3.2635 \(\int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^{9/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{220076 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 \sqrt{3 x+2}}+\frac{3184 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 (3 x+2)^{3/2}}+\frac{74 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^{5/2}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{21 (3 x+2)^{7/2}}-\frac{6584 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36015}-\frac{220076 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36015} \]

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(7/2)) + (74*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(735*(2 + 3*x)^(5/2)) + (3184*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5145*(2 + 3
*x)^(3/2)) + (220076*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36015*Sqrt[2 + 3*x]) - (22007
6*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/36015 - (6584*Sq
rt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/36015

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Rubi [A]  time = 0.426537, antiderivative size = 191, 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 \[ \frac{220076 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 \sqrt{3 x+2}}+\frac{3184 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 (3 x+2)^{3/2}}+\frac{74 \sqrt{1-2 x} \sqrt{5 x+3}}{735 (3 x+2)^{5/2}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{21 (3 x+2)^{7/2}}-\frac{6584 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36015}-\frac{220076 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{36015} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(7/2)) + (74*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(735*(2 + 3*x)^(5/2)) + (3184*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5145*(2 + 3
*x)^(3/2)) + (220076*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36015*Sqrt[2 + 3*x]) - (22007
6*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/36015 - (6584*Sq
rt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/36015

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Rubi in Sympy [A]  time = 37.497, size = 172, normalized size = 0.9 \[ \frac{220076 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{36015 \sqrt{3 x + 2}} + \frac{3184 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{5145 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{74 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{735 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{21 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{220076 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{108045} - \frac{72424 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1260525} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

220076*sqrt(-2*x + 1)*sqrt(5*x + 3)/(36015*sqrt(3*x + 2)) + 3184*sqrt(-2*x + 1)*
sqrt(5*x + 3)/(5145*(3*x + 2)**(3/2)) + 74*sqrt(-2*x + 1)*sqrt(5*x + 3)/(735*(3*
x + 2)**(5/2)) - 2*sqrt(-2*x + 1)*sqrt(5*x + 3)/(21*(3*x + 2)**(7/2)) - 220076*s
qrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/108045 - 72424*sqrt(3
5)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/1260525

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Mathematica [A]  time = 0.31759, size = 106, normalized size = 0.55 \[ \frac{4 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (2971026 x^3+6042348 x^2+4100535 x+926791\right )}{2 (3 x+2)^{7/2}}+\sqrt{2} \left (55019 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-27860 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{108045} \]

Antiderivative was successfully verified.

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

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(926791 + 4100535*x + 6042348*x^2 + 2971026*x
^3))/(2*(2 + 3*x)^(7/2)) + Sqrt[2]*(55019*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]], -33/2] - 27860*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/10804
5

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Maple [C]  time = 0.05, size = 505, normalized size = 2.6 \[{\frac{2}{1080450\,{x}^{2}+108045\,x-324135} \left ( 1504440\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-2971026\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+3008880\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-5942052\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2005920\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3961368\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+445760\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -880304\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-14275797\,{x}^{2}-34124442\,x-8341119 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/108045*(1504440*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)-2971026*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)+3008880*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^2*(3+5*x)^(1/2
)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-5942052*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)+2005920*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)-3961368*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)+445760*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))-880304*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))+8913
0780*x^5+190183518*x^4+114403860*x^3-14275797*x^2-34124442*x-8341119)*(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. \[ \int \frac{\sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(5*x + 3)*sqrt(-2*x + 1)/((81*x^4 + 216*x^3 + 216*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((1-2*x)**(1/2)*(3+5*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. \[ \int \frac{\sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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