3.2821 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{147 (3 x+2)^{7/2}}+\frac{816622 \sqrt{1-2 x} \sqrt{5 x+3}}{2268945 \sqrt{3 x+2}}-\frac{101902 \sqrt{1-2 x} \sqrt{5 x+3}}{324135 (3 x+2)^{3/2}}+\frac{676 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 (3 x+2)^{5/2}}-\frac{265648 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2268945}-\frac{816622 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2268945} \]

[Out]

(676*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15435*(2 + 3*x)^(5/2)) - (101902*Sqrt[1 - 2*x
]*Sqrt[3 + 5*x])/(324135*(2 + 3*x)^(3/2)) + (816622*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
/(2268945*Sqrt[2 + 3*x]) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(147*(2 + 3*x)^(7/2
)) - (816622*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/22689
45 - (265648*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/22689
45

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Rubi [A]  time = 0.408568, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{147 (3 x+2)^{7/2}}+\frac{816622 \sqrt{1-2 x} \sqrt{5 x+3}}{2268945 \sqrt{3 x+2}}-\frac{101902 \sqrt{1-2 x} \sqrt{5 x+3}}{324135 (3 x+2)^{3/2}}+\frac{676 \sqrt{1-2 x} \sqrt{5 x+3}}{15435 (3 x+2)^{5/2}}-\frac{265648 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2268945}-\frac{816622 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2268945} \]

Antiderivative was successfully verified.

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

[Out]

(676*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15435*(2 + 3*x)^(5/2)) - (101902*Sqrt[1 - 2*x
]*Sqrt[3 + 5*x])/(324135*(2 + 3*x)^(3/2)) + (816622*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])
/(2268945*Sqrt[2 + 3*x]) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(147*(2 + 3*x)^(7/2
)) - (816622*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/22689
45 - (265648*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/22689
45

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Rubi in Sympy [A]  time = 40.8204, size = 172, normalized size = 0.9 \[ \frac{816622 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2268945 \sqrt{3 x + 2}} - \frac{101902 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{324135 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{676 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{15435 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{147 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{816622 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{6806835} - \frac{2922128 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{79413075} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

816622*sqrt(-2*x + 1)*sqrt(5*x + 3)/(2268945*sqrt(3*x + 2)) - 101902*sqrt(-2*x +
1)*sqrt(5*x + 3)/(324135*(3*x + 2)**(3/2)) + 676*sqrt(-2*x + 1)*sqrt(5*x + 3)/(
15435*(3*x + 2)**(5/2)) + 2*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/(147*(3*x + 2)**(7/2
)) - 816622*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/6806835
- 2922128*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/79413075

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Mathematica [A]  time = 0.266369, size = 104, normalized size = 0.54 \[ \frac{2 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (11024397 x^3+18838881 x^2+10645545 x+1985537\right )}{(3 x+2)^{7/2}}+\sqrt{2} \left (1783285 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+408311 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{6806835} \]

Antiderivative was successfully verified.

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

[Out]

(2*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1985537 + 10645545*x + 18838881*x^2 + 110243
97*x^3))/(2 + 3*x)^(7/2) + Sqrt[2]*(408311*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2] + 1783285*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/68
06835

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Maple [C]  time = 0.032, size = 505, normalized size = 2.6 \[ -{\frac{2}{68068350\,{x}^{2}+6806835\,x-20420505} \left ( 48148695\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +3266488\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -330731910\,{x}^{5}-598239621\,{x}^{4}-276663420\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+17869833 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/6806835*(48148695*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)+1102439
7*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)[phone]*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)[phone]*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)[phone]*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)+14699
196*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)[phone]*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))+3266488*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))-330731910*x^5-598239621*x^4-276663420*x^3+78047184*x^2+89853294*x+1786983
3)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(7/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)/((81*x^4 + 216*x^3 + 216*x^2 + 96*x +
16)*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)/(2+3*x)**(9/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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