3.2924 \(\int \frac{1}{(1-2 x)^{3/2} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=156 \[ \frac{19480 \sqrt{1-2 x} \sqrt{3 x+2}}{27951 \sqrt{5 x+3}}-\frac{410 \sqrt{1-2 x} \sqrt{3 x+2}}{2541 (5 x+3)^{3/2}}+\frac{4 \sqrt{3 x+2}}{77 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{164 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{847 \sqrt{33}}-\frac{3896 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{847 \sqrt{33}} \]

[Out]

(4*Sqrt[2 + 3*x])/(77*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (410*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x])/(2541*(3 + 5*x)^(3/2)) + (19480*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(27951*Sqr
t[3 + 5*x]) - (3896*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(847*Sqrt
[33]) - (164*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(847*Sqrt[33])

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Rubi [A]  time = 0.344218, 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 \[ \frac{19480 \sqrt{1-2 x} \sqrt{3 x+2}}{27951 \sqrt{5 x+3}}-\frac{410 \sqrt{1-2 x} \sqrt{3 x+2}}{2541 (5 x+3)^{3/2}}+\frac{4 \sqrt{3 x+2}}{77 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{164 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{847 \sqrt{33}}-\frac{3896 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{847 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[2 + 3*x])/(77*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (410*Sqrt[1 - 2*x]*Sqrt[2
+ 3*x])/(2541*(3 + 5*x)^(3/2)) + (19480*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(27951*Sqr
t[3 + 5*x]) - (3896*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(847*Sqrt
[33]) - (164*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(847*Sqrt[33])

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Rubi in Sympy [A]  time = 31.7713, size = 143, normalized size = 0.92 \[ \frac{19480 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{27951 \sqrt{5 x + 3}} - \frac{410 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{2541 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{3896 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{27951} - \frac{164 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{29645} + \frac{4 \sqrt{3 x + 2}}{77 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

19480*sqrt(-2*x + 1)*sqrt(3*x + 2)/(27951*sqrt(5*x + 3)) - 410*sqrt(-2*x + 1)*sq
rt(3*x + 2)/(2541*(5*x + 3)**(3/2)) - 3896*sqrt(33)*elliptic_e(asin(sqrt(21)*sqr
t(-2*x + 1)/7), 35/33)/27951 - 164*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x +
1)/11), 33/35)/29645 + 4*sqrt(3*x + 2)/(77*sqrt(-2*x + 1)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.237936, size = 98, normalized size = 0.63 \[ \frac{2 \left (\frac{\sqrt{3 x+2} \left (-97400 x^2-5230 x+27691\right )}{\sqrt{1-2 x} (5 x+3)^{3/2}}+\sqrt{2} \left (1948 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-595 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{27951} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[2 + 3*x]*(27691 - 5230*x - 97400*x^2))/(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))
+ Sqrt[2]*(1948*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 595*Ellipt
icF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/27951

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Maple [C]  time = 0.034, size = 267, normalized size = 1.7 \[{\frac{2}{167706\,{x}^{2}+27951\,x-55902}\sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 2975\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-9740\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1785\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -5844\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +292200\,{x}^{3}+210490\,{x}^{2}-72613\,x-55382 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/27951*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(2975*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)-9740*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)+1785*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))-5844*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))+292200*x^3+210490*x^2-72613*x-55382)/(3+5*x)^(3/2)/(6*x^2+x-2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-1/((50*x^3 + 35*x^2 - 12*x - 9)*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)**(3/2)/(3+5*x)**(5/2)/(2+3*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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