3.2865 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ \frac{6277760 \sqrt{1-2 x} \sqrt{3 x+2}}{17787 \sqrt{5 x+3}}-\frac{94420 \sqrt{1-2 x} \sqrt{3 x+2}}{1617 (5 x+3)^{3/2}}+\frac{428 \sqrt{1-2 x}}{49 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{37768 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}}-\frac{1255552 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}} \]

[Out]

(2*Sqrt[1 - 2*x])/(7*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (428*Sqrt[1 - 2*x])/(49*
Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (94420*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1617*(3 +
5*x)^(3/2)) + (6277760*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(17787*Sqrt[3 + 5*x]) - (125
5552*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(539*Sqrt[33]) - (37768*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(539*Sqrt[33])

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Rubi [A]  time = 0.436999, 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 \[ \frac{6277760 \sqrt{1-2 x} \sqrt{3 x+2}}{17787 \sqrt{5 x+3}}-\frac{94420 \sqrt{1-2 x} \sqrt{3 x+2}}{1617 (5 x+3)^{3/2}}+\frac{428 \sqrt{1-2 x}}{49 \sqrt{3 x+2} (5 x+3)^{3/2}}+\frac{2 \sqrt{1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}-\frac{37768 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}}-\frac{1255552 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{539 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x])/(7*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (428*Sqrt[1 - 2*x])/(49*
Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (94420*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1617*(3 +
5*x)^(3/2)) + (6277760*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(17787*Sqrt[3 + 5*x]) - (125
5552*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(539*Sqrt[33]) - (37768*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(539*Sqrt[33])

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Rubi in Sympy [A]  time = 39.2292, size = 172, normalized size = 0.92 \[ \frac{6277760 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{17787 \sqrt{5 x + 3}} - \frac{94420 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{1617 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{428 \sqrt{- 2 x + 1}}{49 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{2 \sqrt{- 2 x + 1}}{7 \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{1255552 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{17787} - \frac{37768 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{18865} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6277760*sqrt(-2*x + 1)*sqrt(3*x + 2)/(17787*sqrt(5*x + 3)) - 94420*sqrt(-2*x + 1
)*sqrt(3*x + 2)/(1617*(5*x + 3)**(3/2)) + 428*sqrt(-2*x + 1)/(49*sqrt(3*x + 2)*(
5*x + 3)**(3/2)) + 2*sqrt(-2*x + 1)/(7*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)) - 1255
552*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/17787 - 37768*sq
rt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/18865

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Mathematica [A]  time = 0.232321, size = 104, normalized size = 0.56 \[ \frac{2 \left (\frac{\sqrt{1-2 x} \left (141249600 x^3+268408770 x^2+169778606 x+35747225\right )}{(3 x+2)^{3/2} (5 x+3)^{3/2}}+2 \sqrt{2} \left (313888 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-158095 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{17787} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[1 - 2*x]*(35747225 + 169778606*x + 268408770*x^2 + 141249600*x^3))/((2
+ 3*x)^(3/2)*(3 + 5*x)^(3/2)) + 2*Sqrt[2]*(313888*EllipticE[ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]], -33/2] - 158095*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2
])))/17787

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Maple [C]  time = 0.036, size = 383, normalized size = 2.1 \[{\frac{2}{-17787+35574\,x}\sqrt{1-2\,x} \left ( 4742850\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-9416640\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+6007610\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-11927744\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1897140\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -3766656\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-98284156\,x-35747225 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/17787*(1-2*x)^(1/2)*(4742850*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)-9416640*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)+6007610*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*(
3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-11927744*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)+1897140*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*El
lipticF(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))-3766
656*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^4+395567940*x^3+
71148442*x^2-98284156*x-35747225)/(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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((225*x^4 + 570*x^3 + 541*x^2 + 228*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/(2+3*x)**(5/2)/(3+5*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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