3.2983 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=280 \[ \frac{412810345784 \sqrt{1-2 x} \sqrt{3 x+2}}{738213861 \sqrt{5 x+3}}-\frac{6208896328 \sqrt{1-2 x} \sqrt{3 x+2}}{67110351 (5 x+3)^{3/2}}+\frac{140700876 \sqrt{1-2 x}}{10168235 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{649224 \sqrt{1-2 x}}{1452605 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{3606 \sqrt{1-2 x}}{207515 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{632}{5929 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{12417792656 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{111850585 \sqrt{33}}-\frac{412810345784 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{111850585 \sqrt{33}} \]

[Out]

4/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 632/(5929*Sqrt[1 - 2*x
]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) - (3606*Sqrt[1 - 2*x])/(207515*(2 + 3*x)^(5/2
)*(3 + 5*x)^(3/2)) + (649224*Sqrt[1 - 2*x])/(1452605*(2 + 3*x)^(3/2)*(3 + 5*x)^(
3/2)) + (140700876*Sqrt[1 - 2*x])/(10168235*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (62
08896328*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(67110351*(3 + 5*x)^(3/2)) + (412810345784
*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(738213861*Sqrt[3 + 5*x]) - (412810345784*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(111850585*Sqrt[33]) - (12417792656*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(111850585*Sqrt[33])

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Rubi [A]  time = 0.709606, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{412810345784 \sqrt{1-2 x} \sqrt{3 x+2}}{738213861 \sqrt{5 x+3}}-\frac{6208896328 \sqrt{1-2 x} \sqrt{3 x+2}}{67110351 (5 x+3)^{3/2}}+\frac{140700876 \sqrt{1-2 x}}{10168235 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{649224 \sqrt{1-2 x}}{1452605 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{3606 \sqrt{1-2 x}}{207515 (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{632}{5929 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac{12417792656 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{111850585 \sqrt{33}}-\frac{412810345784 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{111850585 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

4/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 632/(5929*Sqrt[1 - 2*x
]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) - (3606*Sqrt[1 - 2*x])/(207515*(2 + 3*x)^(5/2
)*(3 + 5*x)^(3/2)) + (649224*Sqrt[1 - 2*x])/(1452605*(2 + 3*x)^(3/2)*(3 + 5*x)^(
3/2)) + (140700876*Sqrt[1 - 2*x])/(10168235*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (62
08896328*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(67110351*(3 + 5*x)^(3/2)) + (412810345784
*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(738213861*Sqrt[3 + 5*x]) - (412810345784*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(111850585*Sqrt[33]) - (12417792656*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(111850585*Sqrt[33])

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Rubi in Sympy [A]  time = 61.5598, size = 258, normalized size = 0.92 \[ - \frac{412810345784 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3691069305} - \frac{12417792656 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3691069305} - \frac{825620691568 \sqrt{3 x + 2} \sqrt{5 x + 3}}{3691069305 \sqrt{- 2 x + 1}} + \frac{12149375384 \sqrt{3 x + 2}}{9587193 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} - \frac{185437088 \sqrt{3 x + 2}}{871563 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{4224316}{132055 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{19892}{18865 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{178}{2695 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{4}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-412810345784*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/369106
9305 - 12417792656*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/3
691069305 - 825620691568*sqrt(3*x + 2)*sqrt(5*x + 3)/(3691069305*sqrt(-2*x + 1))
+ 12149375384*sqrt(3*x + 2)/(9587193*sqrt(-2*x + 1)*sqrt(5*x + 3)) - 185437088*
sqrt(3*x + 2)/(871563*sqrt(-2*x + 1)*(5*x + 3)**(3/2)) + 4224316/(132055*sqrt(-2
*x + 1)*sqrt(3*x + 2)*(5*x + 3)**(3/2)) + 19892/(18865*sqrt(-2*x + 1)*(3*x + 2)*
*(3/2)*(5*x + 3)**(3/2)) + 178/(2695*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*(5*x + 3)**
(3/2)) + 4/(231*(-2*x + 1)**(3/2)*(3*x + 2)**(5/2)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.482319, size = 119, normalized size = 0.42 \[ \frac{2 \left (\frac{557293966808400 x^6+873229924799280 x^5+84649478011164 x^4-430611138612568 x^3-149619576926754 x^2+52875828155808 x+23506658680609}{(1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}+4 \sqrt{2} \left (51601293223 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-25989595870 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{3691069305} \]

Antiderivative was successfully verified.

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

[Out]

(2*((23506658680609 + 52875828155808*x - 149619576926754*x^2 - 430611138612568*x
^3 + 84649478011164*x^4 + 873229924799280*x^5 + 557293966808400*x^6)/((1 - 2*x)^
(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + 4*Sqrt[2]*(51601293223*EllipticE[ArcSin
[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 25989595870*EllipticF[ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]], -33/2])))/3691069305

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Maple [C]  time = 0.04, size = 621, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3691069305*(1-2*x)^(1/2)*(18576465560280*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^4*(3+5*x)^(1/2)*(2+3*x)^(1/2
)*(1-2*x)^(1/2)-9356254513200*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^4*(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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)-13410631468
920*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)[phone]*2^(1/2)*El
lipticE(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)-2598959587000*2^(1/2)*EllipticF(1/11*1
1^(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)-6604965532544*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)*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)-5572
93966808400*x^6-2476862074704*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))+12
47500601760*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))-873229924799280*x^5-
84649478011164*x^4+430611138612568*x^3+149619576926754*x^2-52875828155808*x-2350
6658680609)/(2+3*x)^(5/2)/(3+5*x)^(3/2)/(-1+2*x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

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)^(5/2)),x, algorithm="fricas")

[Out]

integral(1/((2700*x^7 + 5940*x^6 + 3087*x^5 - 1828*x^4 - 2045*x^3 - 202*x^2 + 27
6*x + 72)*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)**(5/2)/(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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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