3.2927 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=249 \[ \frac{12071114168 \sqrt{1-2 x} \sqrt{3 x+2}}{9587193 \sqrt{5 x+3}}-\frac{181551856 \sqrt{1-2 x} \sqrt{3 x+2}}{871563 (5 x+3)^{3/2}}+\frac{4115652 \sqrt{1-2 x}}{132055 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{19548 \sqrt{1-2 x}}{18865 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{138 \sqrt{1-2 x}}{2695 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{363103712 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1452605 \sqrt{33}}-\frac{12071114168 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1452605 \sqrt{33}} \]

[Out]

4/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (138*Sqrt[1 - 2*x])/(2695
*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (19548*Sqrt[1 - 2*x])/(18865*(2 + 3*x)^(3/2)
*(3 + 5*x)^(3/2)) + (4115652*Sqrt[1 - 2*x])/(132055*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2
)) - (181551856*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(871563*(3 + 5*x)^(3/2)) + (1207111
4168*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9587193*Sqrt[3 + 5*x]) - (12071114168*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1452605*Sqrt[33]) - (363103712*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1452605*Sqrt[33])

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Rubi [A]  time = 0.610648, 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 \[ \frac{12071114168 \sqrt{1-2 x} \sqrt{3 x+2}}{9587193 \sqrt{5 x+3}}-\frac{181551856 \sqrt{1-2 x} \sqrt{3 x+2}}{871563 (5 x+3)^{3/2}}+\frac{4115652 \sqrt{1-2 x}}{132055 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{19548 \sqrt{1-2 x}}{18865 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{138 \sqrt{1-2 x}}{2695 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{363103712 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1452605 \sqrt{33}}-\frac{12071114168 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1452605 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(5/2)),x]

[Out]

4/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (138*Sqrt[1 - 2*x])/(2695
*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (19548*Sqrt[1 - 2*x])/(18865*(2 + 3*x)^(3/2)
*(3 + 5*x)^(3/2)) + (4115652*Sqrt[1 - 2*x])/(132055*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2
)) - (181551856*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(871563*(3 + 5*x)^(3/2)) + (1207111
4168*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9587193*Sqrt[3 + 5*x]) - (12071114168*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1452605*Sqrt[33]) - (363103712*Elli
pticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1452605*Sqrt[33])

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Rubi in Sympy [A]  time = 54.2296, size = 230, normalized size = 0.92 \[ \frac{12071114168 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{9587193 \sqrt{5 x + 3}} - \frac{181551856 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{871563 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{4115652 \sqrt{- 2 x + 1}}{132055 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{19548 \sqrt{- 2 x + 1}}{18865 \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{138 \sqrt{- 2 x + 1}}{2695 \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{12071114168 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{47935965} - \frac{363103712 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{50841175} + \frac{4}{77 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-2*x)**(3/2)/(2+3*x)**(7/2)/(3+5*x)**(5/2),x)

[Out]

12071114168*sqrt(-2*x + 1)*sqrt(3*x + 2)/(9587193*sqrt(5*x + 3)) - 181551856*sqr
t(-2*x + 1)*sqrt(3*x + 2)/(871563*(5*x + 3)**(3/2)) + 4115652*sqrt(-2*x + 1)/(13
2055*sqrt(3*x + 2)*(5*x + 3)**(3/2)) + 19548*sqrt(-2*x + 1)/(18865*(3*x + 2)**(3
/2)*(5*x + 3)**(3/2)) + 138*sqrt(-2*x + 1)/(2695*(3*x + 2)**(5/2)*(5*x + 3)**(3/
2)) - 12071114168*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/47
935965 - 363103712*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/
50841175 + 4/(77*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.432978, size = 114, normalized size = 0.46 \[ \frac{2 \left (\frac{-8148002063400 x^5-16841199826980 x^4-9658241620704 x^3+1466692421066 x^2+2920885694212 x+687365548973}{\sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}+4 \sqrt{2} \left (1508889271 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-759987865 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{47935965} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(5/2)),x]

[Out]

(2*((687365548973 + 2920885694212*x + 1466692421066*x^2 - 9658241620704*x^3 - 16
841199826980*x^4 - 8148002063400*x^5)/(Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(
3/2)) + 4*Sqrt[2]*(1508889271*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]
- 759987865*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/47935965

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Maple [C]  time = 0.039, size = 502, normalized size = 2. \[{\frac{2}{-47935965+95871930\,x}\sqrt{1-2\,x} \left ( 136797815700\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-271600068780\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-525093466308\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-337991196704\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -72426685008\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-1466692421066\,{x}^{2}-2920885694212\,x-687365548973 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(5/2),x)

[Out]

2/47935965*(1-2*x)^(1/2)*(136797815700*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+
3*x)^(1/2)-271600068780*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)+2644
75777020*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)-525093466308*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)[phone]*2^(1/2)*EllipticF(1/1
1*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)-337991196704*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)[phone]*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellipt
icF(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))-72426685
008*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^5+1684119982
6980*x^4+9658241620704*x^3-1466692421066*x^2-2920885694212*x-687365548973)/(2+3*
x)^(5/2)/(3+5*x)^(3/2)/(-1+2*x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x + 3)^(5/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((5*x + 3)^(5/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (1350 \, x^{6} + 3645 \, x^{5} + 3366 \, x^{4} + 769 \, x^{3} - 638 \, x^{2} - 420 \, x - 72\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x + 3)^(5/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

integral(-1/((1350*x^6 + 3645*x^5 + 3366*x^4 + 769*x^3 - 638*x^2 - 420*x - 72)*s
qrt(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/(1-2*x)**(3/2)/(2+3*x)**(7/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x + 3)^(5/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

integrate(1/((5*x + 3)^(5/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)), x)