3.2675 \(\int \frac{\sqrt{1-2 x} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=156 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}-\frac{326 \sqrt{1-2 x} (3 x+2)^{3/2}}{825 \sqrt{5 x+3}}+\frac{458 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{1375}-\frac{496 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{625 \sqrt{33}}-\frac{169 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{625 \sqrt{33}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(15*(3 + 5*x)^(3/2)) - (326*Sqrt[1 - 2*x]*(2
+ 3*x)^(3/2))/(825*Sqrt[3 + 5*x]) + (458*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*
x])/1375 - (169*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(625*Sqrt[33]
) - (496*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(625*Sqrt[33])

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Rubi [A]  time = 0.334742, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}-\frac{326 \sqrt{1-2 x} (3 x+2)^{3/2}}{825 \sqrt{5 x+3}}+\frac{458 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{1375}-\frac{496 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{625 \sqrt{33}}-\frac{169 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{625 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(15*(3 + 5*x)^(3/2)) - (326*Sqrt[1 - 2*x]*(2
+ 3*x)^(3/2))/(825*Sqrt[3 + 5*x]) + (458*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*
x])/1375 - (169*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(625*Sqrt[33]
) - (496*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(625*Sqrt[33])

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Rubi in Sympy [A]  time = 31.8149, size = 143, normalized size = 0.92 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{326 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}}}{825 \sqrt{5 x + 3}} + \frac{458 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{1375} - \frac{169 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{20625} - \frac{496 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{21875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**(5/2)/(15*(5*x + 3)**(3/2)) - 326*sqrt(-2*x + 1)*(3
*x + 2)**(3/2)/(825*sqrt(5*x + 3)) + 458*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x +
3)/1375 - 169*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/20625
- 496*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/21875

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Mathematica [A]  time = 0.38069, size = 102, normalized size = 0.65 \[ \frac{\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (2475 x^2+1825 x+193\right )}{(5 x+3)^{3/2}}+8015 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+169 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{20625} \]

Antiderivative was successfully verified.

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

[Out]

((10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(193 + 1825*x + 2475*x^2))/(3 + 5*x)^(3/2) + 16
9*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 8015*Sqrt[2]*Elli
pticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/20625

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Maple [C]  time = 0.028, size = 272, normalized size = 1.7 \[ -{\frac{1}{123750\,{x}^{2}+20625\,x-41250} \left ( 40075\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+845\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+24045\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +507\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -148500\,{x}^{4}-134250\,{x}^{3}+19670\,{x}^{2}+34570\,x+3860 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20625*(40075*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)+845*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)+24045*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))+507*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))-148500*x^4-134
250*x^3+19670*x^2+34570*x+3860)*(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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((9*x^2 + 12*x + 4)*sqrt(3*x + 2)*sqrt(-2*x + 1)/((25*x^2 + 30*x + 9)*sq
rt(5*x + 3)), 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)*(1-2*x)**(1/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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