3.2674 \(\int \frac{\sqrt{1-2 x} (2+3 x)^{7/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=189 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}-\frac{458 \sqrt{1-2 x} (3 x+2)^{5/2}}{825 \sqrt{5 x+3}}+\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6875}+\frac{2719 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{34375}-\frac{523 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}-\frac{47342 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625 \sqrt{33}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(15*(3 + 5*x)^(3/2)) - (458*Sqrt[1 - 2*x]*(2
+ 3*x)^(5/2))/(825*Sqrt[3 + 5*x]) + (2719*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5
*x])/34375 + (2818*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/6875 - (47342*El
lipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(15625*Sqrt[33]) - (523*Sqrt[11
/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625

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Rubi [A]  time = 0.41411, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}-\frac{458 \sqrt{1-2 x} (3 x+2)^{5/2}}{825 \sqrt{5 x+3}}+\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{6875}+\frac{2719 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{34375}-\frac{523 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}-\frac{47342 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(3 + 5*x)^(5/2),x]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(15*(3 + 5*x)^(3/2)) - (458*Sqrt[1 - 2*x]*(2
+ 3*x)^(5/2))/(825*Sqrt[3 + 5*x]) + (2719*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5
*x])/34375 + (2818*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/6875 - (47342*El
lipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(15625*Sqrt[33]) - (523*Sqrt[11
/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625

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Rubi in Sympy [A]  time = 39.2212, size = 172, normalized size = 0.91 \[ - \frac{2 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{458 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{825 \sqrt{5 x + 3}} + \frac{2818 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{6875} + \frac{2719 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{34375} - \frac{47342 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{515625} - \frac{5753 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{546875} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(7/2)*(1-2*x)**(1/2)/(3+5*x)**(5/2),x)

[Out]

-2*sqrt(-2*x + 1)*(3*x + 2)**(7/2)/(15*(5*x + 3)**(3/2)) - 458*sqrt(-2*x + 1)*(3
*x + 2)**(5/2)/(825*sqrt(5*x + 3)) + 2818*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5
*x + 3)/6875 + 2719*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/34375 - 47342*sqr
t(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/515625 - 5753*sqrt(35)*
elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/546875

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Mathematica [A]  time = 0.411594, size = 107, normalized size = 0.57 \[ \frac{\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (222750 x^3+398475 x^2+221200 x+37273\right )}{(5 x+3)^{3/2}}+95165 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+94684 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1031250} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(3 + 5*x)^(5/2),x]

[Out]

((10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(37273 + 221200*x + 398475*x^2 + 222750*x^3))/(
3 + 5*x)^(3/2) + 94684*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2
] + 95165*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/1031250

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Maple [C]  time = 0.027, size = 277, normalized size = 1.5 \[ -{\frac{1}{6187500\,{x}^{2}+1031250\,x-2062500} \left ( 475825\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+473420\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+285495\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +284052\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -13365000\,{x}^{5}-26136000\,{x}^{4}-12801750\,{x}^{3}+3521120\,{x}^{2}+4051270\,x+745460 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(7/2)*(1-2*x)^(1/2)/(3+5*x)^(5/2),x)

[Out]

-1/1031250*(475825*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*1
1^(1/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+473420*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)+285495*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))+284052*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(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))-1336
5000*x^5-26136000*x^4-12801750*x^3+3521120*x^2+4051270*x+745460)*(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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(7/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(7/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(7/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="fricas")

[Out]

integral((27*x^3 + 54*x^2 + 36*x + 8)*sqrt(3*x + 2)*sqrt(-2*x + 1)/((25*x^2 + 30
*x + 9)*sqrt(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)**(7/2)*(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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^(7/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(7/2)*sqrt(-2*x + 1)/(5*x + 3)^(5/2), x)