3.2634 \(\int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=160 \[ \frac{4636 \sqrt{1-2 x} \sqrt{5 x+3}}{2205 \sqrt{3 x+2}}+\frac{74 \sqrt{1-2 x} \sqrt{5 x+3}}{315 (3 x+2)^{3/2}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{15 (3 x+2)^{5/2}}-\frac{124 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2205}-\frac{4636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2205} \]

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^(5/2)) + (74*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(315*(2 + 3*x)^(3/2)) + (4636*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2205*Sqrt[2
+ 3*x]) - (4636*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2
205 - (124*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2205

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Rubi [A]  time = 0.342271, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{4636 \sqrt{1-2 x} \sqrt{5 x+3}}{2205 \sqrt{3 x+2}}+\frac{74 \sqrt{1-2 x} \sqrt{5 x+3}}{315 (3 x+2)^{3/2}}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{15 (3 x+2)^{5/2}}-\frac{124 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2205}-\frac{4636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2205} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^(5/2)) + (74*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(315*(2 + 3*x)^(3/2)) + (4636*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2205*Sqrt[2
+ 3*x]) - (4636*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2
205 - (124*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2205

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Rubi in Sympy [A]  time = 30.8231, size = 143, normalized size = 0.89 \[ \frac{4636 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2205 \sqrt{3 x + 2}} + \frac{74 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{315 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{15 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{4636 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{6615} - \frac{1364 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{77175} \]

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)**(7/2),x)

[Out]

4636*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2205*sqrt(3*x + 2)) + 74*sqrt(-2*x + 1)*sqrt(
5*x + 3)/(315*(3*x + 2)**(3/2)) - 2*sqrt(-2*x + 1)*sqrt(5*x + 3)/(15*(3*x + 2)**
(5/2)) - 4636*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/6615 -
1364*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/77175

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Mathematica [A]  time = 0.318054, size = 99, normalized size = 0.62 \[ \frac{2 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (20862 x^2+28593 x+9643\right )}{(3 x+2)^{5/2}}+\sqrt{2} \left (2318 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-1295 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{6615} \]

Antiderivative was successfully verified.

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

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(9643 + 28593*x + 20862*x^2))/(2 + 3*x)^(5/2)
+ Sqrt[2]*(2318*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 1295*Ellip
ticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/6615

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Maple [C]  time = 0.048, size = 386, normalized size = 2.4 \[{\frac{2}{66150\,{x}^{2}+6615\,x-19845} \left ( 11655\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-20862\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+15540\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-27816\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+5180\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -9272\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +625860\,{x}^{4}+920376\,{x}^{3}+187311\,{x}^{2}-228408\,x-86787 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

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)^(7/2),x)

[Out]

2/6615*(11655*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)-20862*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)+15540*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)-27816*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)+5180
*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))-9272*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))+625860*x^4+920376*x^3+187311*x^2-228408*x-86787)*(1-2*x)^(
1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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