3.2926 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=218 \[ \frac{213119320 \sqrt{1-2 x} \sqrt{3 x+2}}{1369599 \sqrt{5 x+3}}-\frac{3205940 \sqrt{1-2 x} \sqrt{3 x+2}}{124509 (5 x+3)^{3/2}}+\frac{14496 \sqrt{1-2 x}}{3773 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{54 \sqrt{1-2 x}}{539 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{1282376 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}}-\frac{42623864 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}} \]

[Out]

4/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (54*Sqrt[1 - 2*x])/(539*(
2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (14496*Sqrt[1 - 2*x])/(3773*Sqrt[2 + 3*x]*(3 +
5*x)^(3/2)) - (3205940*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(124509*(3 + 5*x)^(3/2)) +
(213119320*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1369599*Sqrt[3 + 5*x]) - (42623864*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(41503*Sqrt[33]) - (1282376*Ellip
ticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(41503*Sqrt[33])

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Rubi [A]  time = 0.520714, 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 \[ \frac{213119320 \sqrt{1-2 x} \sqrt{3 x+2}}{1369599 \sqrt{5 x+3}}-\frac{3205940 \sqrt{1-2 x} \sqrt{3 x+2}}{124509 (5 x+3)^{3/2}}+\frac{14496 \sqrt{1-2 x}}{3773 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{54 \sqrt{1-2 x}}{539 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{4}{77 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{1282376 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}}-\frac{42623864 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

4/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (54*Sqrt[1 - 2*x])/(539*(
2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (14496*Sqrt[1 - 2*x])/(3773*Sqrt[2 + 3*x]*(3 +
5*x)^(3/2)) - (3205940*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(124509*(3 + 5*x)^(3/2)) +
(213119320*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1369599*Sqrt[3 + 5*x]) - (42623864*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(41503*Sqrt[33]) - (1282376*Ellip
ticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(41503*Sqrt[33])

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Rubi in Sympy [A]  time = 46.0602, size = 201, normalized size = 0.92 \[ \frac{213119320 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{1369599 \sqrt{5 x + 3}} - \frac{3205940 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{124509 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{14496 \sqrt{- 2 x + 1}}{3773 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{54 \sqrt{- 2 x + 1}}{539 \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{42623864 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1369599} - \frac{1282376 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1369599} + \frac{4}{77 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

213119320*sqrt(-2*x + 1)*sqrt(3*x + 2)/(1369599*sqrt(5*x + 3)) - 3205940*sqrt(-2
*x + 1)*sqrt(3*x + 2)/(124509*(5*x + 3)**(3/2)) + 14496*sqrt(-2*x + 1)/(3773*sqr
t(3*x + 2)*(5*x + 3)**(3/2)) + 54*sqrt(-2*x + 1)/(539*(3*x + 2)**(3/2)*(5*x + 3)
**(3/2)) - 42623864*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/
1369599 - 1282376*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/13
69599 + 4/(77*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.328807, size = 109, normalized size = 0.5 \[ \frac{2 \left (\frac{-9590369400 x^4-13428808080 x^3-2415287594 x^2+3336610202 x+1213551469}{\sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}+2 \sqrt{2} \left (10655966 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5366165 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{1369599} \]

Antiderivative was successfully verified.

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

[Out]

(2*((1213551469 + 3336610202*x - 2415287594*x^2 - 13428808080*x^3 - 9590369400*x
^4)/(Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + 2*Sqrt[2]*(10655966*Ellipt
icE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5366165*EllipticF[ArcSin[Sqrt[2/1
1]*Sqrt[3 + 5*x]], -33/2])))/1369599

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Maple [C]  time = 0.037, size = 383, normalized size = 1.8 \[ -{\frac{2}{-1369599+2739198\,x}\sqrt{1-2\,x} \left ( 319678980\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-160984950\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-203914270\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -64393980\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -9590369400\,{x}^{4}-13428808080\,{x}^{3}-2415287594\,{x}^{2}[phone]\,x+1213551469 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1369599*(1-2*x)^(1/2)*(319678980*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)-160984950*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)[phone]*
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)-203914270*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)[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))-64393980*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(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))-9590369400*x^4-
13428808080*x^3-2415287594*x^2+3336610202*x+1213551469)/(2+3*x)^(3/2)/(3+5*x)^(3
/2)/(-1+2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-1/((450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)*sqrt(5*x + 3)*s
qrt(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)**(5/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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