3.2822 \(\int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^{11/2}} \, dx\)

Optimal. Leaf size=222 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{189 (3 x+2)^{9/2}}+\frac{32098184 \sqrt{1-2 x} \sqrt{5 x+3}}{47647845 \sqrt{3 x+2}}-\frac{43094 \sqrt{1-2 x} \sqrt{5 x+3}}{6806835 (3 x+2)^{3/2}}-\frac{168034 \sqrt{1-2 x} \sqrt{5 x+3}}{972405 (3 x+2)^{5/2}}+\frac{808 \sqrt{1-2 x} \sqrt{5 x+3}}{27783 (3 x+2)^{7/2}}-\frac{2036756 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845}-\frac{32098184 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845} \]

[Out]

(808*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(27783*(2 + 3*x)^(7/2)) - (168034*Sqrt[1 - 2*x
]*Sqrt[3 + 5*x])/(972405*(2 + 3*x)^(5/2)) - (43094*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/
(6806835*(2 + 3*x)^(3/2)) + (32098184*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(47647845*Sqr
t[2 + 3*x]) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(189*(2 + 3*x)^(9/2)) - (3209818
4*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/47647845 - (2036
756*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/47647845

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Rubi [A]  time = 0.498762, 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 \[ \frac{2 \sqrt{1-2 x} (5 x+3)^{3/2}}{189 (3 x+2)^{9/2}}+\frac{32098184 \sqrt{1-2 x} \sqrt{5 x+3}}{47647845 \sqrt{3 x+2}}-\frac{43094 \sqrt{1-2 x} \sqrt{5 x+3}}{6806835 (3 x+2)^{3/2}}-\frac{168034 \sqrt{1-2 x} \sqrt{5 x+3}}{972405 (3 x+2)^{5/2}}+\frac{808 \sqrt{1-2 x} \sqrt{5 x+3}}{27783 (3 x+2)^{7/2}}-\frac{2036756 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845}-\frac{32098184 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845} \]

Antiderivative was successfully verified.

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

[Out]

(808*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(27783*(2 + 3*x)^(7/2)) - (168034*Sqrt[1 - 2*x
]*Sqrt[3 + 5*x])/(972405*(2 + 3*x)^(5/2)) - (43094*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/
(6806835*(2 + 3*x)^(3/2)) + (32098184*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(47647845*Sqr
t[2 + 3*x]) + (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(189*(2 + 3*x)^(9/2)) - (3209818
4*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/47647845 - (2036
756*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/47647845

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Rubi in Sympy [A]  time = 48.6461, size = 201, normalized size = 0.91 \[ \frac{32098184 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{47647845 \sqrt{3 x + 2}} - \frac{43094 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6806835 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{168034 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{972405 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{808 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{27783 \left (3 x + 2\right )^{\frac{7}{2}}} + \frac{2 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}}{189 \left (3 x + 2\right )^{\frac{9}{2}}} - \frac{32098184 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{142943535} - \frac{2036756 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{142943535} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32098184*sqrt(-2*x + 1)*sqrt(5*x + 3)/(47647845*sqrt(3*x + 2)) - 43094*sqrt(-2*x
+ 1)*sqrt(5*x + 3)/(6806835*(3*x + 2)**(3/2)) - 168034*sqrt(-2*x + 1)*sqrt(5*x
+ 3)/(972405*(3*x + 2)**(5/2)) + 808*sqrt(-2*x + 1)*sqrt(5*x + 3)/(27783*(3*x +
2)**(7/2)) + 2*sqrt(-2*x + 1)*(5*x + 3)**(3/2)/(189*(3*x + 2)**(9/2)) - 32098184
*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/142943535 - 2036756
*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/142943535

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Mathematica [A]  time = 0.319356, size = 107, normalized size = 0.48 \[ \frac{\frac{24 \sqrt{2-4 x} \sqrt{5 x+3} \left (1299976452 x^4+3462531489 x^3+3421407609 x^2+1489220097 x+241253543\right )}{(3 x+2)^{9/2}}[phone] F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{571774140 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

((24*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(241253543 + 1489220097*x + 3421407609*x^2 + 34
62531489*x^3 + 1299976452*x^4))/(2 + 3*x)^(9/2) + 256785472*EllipticE[ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]], -33/2] + 12066320*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]], -33/2])/(571774140*Sqrt[2])

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Maple [C]  time = 0.033, size = 624, normalized size = 2.8 \[ -{\frac{2}{1429435350\,{x}^{2}[phone]\,x-428830605} \left ( 61085745\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-38999293560\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -107775874026\,{x}^{5}-101330034669\,{x}^{4}-23778042336\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+2171281887 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/142943535*(61085745*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)+12999
76452*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)[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)[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+5*x)^(1/2)*
(2+3*x)^(1/2)*(1-2*x)^(1/2)-38999293560*x^6+12066320*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*11^(1/2)*3^(1/2)*2^(1/2))-1
07775874026*x^5-101330034669*x^4-23778042336*x^3+19087401900*x^2+12679220244*x+2
171281887)*(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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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