3.2918 \(\int \frac{(2+3 x)^{11/2}}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=218 \[ \frac{7 (3 x+2)^{9/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^{7/2}}{1815 (5 x+3)^{3/2}}-\frac{4553 \sqrt{1-2 x} (3 x+2)^{5/2}}{99825 \sqrt{5 x+3}}+\frac{380188 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{831875}+\frac{17427983 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{8318750}+\frac{18177329 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3781250 \sqrt{33}}+\frac{604915631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3781250 \sqrt{33}} \]

[Out]

(-107*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(9/2)
)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (4553*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(998
25*Sqrt[3 + 5*x]) + (17427983*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/8318750
+ (380188*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/831875 + (604915631*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3781250*Sqrt[33]) + (18177329*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3781250*Sqrt[33])

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Rubi [A]  time = 0.495051, antiderivative size = 218, 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 \[ \frac{7 (3 x+2)^{9/2}}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^{7/2}}{1815 (5 x+3)^{3/2}}-\frac{4553 \sqrt{1-2 x} (3 x+2)^{5/2}}{99825 \sqrt{5 x+3}}+\frac{380188 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{831875}+\frac{17427983 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{8318750}+\frac{18177329 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3781250 \sqrt{33}}+\frac{604915631 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3781250 \sqrt{33}} \]

Antiderivative was successfully verified.

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

[Out]

(-107*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^(9/2)
)/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (4553*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(998
25*Sqrt[3 + 5*x]) + (17427983*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/8318750
+ (380188*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/831875 + (604915631*Elli
pticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3781250*Sqrt[33]) + (18177329*El
lipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(3781250*Sqrt[33])

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Rubi in Sympy [A]  time = 47.6246, size = 201, normalized size = 0.92 \[ - \frac{107 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}}}{1815 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{4553 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}}}{99825 \sqrt{5 x + 3}} + \frac{380188 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{831875} + \frac{17427983 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{8318750} + \frac{604915631 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{124781250} + \frac{18177329 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{124781250} + \frac{7 \left (3 x + 2\right )^{\frac{9}{2}}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-107*sqrt(-2*x + 1)*(3*x + 2)**(7/2)/(1815*(5*x + 3)**(3/2)) - 4553*sqrt(-2*x +
1)*(3*x + 2)**(5/2)/(99825*sqrt(5*x + 3)) + 380188*sqrt(-2*x + 1)*(3*x + 2)**(3/
2)*sqrt(5*x + 3)/831875 + 17427983*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/83
18750 + 604915631*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/12
4781250 + 18177329*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1
24781250 + 7*(3*x + 2)**(9/2)/(11*sqrt(-2*x + 1)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.346092, size = 141, normalized size = 0.65 \[ \frac{10 \sqrt{3 x+2} \left (-242574750 x^4-1255998150 x^3+1267558775 x^2+2667846028 x+904528061\right ) \sqrt{5 x+3}[phone] \sqrt{2-4 x} (5 x+3)^2 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-1209831262 \sqrt{2-4 x} (5 x+3)^2 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{249562500 \sqrt{1-2 x} (5 x+3)^2} \]

Antiderivative was successfully verified.

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

[Out]

(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(904528061 + 2667846028*x + 1267558775*x^2 - 125
5998150*x^3 - 242574750*x^4) - 1209831262*Sqrt[2 - 4*x]*(3 + 5*x)^2*EllipticE[Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 609979405*Sqrt[2 - 4*x]*(3 + 5*x)^2*Ell
ipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(249562500*Sqrt[1 - 2*x]*(3 + 5
*x)^2)

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Maple [C]  time = 0.036, size = 277, normalized size = 1.3 \[ -{\frac{1}{1497375000\,{x}^{2}[phone]\,x-499125000}\sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3049897025\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-6049156310\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -3629493786\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -7277242500\,{x}^{5}-42531439500\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+18090561220 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/249562500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3049897025*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)-6049156310*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))-3629493786*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))-7277242500*x^5-42531439500*x^4+1290680
0250*x^3+105386556340*x^2+80492762390*x+18090561220)/(3+5*x)^(3/2)/(6*x^2+x-2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{3 \, x + 2}}{{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*sqrt(3*x + 2)/((
50*x^3 + 35*x^2 - 12*x - 9)*sqrt(5*x + 3)*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((2+3*x)**(11/2)/(1-2*x)**(3/2)/(3+5*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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