3.2925 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ \frac{2976620 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 \sqrt{5 x+3}}-\frac{45040 \sqrt{1-2 x} \sqrt{3 x+2}}{17787 (5 x+3)^{3/2}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{18016 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}}-\frac{595324 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}} \]

[Out]

4/(77*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + (186*Sqrt[1 - 2*x])/(539*Sq
rt[2 + 3*x]*(3 + 5*x)^(3/2)) - (45040*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(17787*(3 + 5
*x)^(3/2)) + (2976620*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(195657*Sqrt[3 + 5*x]) - (595
324*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5929*Sqrt[33]) - (18016*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5929*Sqrt[33])

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Rubi [A]  time = 0.429534, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2976620 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 \sqrt{5 x+3}}-\frac{45040 \sqrt{1-2 x} \sqrt{3 x+2}}{17787 (5 x+3)^{3/2}}+\frac{186 \sqrt{1-2 x}}{539 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{18016 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}}-\frac{595324 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5929 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

4/(77*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + (186*Sqrt[1 - 2*x])/(539*Sq
rt[2 + 3*x]*(3 + 5*x)^(3/2)) - (45040*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(17787*(3 + 5
*x)^(3/2)) + (2976620*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(195657*Sqrt[3 + 5*x]) - (595
324*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5929*Sqrt[33]) - (18016*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5929*Sqrt[33])

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Rubi in Sympy [A]  time = 39.6028, size = 172, normalized size = 0.92 \[ \frac{2976620 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{195657 \sqrt{5 x + 3}} - \frac{45040 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{17787 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{186 \sqrt{- 2 x + 1}}{539 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{595324 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{195657} - \frac{18016 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{207515} + \frac{4}{77 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2976620*sqrt(-2*x + 1)*sqrt(3*x + 2)/(195657*sqrt(5*x + 3)) - 45040*sqrt(-2*x +
1)*sqrt(3*x + 2)/(17787*(5*x + 3)**(3/2)) + 186*sqrt(-2*x + 1)/(539*sqrt(3*x + 2
)*(5*x + 3)**(3/2)) - 595324*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7)
, 35/33)/195657 - 18016*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33
/35)/207515 + 4/(77*sqrt(-2*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.29631, size = 104, normalized size = 0.56 \[ \frac{2 \left (\frac{-44649300 x^3-32744810 x^2+10598372 x+8473261}{\sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}+2 \sqrt{2} \left (148831 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-74515 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{195657} \]

Antiderivative was successfully verified.

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

[Out]

(2*((8473261 + 10598372*x - 32744810*x^2 - 44649300*x^3)/(Sqrt[1 - 2*x]*Sqrt[2 +
3*x]*(3 + 5*x)^(3/2)) + 2*Sqrt[2]*(148831*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2] - 74515*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/1956
57

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Maple [C]  time = 0.037, size = 267, normalized size = 1.4 \[{\frac{2}{1173942\,{x}^{2}+195657\,x-391314}\sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 745150\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1488310\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+447090\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -892986\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{3}[phone]\,{x}^{2}-10598372\,x-8473261 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/195657*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(745150*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)-1488310*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)+447090*
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))-892986*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^3+32744810*x^2-10598372*x-8473261)/(3+5*x)^(3/
2)/(6*x^2+x-2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

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)^(3/2)),x, algorithm="fricas")

[Out]

integral(-1/((150*x^4 + 205*x^3 + 34*x^2 - 51*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/(1-2*x)**(3/2)/(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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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