3.2861 \(\int \frac{(2+3 x)^{3/2}}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{272 \sqrt{1-2 x} \sqrt{3 x+2}}{1815 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x} \sqrt{3 x+2}}{165 (5 x+3)^{3/2}}-\frac{202 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{275 \sqrt{33}}+\frac{272 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{275 \sqrt{33}} \]

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(165*(3 + 5*x)^(3/2)) - (272*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x])/(1815*Sqrt[3 + 5*x]) + (272*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/(275*Sqrt[33]) - (202*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/(275*Sqrt[33])

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Rubi [A]  time = 0.267153, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{272 \sqrt{1-2 x} \sqrt{3 x+2}}{1815 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x} \sqrt{3 x+2}}{165 (5 x+3)^{3/2}}-\frac{202 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{275 \sqrt{33}}+\frac{272 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{275 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(165*(3 + 5*x)^(3/2)) - (272*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x])/(1815*Sqrt[3 + 5*x]) + (272*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/(275*Sqrt[33]) - (202*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/(275*Sqrt[33])

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Rubi in Sympy [A]  time = 25.8418, size = 114, normalized size = 0.91 \[ - \frac{272 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{1815 \sqrt{5 x + 3}} - \frac{2 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{165 \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{272 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9075} - \frac{202 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{9625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-272*sqrt(-2*x + 1)*sqrt(3*x + 2)/(1815*sqrt(5*x + 3)) - 2*sqrt(-2*x + 1)*sqrt(3
*x + 2)/(165*(5*x + 3)**(3/2)) + 272*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x
+ 1)/7), 35/33)/9075 - 202*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11)
, 33/35)/9625

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Mathematica [A]  time = 0.355548, size = 97, normalized size = 0.78 \[ \frac{-\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} (680 x+419)}{(5 x+3)^{3/2}}+3605 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-272 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{9075} \]

Antiderivative was successfully verified.

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

[Out]

((-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(419 + 680*x))/(3 + 5*x)^(3/2) - 272*Sqrt[2]*E
llipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 3605*Sqrt[2]*EllipticF[ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/9075

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Maple [C]  time = 0.03, size = 267, normalized size = 2.1 \[ -{\frac{1}{54450\,{x}^{2}+9075\,x-18150} \left ( 18025\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1360\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+10815\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -816\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +40800\,{x}^{3}+31940\,{x}^{2}-9410\,x-8380 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9075*(18025*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)-1360*2^(1/2)*El
lipticE(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)+10815*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))-816*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))+40800*x^3+3194
0*x^2-9410*x-8380)*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((3*x + 2)^(3/2)/((25*x^2 + 30*x + 9)*sqrt(5*x + 3)*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((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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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