3.2956 \(\int \frac{(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=125 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{3/2}}{33 (1-2 x)^{3/2}}-\frac{448 \sqrt{5 x+3} \sqrt{3 x+2}}{363 \sqrt{1-2 x}}-\frac{67 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55 \sqrt{33}}-\frac{4451 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{110 \sqrt{33}} \]

[Out]

(-448*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x]) + (7*(2 + 3*x)^(3/2)*Sqrt
[3 + 5*x])/(33*(1 - 2*x)^(3/2)) - (4451*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/(110*Sqrt[33]) - (67*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/(55*Sqrt[33])

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Rubi [A]  time = 0.260752, 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 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{3/2}}{33 (1-2 x)^{3/2}}-\frac{448 \sqrt{5 x+3} \sqrt{3 x+2}}{363 \sqrt{1-2 x}}-\frac{67 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55 \sqrt{33}}-\frac{4451 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{110 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-448*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x]) + (7*(2 + 3*x)^(3/2)*Sqrt
[3 + 5*x])/(33*(1 - 2*x)^(3/2)) - (4451*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/(110*Sqrt[33]) - (67*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/(55*Sqrt[33])

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Rubi in Sympy [A]  time = 23.8243, size = 114, normalized size = 0.91 \[ - \frac{4451 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3630} - \frac{67 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1925} - \frac{448 \sqrt{3 x + 2} \sqrt{5 x + 3}}{363 \sqrt{- 2 x + 1}} + \frac{7 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4451*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/3630 - 67*sqrt
(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/1925 - 448*sqrt(3*x + 2
)*sqrt(5*x + 3)/(363*sqrt(-2*x + 1)) + 7*(3*x + 2)**(3/2)*sqrt(5*x + 3)/(33*(-2*
x + 1)**(3/2))

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Mathematica [A]  time = 0.358694, size = 120, normalized size = 0.96 \[ -\frac{1}{2} \sqrt{11-5 (1-2 x)} \sqrt{7-3 (1-2 x)} \left (\frac{1127}{726 \sqrt{1-2 x}}-\frac{49}{66 (1-2 x)^{3/2}}\right )-\frac{2240 F\left (\sin ^{-1}\left (\frac{\sqrt{11-5 (1-2 x)}}{\sqrt{11}}\right )|-\frac{33}{2}\right )-4451 E\left (\sin ^{-1}\left (\frac{\sqrt{11-5 (1-2 x)}}{\sqrt{11}}\right )|-\frac{33}{2}\right )}{1815 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

-((-49/(66*(1 - 2*x)^(3/2)) + 1127/(726*Sqrt[1 - 2*x]))*Sqrt[11 - 5*(1 - 2*x)]*S
qrt[7 - 3*(1 - 2*x)])/2 - (-4451*EllipticE[ArcSin[Sqrt[11 - 5*(1 - 2*x)]/Sqrt[11
]], -33/2] + 2240*EllipticF[ArcSin[Sqrt[11 - 5*(1 - 2*x)]/Sqrt[11]], -33/2])/(18
15*Sqrt[2])

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Maple [C]  time = 0.03, size = 276, normalized size = 2.2 \[{\frac{1}{3630\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 4480\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-8902\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2240\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +4451\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +169050\,{x}^{3}+170030\,{x}^{2}+11760\,x-17640 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}\sqrt{2+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3630*(4480*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)-8902*2^(1/2)*Elli
pticE(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)-2240*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))+4451*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))+169050*x^3+17003
0*x^2+11760*x-17640)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*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. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((9*x^2 + 12*x + 4)*sqrt(3*x + 2)/((4*x^2 - 4*x + 1)*sqrt(5*x + 3)*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)**(5/2)/(1-2*x)**(5/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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