∖int 1___________________ (1−2x)5∕2(2+3x)5∕2(3+5x)5∕2 ∖,dx

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3.2982 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=249 \[ \frac{7231789120 \sqrt{1-2 x} \sqrt{3 x+2}}{105459123 \sqrt{5 x+3}}-\frac{108842540 \sqrt{1-2 x} \sqrt{3 x+2}}{9587193 (5 x+3)^{3/2}}+\frac{488436 \sqrt{1-2 x}}{290521 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{414 \sqrt{1-2 x}}{41503 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{544}{5929 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{43537016 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3195731 \sqrt{33}}-\frac{1446357824 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3195731 \sqrt{33}} \]

[Out]

4/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + 544/(5929*Sqrt[1 - 2*x

]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (414*Sqrt[1 - 2*x])/(41503*(2 + 3*x)^(3/2)*

(3 + 5*x)^(3/2)) + (488436*Sqrt[1 - 2*x])/(290521*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))

- (108842540*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9587193*(3 + 5*x)^(3/2)) + (72317891

20*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(105459123*Sqrt[3 + 5*x]) - (1446357824*Elliptic

E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3195731*Sqrt[33]) - (43537016*Ellipt

icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3195731*Sqrt[33])

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Rubi [A] time = 0.615761, 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 \[ \frac{7231789120 \sqrt{1-2 x} \sqrt{3 x+2}}{105459123 \sqrt{5 x+3}}-\frac{108842540 \sqrt{1-2 x} \sqrt{3 x+2}}{9587193 (5 x+3)^{3/2}}+\frac{488436 \sqrt{1-2 x}}{290521 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{414 \sqrt{1-2 x}}{41503 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{544}{5929 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac{4}{231 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{43537016 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3195731 \sqrt{33}}-\frac{1446357824 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3195731 \sqrt{33}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

4/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + 544/(5929*Sqrt[1 - 2*x

]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (414*Sqrt[1 - 2*x])/(41503*(2 + 3*x)^(3/2)*

(3 + 5*x)^(3/2)) + (488436*Sqrt[1 - 2*x])/(290521*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))

- (108842540*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(9587193*(3 + 5*x)^(3/2)) + (72317891

20*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(105459123*Sqrt[3 + 5*x]) - (1446357824*Elliptic

E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3195731*Sqrt[33]) - (43537016*Ellipt

icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3195731*Sqrt[33])

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Rubi in Sympy [A] time = 53.7012, size = 230, normalized size = 0.92 \[ - \frac{1446357824 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{105459123} - \frac{43537016 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{105459123} - \frac{2892715648 \sqrt{3 x + 2} \sqrt{5 x + 3}}{105459123 \sqrt{- 2 x + 1}} + \frac{212842120 \sqrt{3 x + 2}}{1369599 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} - \frac{3249340 \sqrt{3 x + 2}}{124509 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{14776}{3773 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{62}{539 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{4}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-1446357824*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/10545912

3 - 43537016*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1054591

23 - 2892715648*sqrt(3*x + 2)*sqrt(5*x + 3)/(105459123*sqrt(-2*x + 1)) + 2128421

20*sqrt(3*x + 2)/(1369599*sqrt(-2*x + 1)*sqrt(5*x + 3)) - 3249340*sqrt(3*x + 2)/

(124509*sqrt(-2*x + 1)*(5*x + 3)**(3/2)) + 14776/(3773*sqrt(-2*x + 1)*sqrt(3*x +

2)*(5*x + 3)**(3/2)) + 62/(539*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)

) + 4/(231*(-2*x + 1)**(3/2)*(3*x + 2)**(3/2)*(5*x + 3)**(3/2))

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Mathematica [A] time = 0.365125, size = 114, normalized size = 0.46 \[ \frac{2 \left (\frac{650861020800 x^5+585919463160 x^4-291775464272 x^3-308398535118 x^2+30866656614 x+41179778225}{(1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2}}+2 \sqrt{2} \left (361589456 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-181999265 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{105459123} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(2*((41179778225 + 30866656614*x - 308398535118*x^2 - 291775464272*x^3 + 5859194

63160*x^4 + 650861020800*x^5)/((1 - 2*x)^(3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))

+ 2*Sqrt[2]*(361589456*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 1819

99265*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/105459123

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Maple [C] time = 0.038, size = 502, normalized size = 2. \[ -{\frac{2}{105459123\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 21695367360\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-10919955900\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+16633114976\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-8371966190\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-5062252384\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2547989710\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4339073472\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +2183991180\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -650861020800\,{x}^{5}-585919463160\,{x}^{4}+291775464272\,{x}^{3}+308398535118\,{x}^{2}-30866656614\,x-41179778225 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105459123*(1-2*x)^(1/2)*(21695367360*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)-10919955900*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)+1663

3114976*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)-8371966190*2^(1/2)*E

llipticF(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)-5062252384*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)+2547989710*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)-4339073472*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))+2183991180*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))-650861020800*x^5-585919463160*x^4+291

775464272*x^3+308398535118*x^2-30866656614*x-41179778225)/(2+3*x)^(3/2)/(3+5*x)^

(3/2)/(-1+2*x)^2

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

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)^(5/2)),x, algorithm="fricas")

[Out]

integral(1/((900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)*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)**(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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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