3.2981 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=218 \[ \frac{99425780 \sqrt{1-2 x} \sqrt{3 x+2}}{15065589 \sqrt{5 x+3}}-\frac{1523260 \sqrt{1-2 x} \sqrt{3 x+2}}{1369599 (5 x+3)^{3/2}}+\frac{5034 \sqrt{1-2 x}}{41503 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{456}{5929 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{609304 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{456533 \sqrt{33}}-\frac{19885156 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{456533 \sqrt{33}} \]

[Out]

4/(231*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + 456/(5929*Sqrt[1 - 2*x]*
Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + (5034*Sqrt[1 - 2*x])/(41503*Sqrt[2 + 3*x]*(3 +
5*x)^(3/2)) - (1523260*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1369599*(3 + 5*x)^(3/2)) +
(99425780*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(15065589*Sqrt[3 + 5*x]) - (19885156*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(456533*Sqrt[33]) - (609304*Ellip
ticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(456533*Sqrt[33])

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Rubi [A]  time = 0.521696, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{99425780 \sqrt{1-2 x} \sqrt{3 x+2}}{15065589 \sqrt{5 x+3}}-\frac{1523260 \sqrt{1-2 x} \sqrt{3 x+2}}{1369599 (5 x+3)^{3/2}}+\frac{5034 \sqrt{1-2 x}}{41503 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{456}{5929 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{609304 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{456533 \sqrt{33}}-\frac{19885156 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{456533 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

4/(231*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + 456/(5929*Sqrt[1 - 2*x]*
Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + (5034*Sqrt[1 - 2*x])/(41503*Sqrt[2 + 3*x]*(3 +
5*x)^(3/2)) - (1523260*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1369599*(3 + 5*x)^(3/2)) +
(99425780*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(15065589*Sqrt[3 + 5*x]) - (19885156*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(456533*Sqrt[33]) - (609304*Ellip
ticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(456533*Sqrt[33])

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Rubi in Sympy [A]  time = 47.9497, size = 201, normalized size = 0.92 \[ - \frac{19885156 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{15065589} - \frac{609304 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{15978655} - \frac{39770312 \sqrt{3 x + 2} \sqrt{5 x + 3}}{15065589 \sqrt{- 2 x + 1}} + \frac{2927780 \sqrt{3 x + 2}}{195657 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} - \frac{44960 \sqrt{3 x + 2}}{17787 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{194}{539 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{4}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-19885156*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/15065589 -
609304*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/15978655 -
39770312*sqrt(3*x + 2)*sqrt(5*x + 3)/(15065589*sqrt(-2*x + 1)) + 2927780*sqrt(3*
x + 2)/(195657*sqrt(-2*x + 1)*sqrt(5*x + 3)) - 44960*sqrt(3*x + 2)/(17787*sqrt(-
2*x + 1)*(5*x + 3)**(3/2)) + 194/(539*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3
/2)) + 4/(231*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.304142, size = 109, normalized size = 0.5 \[ \frac{\frac{5965546800 x^4+1389742160 x^3-3604421052 x^2-422976360 x+566289874}{(1-2 x)^{3/2} \sqrt{3 x+2} (5 x+3)^{3/2}}+4 \sqrt{2} \left (4971289 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2457910 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{15065589} \]

Antiderivative was successfully verified.

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

[Out]

((566289874 - 422976360*x - 3604421052*x^2 + 1389742160*x^3 + 5965546800*x^4)/((
1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + 4*Sqrt[2]*(4971289*EllipticE[Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2457910*EllipticF[ArcSin[Sqrt[2/11]*Sqrt
[3 + 5*x]], -33/2]))/15065589

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Maple [C]  time = 0.037, size = 383, normalized size = 1.8 \[{\frac{2}{15065589\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 49158200\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-99425780\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+4915820\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-9942578\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-14747460\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}-1802210526\,{x}^{2}-211488180\,x+283144937 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15065589*(1-2*x)^(1/2)*(49158200*2^(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))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)
^(1/2)-99425780*2^(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))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+4915820*2^(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)
)*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-9942578*2^(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))*x*(3+5*x)^(1/2)*(2+3
*x)^(1/2)*(1-2*x)^(1/2)-14747460*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))
[phone]*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))[phone]*x^4+6948710
80*x^3-1802210526*x^2-211488180*x+283144937)/(3+5*x)^(3/2)/(-1+2*x)^2/(2+3*x)^(1
/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((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. \[{\rm integral}\left (\frac{1}{{\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((300*x^5 + 260*x^4 - 137*x^3 - 136*x^2 + 15*x + 18)*sqrt(5*x + 3)*sq
rt(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/(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 [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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