3.2653 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{11/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{1323 (3 x+2)^{7/2}}+\frac{27198452 \sqrt{1-2 x} \sqrt{5 x+3}}{20420505 \sqrt{3 x+2}}+\frac{568318 \sqrt{1-2 x} \sqrt{5 x+3}}{2917215 (3 x+2)^{3/2}}-\frac{12934 \sqrt{1-2 x} \sqrt{5 x+3}}{138915 (3 x+2)^{5/2}}-\frac{442868 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{20420505}-\frac{27198452 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{20420505} \]

[Out]

(-12934*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(138915*(2 + 3*x)^(5/2)) + (568318*Sqrt[1 -
2*x]*Sqrt[3 + 5*x])/(2917215*(2 + 3*x)^(3/2)) + (27198452*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(20420505*Sqrt[2 + 3*x]) - (118*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(1323*(2
+ 3*x)^(7/2)) - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(27*(2 + 3*x)^(9/2)) - (271984
52*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/20420505 - (442
868*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/20420505

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Rubi [A]  time = 0.499168, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{1323 (3 x+2)^{7/2}}+\frac{27198452 \sqrt{1-2 x} \sqrt{5 x+3}}{20420505 \sqrt{3 x+2}}+\frac{568318 \sqrt{1-2 x} \sqrt{5 x+3}}{2917215 (3 x+2)^{3/2}}-\frac{12934 \sqrt{1-2 x} \sqrt{5 x+3}}{138915 (3 x+2)^{5/2}}-\frac{442868 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{20420505}-\frac{27198452 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{20420505} \]

Antiderivative was successfully verified.

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

[Out]

(-12934*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(138915*(2 + 3*x)^(5/2)) + (568318*Sqrt[1 -
2*x]*Sqrt[3 + 5*x])/(2917215*(2 + 3*x)^(3/2)) + (27198452*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(20420505*Sqrt[2 + 3*x]) - (118*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(1323*(2
+ 3*x)^(7/2)) - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(27*(2 + 3*x)^(9/2)) - (271984
52*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/20420505 - (442
868*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/20420505

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Rubi in Sympy [A]  time = 45.5805, size = 201, normalized size = 0.91 \[ \frac{27198452 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{20420505 \sqrt{3 x + 2}} + \frac{568318 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{2917215 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{12934 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{138915 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{118 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{1323 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{5}{2}}}{27 \left (3 x + 2\right )^{\frac{9}{2}}} - \frac{27198452 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{61261515} - \frac{442868 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{61261515} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27198452*sqrt(-2*x + 1)*sqrt(5*x + 3)/(20420505*sqrt(3*x + 2)) + 568318*sqrt(-2*
x + 1)*sqrt(5*x + 3)/(2917215*(3*x + 2)**(3/2)) - 12934*sqrt(-2*x + 1)*sqrt(5*x
+ 3)/(138915*(3*x + 2)**(5/2)) - 118*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/(1323*(3*x
+ 2)**(7/2)) - 2*sqrt(-2*x + 1)*(5*x + 3)**(5/2)/(27*(3*x + 2)**(9/2)) - 2719845
2*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/61261515 - 442868*
sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/61261515

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Mathematica [A]  time = 0.421964, size = 110, normalized size = 0.5 \[ \frac{\frac{24 \sqrt{1-2 x} \sqrt{5 x+3} \left (1101537306 x^4+2991138867 x^3+3003721227 x^2+1325733891 x+217427099\right )}{(3 x+2)^{9/2}}+8 \sqrt{2} \left (13599226 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-9945565 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{245046060} \]

Antiderivative was successfully verified.

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

[Out]

((24*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(217427099 + 1325733891*x + 3003721227*x^2 + 29
91138867*x^3 + 1101537306*x^4))/(2 + 3*x)^(9/2) + 8*Sqrt[2]*(13599226*EllipticE[
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 9945565*EllipticF[ArcSin[Sqrt[2/11]*S
qrt[3 + 5*x]], -33/2]))/245046060

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Maple [C]  time = 0.029, size = 624, normalized size = 2.8 \[{\frac{2}{612615150\,{x}^{2}[phone]\,x-183784545} \left ( 805590765\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1101537306\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-2937432816\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2937432816\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1305525696\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -217587616\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-16533476400\,{x}^{2}-11279323722\,x-1956843891 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/61261515*(805590765*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)-110153
7306*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)*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^3*(1
-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)-2937432816*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)-2937432816*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)-1305525696*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]*x^6+159129040*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))-217587616*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^5+89171217657*x^4+21862930608*x^3-16533476400*x^2-11279323722*x-
1956843891)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(9/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)*sqrt(-2*x + 1)/((243*x^5 + 810*x^4 +
1080*x^3 + 720*x^2 + 240*x + 32)*sqrt(3*x + 2)), 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)**(1/2)/(2+3*x)**(11/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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