3.2970 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=187 \[ -\frac{2377960 \sqrt{1-2 x} \sqrt{3 x+2}}{1369599 \sqrt{5 x+3}}+\frac{5314 \sqrt{1-2 x}}{41503 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{1088}{17787 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4}{231 (1-2 x)^{3/2} \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{10628 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}}+\frac{475592 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}} \]

[Out]

4/(231*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + 1088/(17787*Sqrt[1 - 2*x]*
Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + (5314*Sqrt[1 - 2*x])/(41503*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x]) - (2377960*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1369599*Sqrt[3 + 5*x]) + (47559
2*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(41503*Sqrt[33]) + (10628*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(41503*Sqrt[33])

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Rubi [A]  time = 0.438862, 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 \[ -\frac{2377960 \sqrt{1-2 x} \sqrt{3 x+2}}{1369599 \sqrt{5 x+3}}+\frac{5314 \sqrt{1-2 x}}{41503 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{1088}{17787 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{4}{231 (1-2 x)^{3/2} \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{10628 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}}+\frac{475592 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{41503 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

4/(231*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + 1088/(17787*Sqrt[1 - 2*x]*
Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) + (5314*Sqrt[1 - 2*x])/(41503*Sqrt[2 + 3*x]*Sqrt[3
+ 5*x]) - (2377960*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1369599*Sqrt[3 + 5*x]) + (47559
2*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(41503*Sqrt[33]) + (10628*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(41503*Sqrt[33])

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Rubi in Sympy [A]  time = 37.3872, size = 172, normalized size = 0.92 \[ \frac{475592 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1369599} + \frac{10628 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1452605} + \frac{951184 \sqrt{3 x + 2} \sqrt{5 x + 3}}{1369599 \sqrt{- 2 x + 1}} - \frac{69460 \sqrt{3 x + 2}}{17787 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} + \frac{194}{539 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{4}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \sqrt{5 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

475592*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1369599 + 106
28*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/1452605 + 951184
*sqrt(3*x + 2)*sqrt(5*x + 3)/(1369599*sqrt(-2*x + 1)) - 69460*sqrt(3*x + 2)/(177
87*sqrt(-2*x + 1)*sqrt(5*x + 3)) + 194/(539*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*
x + 3)) + 4/(231*(-2*x + 1)**(3/2)*sqrt(3*x + 2)*sqrt(5*x + 3))

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Mathematica [A]  time = 0.322208, size = 103, normalized size = 0.55 \[ \frac{2 \left (\frac{-14267760 x^3+5106644 x^2+5510400 x-2236533}{(1-2 x)^{3/2} \sqrt{3 x+2} \sqrt{5 x+3}}+\sqrt{2} \left (150115 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-237796 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{1369599} \]

Antiderivative was successfully verified.

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

[Out]

(2*((-2236533 + 5510400*x + 5106644*x^2 - 14267760*x^3)/((1 - 2*x)^(3/2)*Sqrt[2
+ 3*x]*Sqrt[3 + 5*x]) + Sqrt[2]*(-237796*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]], -33/2] + 150115*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/13695
99

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Maple [C]  time = 0.039, size = 276, normalized size = 1.5 \[ -{\frac{2}{ \left ( 20543985\,{x}^{2}[phone]\,x+8217594 \right ) \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 300230\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-475592\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-150115\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +237796\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{3}-5106644\,{x}^{2}-5510400\,x+2236533 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1369599*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(300230*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)-475592*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)-150115*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))+237796*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^3-5106644*x^2-5510400*x+2236533
)/(15*x^2+19*x+6)/(-1+2*x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqr
t(-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)**(3/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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