3.2949 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{5/2} \sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=127 \[ \frac{11 \sqrt{3 x+2} (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{264 \sqrt{3 x+2} \sqrt{5 x+3}}{49 \sqrt{1-2 x}}-\frac{8}{49} \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{1597}{98} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-264*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]) + (11*Sqrt[2 + 3*x]*(3 + 5
*x)^(3/2))/(21*(1 - 2*x)^(3/2)) - (1597*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sq
rt[1 - 2*x]], 35/33])/98 - (8*Sqrt[33]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/49

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Rubi [A]  time = 0.261373, antiderivative size = 127, 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 \[ \frac{11 \sqrt{3 x+2} (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac{264 \sqrt{3 x+2} \sqrt{5 x+3}}{49 \sqrt{1-2 x}}-\frac{8}{49} \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{1597}{98} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-264*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]) + (11*Sqrt[2 + 3*x]*(3 + 5
*x)^(3/2))/(21*(1 - 2*x)^(3/2)) - (1597*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sq
rt[1 - 2*x]], 35/33])/98 - (8*Sqrt[33]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/49

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Rubi in Sympy [A]  time = 23.6192, size = 114, normalized size = 0.9 \[ - \frac{1597 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{294} - \frac{264 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1715} - \frac{264 \sqrt{3 x + 2} \sqrt{5 x + 3}}{49 \sqrt{- 2 x + 1}} + \frac{11 \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1597*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/294 - 264*sqrt
(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/1715 - 264*sqrt(3*x + 2
)*sqrt(5*x + 3)/(49*sqrt(-2*x + 1)) + 11*sqrt(3*x + 2)*(5*x + 3)**(3/2)/(21*(-2*
x + 1)**(3/2))

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Mathematica [A]  time = 0.200791, size = 115, normalized size = 0.91 \[ -\frac{22 \sqrt{3 x+2} \sqrt{5 x+3} (51-179 x)-805 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+1597 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{294 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(22*(51 - 179*x)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + 1597*Sqrt[2 - 4*x]*(-1 + 2*x)*El
lipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 805*Sqrt[2 - 4*x]*(-1 + 2*x)*
EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(294*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.03, size = 276, normalized size = 2.2 \[{\frac{1}{294\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 1610\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3194\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-805\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1597\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +59070\,{x}^{3}+57992\,{x}^{2}+2310\,x-6732 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/294*(1610*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)-3194*2^(1/2)*Ellip
ticE(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)-805*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))+1597*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/11*1
1^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))+59070*x^3+57992*x^
2+2310*x-6732)*(2+3*x)^(1/2)*(1-2*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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{\sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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