3.2758 \(\int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=280 \[ \frac{2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac{106 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}}{4875}+\frac{8038 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}+\frac{364267 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{36196875}-\frac{26534891 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{760134375}-\frac{359748241 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{1520268750}-\frac{23763809947 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{13682418750}-\frac{23763809947 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6219281250 \sqrt{33}}-\frac{1580201444291 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12438562500 \sqrt{33}} \]

[Out]

(-23763809947*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/13682418750 - (35974824
1*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/1520268750 - (26534891*Sqrt[1 - 2
*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/760134375 + (364267*Sqrt[1 - 2*x]*Sqrt[2 + 3*
x]*(3 + 5*x)^(7/2))/36196875 + (8038*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/
2))/804375 + (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/4875 + (2*(1
- 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/75 - (1580201444291*EllipticE[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(12438562500*Sqrt[33]) - (23763809947*Ellip
ticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6219281250*Sqrt[33])

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Rubi [A]  time = 0.645549, 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 \[ \frac{2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac{106 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}}{4875}+\frac{8038 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}+\frac{364267 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{36196875}-\frac{26534891 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{760134375}-\frac{359748241 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{1520268750}-\frac{23763809947 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{13682418750}-\frac{23763809947 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{6219281250 \sqrt{33}}-\frac{1580201444291 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12438562500 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-23763809947*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/13682418750 - (35974824
1*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/1520268750 - (26534891*Sqrt[1 - 2
*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/760134375 + (364267*Sqrt[1 - 2*x]*Sqrt[2 + 3*
x]*(3 + 5*x)^(7/2))/36196875 + (8038*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/
2))/804375 + (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/4875 + (2*(1
- 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/75 - (1580201444291*EllipticE[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(12438562500*Sqrt[33]) - (23763809947*Ellip
ticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6219281250*Sqrt[33])

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Rubi in Sympy [A]  time = 63.5606, size = 258, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{45} - \frac{37 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{351} + \frac{8318 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{57915} + \frac{21608 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{200475} - \frac{4239971 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{54729675} - \frac{978675493 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{2736483750} - \frac{11371367372 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{6841209375} - \frac{1580201444291 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{410472562500} - \frac{23763809947 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{205236281250} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(5/2)*(3*x + 2)**(5/2)*(5*x + 3)**(5/2)/45 - 37*(-2*x + 1)**(5/2)*
(3*x + 2)**(5/2)*(5*x + 3)**(3/2)/351 + 8318*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2)*
(5*x + 3)**(3/2)/57915 + 21608*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*(5*x + 3)**(3/2)/
200475 - 4239971*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/54729675 - 978675
493*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/2736483750 - 11371367372*sqrt(
-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/6841209375 - 1580201444291*sqrt(33)*ellipt
ic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/410472562500 - 23763809947*sqrt(33)
*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/205236281250

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Mathematica [A]  time = 0.34304, size = 119, normalized size = 0.42 \[ \frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (547296750000 x^6+579573225000 x^5-352885207500 x^4-487924998750 x^3+59959633500 x^2+157612390605 x+9093216326\right )+\sqrt{2} \left (1580201444291 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-795995716040 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{410472562500} \]

Antiderivative was successfully verified.

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

[Out]

(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(9093216326 + 157612390605*x + 599
59633500*x^2 - 487924998750*x^3 - 352885207500*x^4 + 579573225000*x^5 + 54729675
0000*x^6) + Sqrt[2]*(1580201444291*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -
33/2] - 795995716040*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/410472
562500

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Maple [C]  time = 0.018, size = 194, normalized size = 0.7 \[{\frac{1}{12314176875000\,{x}^{3}[phone]\,{x}^{2}-2873307937500\,x-2462835375000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 492567075000000\,{x}^{9}[phone]\,{x}^{8}-32623479000000\,{x}^{7}-902847084300000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1580201444291\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -312921865912500\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-37617016792110\,{x}^{2}-30279805737360\,x-1636778938680 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/410472562500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(492567075000000*x^9+89
9250660000000*x^8-32623479000000*x^7-902847084300000*x^6+795995716040*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))-1580201444291*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))-312921865912500*x^5+349206885747000*x^4+192171420950850*x^3-
37617016792110*x^2-30279805737360*x-1636778938680)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)*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)**(3/2)*(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.47386, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done