3.2820 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx\)

Optimal. Leaf size=160 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^{5/2}}-\frac{53194 \sqrt{1-2 x} \sqrt{5 x+3}}{46305 \sqrt{3 x+2}}+\frac{544 \sqrt{1-2 x} \sqrt{5 x+3}}{6615 (3 x+2)^{3/2}}-\frac{34154 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305}+\frac{53194 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305} \]

[Out]

(544*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6615*(2 + 3*x)^(3/2)) - (53194*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/(46305*Sqrt[2 + 3*x]) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(105*(2
+ 3*x)^(5/2)) + (53194*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/46305 - (34154*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/46305

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Rubi [A]  time = 0.338527, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{105 (3 x+2)^{5/2}}-\frac{53194 \sqrt{1-2 x} \sqrt{5 x+3}}{46305 \sqrt{3 x+2}}+\frac{544 \sqrt{1-2 x} \sqrt{5 x+3}}{6615 (3 x+2)^{3/2}}-\frac{34154 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305}+\frac{53194 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)),x]

[Out]

(544*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6615*(2 + 3*x)^(3/2)) - (53194*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/(46305*Sqrt[2 + 3*x]) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(105*(2
+ 3*x)^(5/2)) + (53194*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/46305 - (34154*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33
])/46305

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Rubi in Sympy [A]  time = 33.1677, size = 143, normalized size = 0.89 \[ - \frac{53194 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{46305 \sqrt{3 x + 2}} + \frac{544 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6615 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{105 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{53194 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{138915} - \frac{375694 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1620675} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**(5/2)/(2+3*x)**(7/2)/(1-2*x)**(1/2),x)

[Out]

-53194*sqrt(-2*x + 1)*sqrt(5*x + 3)/(46305*sqrt(3*x + 2)) + 544*sqrt(-2*x + 1)*s
qrt(5*x + 3)/(6615*(3*x + 2)**(3/2)) + 2*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/(105*(3
*x + 2)**(5/2)) + 53194*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/
33)/138915 - 375694*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)
/1620675

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Mathematica [A]  time = 0.245362, size = 99, normalized size = 0.62 \[ \frac{\sqrt{2} \left (616735 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-53194 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} \left (239373 x^2+311247 x+101257\right )}{(3 x+2)^{5/2}}}{138915} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 5*x)^(5/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)),x]

[Out]

((-6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(101257 + 311247*x + 239373*x^2))/(2 + 3*x)^(5/
2) + Sqrt[2]*(-53194*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 616735
*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/138915

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Maple [C]  time = 0.03, size = 386, normalized size = 2.4 \[ -{\frac{1}{1389150\,{x}^{2}+138915\,x-416745} \left ( 5550615\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-478746\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+7400820\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-638328\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2466940\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -212776\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}+3634188\,{x}^{2}-4994904\,x-1822626 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^(5/2)/(2+3*x)^(7/2)/(1-2*x)^(1/2),x)

[Out]

-1/138915*(5550615*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*1
1^(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)-478746*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)+7400820*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)-638328*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)+2466940*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))-212776*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))[phone]*x^4+20111058*x^3+3634188*x^2-499490
4*x-1822626)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^(7/2)*sqrt(-2*x + 1)),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(7/2)*sqrt(-2*x + 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}{{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^(7/2)*sqrt(-2*x + 1)),x, algorithm="fricas")

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)/((27*x^3 + 54*x^2 + 36*x + 8)*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)**(7/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{7}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^(7/2)*sqrt(-2*x + 1)),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(7/2)*sqrt(-2*x + 1)), x)