3.2864 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=156 \[ \frac{89020 \sqrt{1-2 x} \sqrt{3 x+2}}{2541 \sqrt{5 x+3}}-\frac{1340 \sqrt{1-2 x} \sqrt{3 x+2}}{231 (5 x+3)^{3/2}}+\frac{6 \sqrt{1-2 x}}{7 \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{536 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}}-\frac{17804 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}} \]

[Out]

(6*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (1340*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x])/(231*(3 + 5*x)^(3/2)) + (89020*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(2541*Sqrt[
3 + 5*x]) - (17804*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(77*Sqrt[3
3]) - (536*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(77*Sqrt[33])

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Rubi [A]  time = 0.349413, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{89020 \sqrt{1-2 x} \sqrt{3 x+2}}{2541 \sqrt{5 x+3}}-\frac{1340 \sqrt{1-2 x} \sqrt{3 x+2}}{231 (5 x+3)^{3/2}}+\frac{6 \sqrt{1-2 x}}{7 \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{536 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}}-\frac{17804 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(6*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (1340*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x])/(231*(3 + 5*x)^(3/2)) + (89020*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(2541*Sqrt[
3 + 5*x]) - (17804*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(77*Sqrt[3
3]) - (536*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(77*Sqrt[33])

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Rubi in Sympy [A]  time = 31.771, size = 143, normalized size = 0.92 \[ \frac{89020 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{2541 \sqrt{5 x + 3}} - \frac{1340 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{231 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{6 \sqrt{- 2 x + 1}}{7 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{17804 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2541} - \frac{536 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{2695} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

89020*sqrt(-2*x + 1)*sqrt(3*x + 2)/(2541*sqrt(5*x + 3)) - 1340*sqrt(-2*x + 1)*sq
rt(3*x + 2)/(231*(5*x + 3)**(3/2)) + 6*sqrt(-2*x + 1)/(7*sqrt(3*x + 2)*(5*x + 3)
**(3/2)) - 17804*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/254
1 - 536*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/2695

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Mathematica [A]  time = 0.194814, size = 99, normalized size = 0.63 \[ \frac{2 \left (\frac{\sqrt{1-2 x} \left (667650 x^2+823580 x+253409\right )}{\sqrt{3 x+2} (5 x+3)^{3/2}}+2 \sqrt{2} \left (4451 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2240 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{2541} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[1 - 2*x]*(253409 + 823580*x + 667650*x^2))/(Sqrt[2 + 3*x]*(3 + 5*x)^(3
/2)) + 2*Sqrt[2]*(4451*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2240
*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/2541

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Maple [C]  time = 0.034, size = 267, normalized size = 1.7 \[{\frac{2}{15246\,{x}^{2}+2541\,x-5082}\sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 22400\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-44510\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+13440\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -26706\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1335300\,{x}^{3}+979510\,{x}^{2}-316762\,x-253409 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/2541*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(22400*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)-44510*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)+13440*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))-26706*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))+1335300*x^3+979510*x^2-316762*x-253409)/(3+5*x)^(3/2)/(6*x^2+x-2
)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((75*x^3 + 140*x^2 + 87*x + 18)*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)**(3/2)/(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. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{3}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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