3.2874 \(\int \frac{(2+3 x)^{7/2} \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=186 \[ \frac{\sqrt{5 x+3} (3 x+2)^{7/2}}{\sqrt{1-2 x}}+\frac{12}{7} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{2517}{350} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{29293}{875} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{673523 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750 \sqrt{33}}+\frac{4071079 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17500} \]

[Out]

(29293*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/875 + (2517*Sqrt[1 - 2*x]*(2 +
3*x)^(3/2)*Sqrt[3 + 5*x])/350 + (12*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]
)/7 + ((2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (4071079*Sqrt[11/3]*Ellipt
icE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/17500 + (673523*EllipticF[ArcSin[Sq
rt[3/7]*Sqrt[1 - 2*x]], 35/33])/(8750*Sqrt[33])

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Rubi [A]  time = 0.399235, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\sqrt{5 x+3} (3 x+2)^{7/2}}{\sqrt{1-2 x}}+\frac{12}{7} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}+\frac{2517}{350} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{29293}{875} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{673523 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8750 \sqrt{33}}+\frac{4071079 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17500} \]

Antiderivative was successfully verified.

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

[Out]

(29293*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/875 + (2517*Sqrt[1 - 2*x]*(2 +
3*x)^(3/2)*Sqrt[3 + 5*x])/350 + (12*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]
)/7 + ((2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (4071079*Sqrt[11/3]*Ellipt
icE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/17500 + (673523*EllipticF[ArcSin[Sq
rt[3/7]*Sqrt[1 - 2*x]], 35/33])/(8750*Sqrt[33])

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Rubi in Sympy [A]  time = 39.1486, size = 168, normalized size = 0.9 \[ \frac{12 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{7} + \frac{2517 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{350} + \frac{29293 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{875} + \frac{4071079 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{52500} + \frac{673523 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{306250} + \frac{\left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/7 + 2517*sqrt(-2*x + 1)*(3*x +
2)**(3/2)*sqrt(5*x + 3)/350 + 29293*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/8
75 + 4071079*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/52500 +
673523*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/306250 + (3
*x + 2)**(7/2)*sqrt(5*x + 3)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.296917, size = 115, normalized size = 0.62 \[ \frac{-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (6750 x^3+26010 x^2+54757 x-109756\right )+2050510 \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4071079 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{52500 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-109756 + 54757*x + 26010*x^2 + 6750*x^3) - 40
71079*Sqrt[2 - 4*x]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 2050510
*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(52500*Sqrt[1
- 2*x])

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Maple [C]  time = 0.049, size = 174, normalized size = 0.9 \[ -{\frac{1}{1575000\,{x}^{3}+1207500\,{x}^{2}-367500\,x-315000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 2050510\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -4071079\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -3037500\,{x}^{5}-15552000\,{x}^{4}-40681350\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+19756080 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/52500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2050510*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))-4071079*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))-3037500*x^5-15552000*x^4-40681350*x^3+13496910*x^2+52704660*x+19756080)/(30*
x^3+23*x^2-7*x-6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(27*x^3 + 54*x^2 + 36*x + 8)*sqrt(5*x + 3)*sqrt(3*x + 2)/((2*x - 1)*sq
rt(-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)**(7/2)*(3+5*x)**(1/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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