3.2932 \(\int \frac{\sqrt{2+3 x} \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=125 \[ -\frac{68 \sqrt{3 x+2} \sqrt{5 x+3}}{231 \sqrt{1-2 x}}+\frac{\sqrt{3 x+2} \sqrt{5 x+3}}{3 (1-2 x)^{3/2}}-\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}}-\frac{34 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}} \]

[Out]

(Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(3*(1 - 2*x)^(3/2)) - (68*Sqrt[2 + 3*x]*Sqrt[3 + 5
*x])/(231*Sqrt[1 - 2*x]) - (34*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]
)/(7*Sqrt[33]) - EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]/(7*Sqrt[33])

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Rubi [A]  time = 0.258712, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{68 \sqrt{3 x+2} \sqrt{5 x+3}}{231 \sqrt{1-2 x}}+\frac{\sqrt{3 x+2} \sqrt{5 x+3}}{3 (1-2 x)^{3/2}}-\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}}-\frac{34 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(3*(1 - 2*x)^(3/2)) - (68*Sqrt[2 + 3*x]*Sqrt[3 + 5
*x])/(231*Sqrt[1 - 2*x]) - (34*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]
)/(7*Sqrt[33]) - EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]/(7*Sqrt[33])

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Rubi in Sympy [A]  time = 23.841, size = 110, normalized size = 0.88 \[ - \frac{34 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{231} - \frac{\sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{231} - \frac{68 \sqrt{3 x + 2} \sqrt{5 x + 3}}{231 \sqrt{- 2 x + 1}} + \frac{\sqrt{3 x + 2} \sqrt{5 x + 3}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-34*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/231 - sqrt(33)*e
lliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/231 - 68*sqrt(3*x + 2)*sqrt(5*
x + 3)/(231*sqrt(-2*x + 1)) + sqrt(3*x + 2)*sqrt(5*x + 3)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.241114, size = 115, normalized size = 0.92 \[ \frac{2 \sqrt{3 x+2} \sqrt{5 x+3} (136 x+9)+35 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-68 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{462 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(9 + 136*x) - 68*Sqrt[2 - 4*x]*(-1 + 2*x)*Ellipti
cE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 35*Sqrt[2 - 4*x]*(-1 + 2*x)*Ellipt
icF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(462*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.024, size = 276, normalized size = 2.2 \[{\frac{1}{462\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 70\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-136\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +68\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +4080\,{x}^{3}+5438\,{x}^{2}+1974\,x+108 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/462*(70*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)-136*2^(1/2)*Elliptic
E(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)-35*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))+68*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))+4080*x^3+5438*x^2+1974*x
+108)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)/(-1+2*x)^2/(15*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(5*x + 3)*sqrt(3*x + 2)/((4*x^2 - 4*x + 1)*sqrt(-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)**(1/2)*(3+5*x)**(1/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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