3.2691 \(\int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=249 \[ \frac{2}{65} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}+\frac{178 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}{10725}+\frac{601 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{160875}-\frac{18034 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{625625}-\frac{11725073 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{56306250}-\frac{776112041 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{506756250}-\frac{776112041 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{230343750 \sqrt{33}}-\frac{51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{460687500 \sqrt{33}} \]

[Out]

(-776112041*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/506756250 - (11725073*Sqr
t[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/56306250 - (18034*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(5/2))/625625 + (601*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(
5/2))/160875 + (178*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/10725 + (2*(1
- 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/65 - (51601293223*EllipticE[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(460687500*Sqrt[33]) - (776112041*EllipticF[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(230343750*Sqrt[33])

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Rubi [A]  time = 0.568609, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2}{65} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}+\frac{178 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}{10725}+\frac{601 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{160875}-\frac{18034 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{625625}-\frac{11725073 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{56306250}-\frac{776112041 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{506756250}-\frac{776112041 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{230343750 \sqrt{33}}-\frac{51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{460687500 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-776112041*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/506756250 - (11725073*Sqr
t[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/56306250 - (18034*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x]*(3 + 5*x)^(5/2))/625625 + (601*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(
5/2))/160875 + (178*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/10725 + (2*(1
- 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/65 - (51601293223*EllipticE[ArcSi
n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(460687500*Sqrt[33]) - (776112041*EllipticF[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(230343750*Sqrt[33])

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Rubi in Sympy [A]  time = 54.3711, size = 230, normalized size = 0.92 \[ \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{39} - \frac{37 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{429} + \frac{8746 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{57915} - \frac{150812 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{2027025} - \frac{31887029 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{101351250} - \frac{371279941 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{253378125} - \frac{51601293223 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{15202687500} - \frac{776112041 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{8062031250} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*x + 1)**(3/2)*(3*x + 2)**(7/2)*(5*x + 3)**(3/2)/39 - 37*(-2*x + 1)**(3/2)*
(3*x + 2)**(7/2)*sqrt(5*x + 3)/429 + 8746*sqrt(-2*x + 1)*(3*x + 2)**(7/2)*sqrt(5
*x + 3)/57915 - 150812*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/2027025 - 3
1887029*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x + 3)/101351250 - 371279941*sqrt
(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/253378125 - 51601293223*sqrt(33)*elliptic
_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/15202687500 - 776112041*sqrt(35)*elli
ptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/8062031250

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Mathematica [A]  time = 0.441935, size = 115, normalized size = 0.46 \[ \frac{51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (7016625000 x^5+12374775000 x^4+3047388750 x^3-5775295500 x^2-3548873565 x+325972172\right )[phone] F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{7601343750 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(51601293223*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 -
4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(325972172 - 3548873565*x - 5775295500*x^2 + 30
47388750*x^3 + 12374775000*x^4 + 7016625000*x^5) + 5197919174*EllipticF[ArcSin[S
qrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(7601343750*Sqrt[2])

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Maple [C]  time = 0.016, size = 189, normalized size = 0.8 \[{\frac{1}{456080625000\,{x}^{3}[phone]\,{x}^{2}-106418812500\,x-91216125000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -6314962500000\,{x}^{8}-15978768750000\,{x}^{7}-9807753375000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -51601293223\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-2009737437330\,{x}^{2}-570343085580\,x+58674990960 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15202687500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-6314962500000*x^8-1597
8768750000*x^7-9807753375000*x^6+25989595870*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))-51601293223*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellip
ticE(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))+6956762
962500*x^5+10046351241000*x^4+1491065725050*x^3-2009737437330*x^2-570343085580*x
[phone])/(30*x^3+23*x^2-7*x-6)

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

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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \]

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)^(3/2),x, algorithm="fricas")

[Out]

integral(-(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)*sqrt(5*x + 3)*sqrt(3*x + 2)*sq
rt(-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)**(3/2)*(2+3*x)**(5/2)*(3+5*x)**(3/2),x)

[Out]

Timed out

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

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)^(3/2),x, algorithm="giac")

[Out]

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