3.2900 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx\)

Optimal. Leaf size=253 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{9/2}}+\frac{6036028 \sqrt{1-2 x} \sqrt{5 x+3}}{22235661 \sqrt{3 x+2}}-\frac{392998 \sqrt{1-2 x} \sqrt{5 x+3}}{3176523 (3 x+2)^{3/2}}-\frac{167228 \sqrt{1-2 x} \sqrt{5 x+3}}{453789 (3 x+2)^{5/2}}-\frac{67345 \sqrt{1-2 x} \sqrt{5 x+3}}{64827 (3 x+2)^{7/2}}+\frac{295 \sqrt{1-2 x} \sqrt{5 x+3}}{1323 (3 x+2)^{9/2}}-\frac{1199452 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{22235661}-\frac{6036028 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{22235661} \]

[Out]

(295*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1323*(2 + 3*x)^(9/2)) - (67345*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/(64827*(2 + 3*x)^(7/2)) - (167228*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4
53789*(2 + 3*x)^(5/2)) - (392998*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3176523*(2 + 3*x)
^(3/2)) + (6036028*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(22235661*Sqrt[2 + 3*x]) + (11*(
3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(9/2)) - (6036028*Sqrt[11/3]*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/22235661 - (1199452*Sqrt[11/3]*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/22235661

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Rubi [A]  time = 0.603719, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{9/2}}+\frac{6036028 \sqrt{1-2 x} \sqrt{5 x+3}}{22235661 \sqrt{3 x+2}}-\frac{392998 \sqrt{1-2 x} \sqrt{5 x+3}}{3176523 (3 x+2)^{3/2}}-\frac{167228 \sqrt{1-2 x} \sqrt{5 x+3}}{453789 (3 x+2)^{5/2}}-\frac{67345 \sqrt{1-2 x} \sqrt{5 x+3}}{64827 (3 x+2)^{7/2}}+\frac{295 \sqrt{1-2 x} \sqrt{5 x+3}}{1323 (3 x+2)^{9/2}}-\frac{1199452 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{22235661}-\frac{6036028 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{22235661} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^(11/2)),x]

[Out]

(295*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1323*(2 + 3*x)^(9/2)) - (67345*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/(64827*(2 + 3*x)^(7/2)) - (167228*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4
53789*(2 + 3*x)^(5/2)) - (392998*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3176523*(2 + 3*x)
^(3/2)) + (6036028*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(22235661*Sqrt[2 + 3*x]) + (11*(
3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(9/2)) - (6036028*Sqrt[11/3]*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/22235661 - (1199452*Sqrt[11/3]*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/22235661

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Rubi in Sympy [A]  time = 53.7903, size = 230, normalized size = 0.91 \[ \frac{6036028 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{22235661 \sqrt{3 x + 2}} - \frac{392998 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3176523 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{167228 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{453789 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{67345 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{64827 \left (3 x + 2\right )^{\frac{7}{2}}} + \frac{295 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1323 \left (3 x + 2\right )^{\frac{9}{2}}} - \frac{6036028 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{66706983} - \frac{13193972 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{778248135} + \frac{11 \left (5 x + 3\right )^{\frac{3}{2}}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**(11/2),x)

[Out]

6036028*sqrt(-2*x + 1)*sqrt(5*x + 3)/(22235661*sqrt(3*x + 2)) - 392998*sqrt(-2*x
+ 1)*sqrt(5*x + 3)/(3176523*(3*x + 2)**(3/2)) - 167228*sqrt(-2*x + 1)*sqrt(5*x
+ 3)/(453789*(3*x + 2)**(5/2)) - 67345*sqrt(-2*x + 1)*sqrt(5*x + 3)/(64827*(3*x
+ 2)**(7/2)) + 295*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1323*(3*x + 2)**(9/2)) - 603602
8*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/66706983 - 1319397
2*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/778248135 + 11*(5
*x + 3)**(3/2)/(7*sqrt(-2*x + 1)*(3*x + 2)**(9/2))

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Mathematica [A]  time = 0.386087, size = 115, normalized size = 0.45 \[ \frac{8 \sqrt{2} \left (6877465 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+3018014 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{24 \sqrt{5 x+3} \left (488918268 x^5+985046292 x^4+466728543 x^3-227945505 x^2-243200677 x-52688263\right )}{\sqrt{1-2 x} (3 x+2)^{9/2}}}{266827932} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^(11/2)),x]

[Out]

((-24*Sqrt[3 + 5*x]*(-52688263 - 243200677*x - 227945505*x^2 + 466728543*x^3 + 9
85046292*x^4 + 488918268*x^5))/(Sqrt[1 - 2*x]*(2 + 3*x)^(9/2)) + 8*Sqrt[2]*(3018
014*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 6877465*EllipticF[ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/266827932

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Maple [C]  time = 0.039, size = 624, normalized size = 2.5 \[ -{\frac{2}{667069830\,{x}^{2}[phone]\,x-200120949}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 557074665\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-7333774020\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -19175958792\,{x}^{5}-15866344773\,{x}^{4}-781374312\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+474194367 \right ) \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(11/2),x)

[Out]

-2/66706983*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(557074665*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^4*(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^4*(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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*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^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)+28972
9344*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)-7333774020*x^6+110039440*
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))[phone]*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*1
1^(1/2)*3^(1/2)*2^(1/2))-19175958792*x^5-15866344773*x^4-781374312*x^3+569951970
0*x^2+2979130038*x+474194367)/(2+3*x)^(9/2)/(10*x^2+x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^(11/2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(11/2)*(-2*x + 1)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3}}{{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^(11/2)*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

integral(-(25*x^2 + 30*x + 9)*sqrt(5*x + 3)/((486*x^6 + 1377*x^5 + 1350*x^4 + 36
0*x^3 - 240*x^2 - 176*x - 32)*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)**(3/2)/(2+3*x)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)/((3*x + 2)^(11/2)*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)/((3*x + 2)^(11/2)*(-2*x + 1)^(3/2)), x)