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3.2890 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx\)

Optimal. Leaf size=222 \[ \frac{189368 \sqrt{1-2 x} \sqrt{5 x+3}}{588245 \sqrt{3 x+2}}-\frac{5438 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 (3 x+2)^{3/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{5/2}}-\frac{229 \sqrt{1-2 x} \sqrt{5 x+3}}{343 (3 x+2)^{7/2}}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{7/2}}-\frac{23012 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{588245}-\frac{189368 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{588245} \]

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) - (229*Sqrt[1 - 2*x]*Sqrt[3
 + 5*x])/(343*(2 + 3*x)^(7/2)) - (2818*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12005*(2 +
3*x)^(5/2)) - (5438*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84035*(2 + 3*x)^(3/2)) + (1893
68*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(588245*Sqrt[2 + 3*x]) - (189368*Sqrt[11/3]*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/588245 - (23012*Sqrt[11/3]*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/588245

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Rubi [A]  time = 0.519858, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{189368 \sqrt{1-2 x} \sqrt{5 x+3}}{588245 \sqrt{3 x+2}}-\frac{5438 \sqrt{1-2 x} \sqrt{5 x+3}}{84035 (3 x+2)^{3/2}}-\frac{2818 \sqrt{1-2 x} \sqrt{5 x+3}}{12005 (3 x+2)^{5/2}}-\frac{229 \sqrt{1-2 x} \sqrt{5 x+3}}{343 (3 x+2)^{7/2}}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{7/2}}-\frac{23012 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{588245}-\frac{189368 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{588245} \]

Antiderivative was successfully verified.

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

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) - (229*Sqrt[1 - 2*x]*Sqrt[3
 + 5*x])/(343*(2 + 3*x)^(7/2)) - (2818*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12005*(2 +
3*x)^(5/2)) - (5438*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84035*(2 + 3*x)^(3/2)) + (1893
68*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(588245*Sqrt[2 + 3*x]) - (189368*Sqrt[11/3]*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/588245 - (23012*Sqrt[11/3]*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/588245

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Rubi in Sympy [A]  time = 46.2887, size = 201, normalized size = 0.91 \[ \frac{189368 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{588245 \sqrt{3 x + 2}} - \frac{5438 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{84035 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{2818 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{12005 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{229 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{343 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{189368 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1764735} - \frac{23012 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1764735} + \frac{11 \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

189368*sqrt(-2*x + 1)*sqrt(5*x + 3)/(588245*sqrt(3*x + 2)) - 5438*sqrt(-2*x + 1)
*sqrt(5*x + 3)/(84035*(3*x + 2)**(3/2)) - 2818*sqrt(-2*x + 1)*sqrt(5*x + 3)/(120
05*(3*x + 2)**(5/2)) - 229*sqrt(-2*x + 1)*sqrt(5*x + 3)/(343*(3*x + 2)**(7/2)) -
 189368*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1764735 - 23
012*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1764735 + 11*sqr
t(5*x + 3)/(7*sqrt(-2*x + 1)*(3*x + 2)**(7/2))

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Mathematica [A]  time = 0.31307, size = 109, normalized size = 0.49 \[ \frac{2 \left (\sqrt{2} \left (95165 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+94684 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{3 \sqrt{5 x+3} \left (5112936 x^4+7326810 x^3+1004571 x^2-2279324 x-809083\right )}{\sqrt{1-2 x} (3 x+2)^{7/2}}\right )}{1764735} \]

Antiderivative was successfully verified.

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

[Out]

(2*((-3*Sqrt[3 + 5*x]*(-809083 - 2279324*x + 1004571*x^2 + 7326810*x^3 + 5112936
*x^4))/(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) + Sqrt[2]*(94684*EllipticE[ArcSin[Sqrt[2/
11]*Sqrt[3 + 5*x]], -33/2] + 95165*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -
33/2])))/1764735

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Maple [C]  time = 0.036, size = 505, normalized size = 2.3 \[ -{\frac{2}{17647350\,{x}^{2}+1764735\,x-5294205}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2556468\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+2569455\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+5112936\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+5138910\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+3408624\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+3425940\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+757472\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +761320\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -76694040\,{x}^{5}-155918574\,{x}^{4}-81009855\,{x}^{3}+25148721\,{x}^{2}+32650161\,x+7281747 \right ) \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1764735*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2556468*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)+2569455*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)+
5112936*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)+5138910*2^(1/2)*Elli
pticF(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)+3408624*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)+3425940*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)+7574
72*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))+761320*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))-76694040*x^5-155918574*x^4-81009855*x^3+25148721*x^2+3
2650161*x+7281747)/(2+3*x)^(7/2)/(10*x^2+x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(5*x + 3)^(3/2)/((162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*sq
rt(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)**(3/2)/(1-2*x)**(3/2)/(2+3*x)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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