3.2652 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{735 (3 x+2)^{5/2}}+\frac{173482 \sqrt{1-2 x} \sqrt{5 x+3}}{108045 \sqrt{3 x+2}}-\frac{4282 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 (3 x+2)^{3/2}}+\frac{23612 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045}-\frac{173482 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045} \]

[Out]

(-4282*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15435*(2 + 3*x)^(3/2)) + (173482*Sqrt[1 - 2
*x]*Sqrt[3 + 5*x])/(108045*Sqrt[2 + 3*x]) - (118*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/
(735*(2 + 3*x)^(5/2)) - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(21*(2 + 3*x)^(7/2)) -
(173482*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/108045 +
(23612*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/108045

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Rubi [A]  time = 0.421032, antiderivative size = 191, 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 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{735 (3 x+2)^{5/2}}+\frac{173482 \sqrt{1-2 x} \sqrt{5 x+3}}{108045 \sqrt{3 x+2}}-\frac{4282 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 (3 x+2)^{3/2}}+\frac{23612 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045}-\frac{173482 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{108045} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^(9/2),x]

[Out]

(-4282*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15435*(2 + 3*x)^(3/2)) + (173482*Sqrt[1 - 2
*x]*Sqrt[3 + 5*x])/(108045*Sqrt[2 + 3*x]) - (118*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/
(735*(2 + 3*x)^(5/2)) - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(21*(2 + 3*x)^(7/2)) -
(173482*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/108045 +
(23612*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/108045

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Rubi in Sympy [A]  time = 38.3525, size = 172, normalized size = 0.9 \[ \frac{173482 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{108045 \sqrt{3 x + 2}} - \frac{4282 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{15435 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{118 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{735 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{21 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{173482 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{324135} + \frac{23612 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{324135} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**(5/2)*(1-2*x)**(1/2)/(2+3*x)**(9/2),x)

[Out]

173482*sqrt(-2*x + 1)*sqrt(5*x + 3)/(108045*sqrt(3*x + 2)) - 4282*sqrt(-2*x + 1)
*sqrt(5*x + 3)/(15435*(3*x + 2)**(3/2)) - 118*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/(7
35*(3*x + 2)**(5/2)) - 2*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/(21*(3*x + 2)**(7/2)) -
173482*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/324135 + 236
12*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/324135

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Mathematica [A]  time = 0.348585, size = 104, normalized size = 0.54 \[ \frac{2 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (2342007 x^3+4290411 x^2+2623695 x+535637\right )}{(3 x+2)^{7/2}}+\sqrt{2} \left (86741 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-281540 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{324135} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^(9/2),x]

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(535637 + 2623695*x + 4290411*x^2 + 2342007*x
^3))/(2 + 3*x)^(7/2) + Sqrt[2]*(86741*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]
, -33/2] - 281540*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/324135

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Maple [C]  time = 0.03, size = 505, normalized size = 2.6 \[{\frac{2}{3241350\,{x}^{2}+324135\,x-972405} \left ( 7601580\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-2342007\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4684014\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3122676\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2252320\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -693928\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-14673504\,{x}^{2}-22006344\,x-4820733 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(9/2),x)

[Out]

2/324135*(7601580*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)-2342007*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/1
1*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^2*(3+5*x)^(1/
2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-4684014*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^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-
2*x)^(1/2)[phone]*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)-3122676*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)+2252320*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))-693928*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))+7
0260210*x^5+135738351*x^4+70504020*x^3-14673504*x^2-22006344*x-4820733)*(1-2*x)^
(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^(9/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^(9/2),x, algorithm="fricas")

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((81*x^4 + 216*x^3 + 2
16*x^2 + 96*x + 16)*sqrt(3*x + 2)), 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)*(1-2*x)**(1/2)/(2+3*x)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^(9/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(3*x + 2)^(9/2), x)