3.2860 \(\int \frac{(2+3 x)^{5/2}}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{3/2}}{165 (5 x+3)^{3/2}}-\frac{404 \sqrt{1-2 x} \sqrt{3 x+2}}{9075 \sqrt{5 x+3}}-\frac{598 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1375 \sqrt{33}}-\frac{2797 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1375 \sqrt{33}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(165*(3 + 5*x)^(3/2)) - (404*Sqrt[1 - 2*x]*Sq
rt[2 + 3*x])/(9075*Sqrt[3 + 5*x]) - (2797*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*
x]], 35/33])/(1375*Sqrt[33]) - (598*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3
5/33])/(1375*Sqrt[33])

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Rubi [A]  time = 0.261281, 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 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{3/2}}{165 (5 x+3)^{3/2}}-\frac{404 \sqrt{1-2 x} \sqrt{3 x+2}}{9075 \sqrt{5 x+3}}-\frac{598 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1375 \sqrt{33}}-\frac{2797 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1375 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(165*(3 + 5*x)^(3/2)) - (404*Sqrt[1 - 2*x]*Sq
rt[2 + 3*x])/(9075*Sqrt[3 + 5*x]) - (2797*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*
x]], 35/33])/(1375*Sqrt[33]) - (598*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3
5/33])/(1375*Sqrt[33])

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Rubi in Sympy [A]  time = 25.6683, size = 116, normalized size = 0.93 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}}{165 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{404 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{9075 \sqrt{5 x + 3}} - \frac{2797 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{45375} - \frac{598 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{48125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**(3/2)/(165*(5*x + 3)**(3/2)) - 404*sqrt(-2*x + 1)*s
qrt(3*x + 2)/(9075*sqrt(5*x + 3)) - 2797*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(
-2*x + 1)/7), 35/33)/45375 - 598*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1
)/11), 33/35)/48125

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Mathematica [A]  time = 0.384675, size = 97, normalized size = 0.78 \[ \frac{-\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} (1175 x+716)}{(5 x+3)^{3/2}}+7070 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+2797 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{45375} \]

Antiderivative was successfully verified.

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

[Out]

((-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(716 + 1175*x))/(3 + 5*x)^(3/2) + 2797*Sqrt[2]
*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 7070*Sqrt[2]*EllipticF[Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/45375

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Maple [C]  time = 0.03, size = 267, normalized size = 2.1 \[ -{\frac{1}{272250\,{x}^{2}+45375\,x-90750} \left ( 35350\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+13985\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+21210\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +8391\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +70500\,{x}^{3}+54710\,{x}^{2}-16340\,x-14320 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/45375*(35350*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)+13985*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)+21210*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))+8391*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))+70500*x^3+5
4710*x^2-16340*x-14320)*(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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((9*x^2 + 12*x + 4)*sqrt(3*x + 2)/((25*x^2 + 30*x + 9)*sqrt(5*x + 3)*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((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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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