}-4788846562500\,x-4104725625000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 492567075000000\,{x}^{9}[phone]\,{x}^{8}[phone]\,{x}^{7}-937455630300000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1764163292393\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -362238910312500\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-36835025076780\,{x}^{2}-33779897017530\,x-2188003378140 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(2+3*x)^(5/2)*(3+5*x)^(3/2),x)

[Out]

1/684120937500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(492567075000000*x^9+93
7140435000000*x^8+11007171000000*x^7-937455630300000*x^6+888487137545*2^(1/2)*(3
+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^
(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))-1764163292393*2^(1/2)*(3+5*x)^(1/2)*(2+3*x
)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/
2)*3^(1/2)*2^(1/2))-362238910312500*x^5+361521647968500*x^4+215604575302050*x^3-
36835025076780*x^2-33779897017530*x-2188003378140)/(30*x^3+23*x^2-7*x-6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(3*x + 2)^(5/2)*(-2*x + 1)^(5/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(3/2)*(3*x + 2)^(5/2)*(-2*x + 1)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(3*x + 2)^(5/2)*(-2*x + 1)^(5/2),x, algorithm="fricas")

[Out]

integral((180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)*sqrt(5*x + 3)*sqrt(3*x
+ 2)*sqrt(-2*x + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(5/2)*(2+3*x)**(5/2)*(3+5*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.484079, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(3/2)*(3*x + 2)^(5/2)*(-2*x + 1)^(5/2),x, algorithm="giac")

[Out]

Done