3.2819 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=129 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{3/2}}+\frac{412 \sqrt{1-2 x} \sqrt{5 x+3}}{1323 \sqrt{3 x+2}}+\frac{412 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323}-\frac{4157 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323} \]

[Out]

(412*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1323*Sqrt[2 + 3*x]) + (2*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2))/(63*(2 + 3*x)^(3/2)) - (4157*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sq
rt[1 - 2*x]], 35/33])/1323 + (412*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/1323

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Rubi [A]  time = 0.259003, antiderivative size = 129, 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 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{3/2}}+\frac{412 \sqrt{1-2 x} \sqrt{5 x+3}}{1323 \sqrt{3 x+2}}+\frac{412 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323}-\frac{4157 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1323} \]

Antiderivative was successfully verified.

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

[Out]

(412*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1323*Sqrt[2 + 3*x]) + (2*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2))/(63*(2 + 3*x)^(3/2)) - (4157*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sq
rt[1 - 2*x]], 35/33])/1323 + (412*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/1323

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Rubi in Sympy [A]  time = 25.4333, size = 114, normalized size = 0.88 \[ \frac{412 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1323 \sqrt{3 x + 2}} + \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{63 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{4157 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3969} + \frac{4532 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{46305} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

412*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1323*sqrt(3*x + 2)) + 2*sqrt(-2*x + 1)*(5*x +
3)**(3/2)/(63*(3*x + 2)**(3/2)) - 4157*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2
*x + 1)/7), 35/33)/3969 + 4532*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/
11), 33/35)/46305

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Mathematica [A]  time = 0.261942, size = 97, normalized size = 0.75 \[ \frac{\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} (723 x+475)}{(3 x+2)^{3/2}}-10955 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+4157 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{3969} \]

Antiderivative was successfully verified.

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

[Out]

((6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(475 + 723*x))/(2 + 3*x)^(3/2) + 4157*Sqrt[2]*El
lipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 10955*Sqrt[2]*EllipticF[ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/3969

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Maple [C]  time = 0.03, size = 267, normalized size = 2.1 \[{\frac{1}{39690\,{x}^{2}+3969\,x-11907} \left ( 32865\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-12471\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+21910\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -8314\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +43380\,{x}^{3}+32838\,{x}^{2}-10164\,x-8550 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3969*(32865*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)-12471*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)+21910*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))-8314*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))+43380*x^3+328
38*x^2-10164*x-8550)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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