3.2946 \(\int \frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=220 \[ \frac{(3 x+2)^{5/2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{170 (3 x+2)^{3/2} (5 x+3)^{5/2}}{33 \sqrt{1-2 x}}-\frac{1355}{154} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{28283}{462} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{12601}{28} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{12601}{140} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{69819}{70} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(-12601*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/28 - (28283*Sqrt[1 - 2*x]*Sqr
t[2 + 3*x]*(3 + 5*x)^(3/2))/462 - (1355*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5
/2))/154 - (170*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/(33*Sqrt[1 - 2*x]) + ((2 + 3*x)
^(5/2)*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) - (69819*Sqrt[33]*EllipticE[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/70 - (12601*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/
7]*Sqrt[1 - 2*x]], 35/33])/140

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Rubi [A]  time = 0.489052, antiderivative size = 220, 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 \[ \frac{(3 x+2)^{5/2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{170 (3 x+2)^{3/2} (5 x+3)^{5/2}}{33 \sqrt{1-2 x}}-\frac{1355}{154} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}-\frac{28283}{462} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}-\frac{12601}{28} \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}-\frac{12601}{140} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{69819}{70} \sqrt{33} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-12601*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/28 - (28283*Sqrt[1 - 2*x]*Sqr
t[2 + 3*x]*(3 + 5*x)^(3/2))/462 - (1355*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5
/2))/154 - (170*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/(33*Sqrt[1 - 2*x]) + ((2 + 3*x)
^(5/2)*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) - (69819*Sqrt[33]*EllipticE[ArcSin[S
qrt[3/7]*Sqrt[1 - 2*x]], 35/33])/70 - (12601*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/
7]*Sqrt[1 - 2*x]], 35/33])/140

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Rubi in Sympy [A]  time = 46.116, size = 199, normalized size = 0.9 \[ - \frac{325 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{14} - \frac{185 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{2} - \frac{12057 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{28} - \frac{69819 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{70} - \frac{138611 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{4900} - \frac{170 \left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{21 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-325*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/14 - 185*sqrt(-2*x + 1)*(3*x
+ 2)**(3/2)*sqrt(5*x + 3)/2 - 12057*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/2
8 - 69819*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/70 - 13861
1*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/4900 - 170*(3*x +
2)**(5/2)*(5*x + 3)**(3/2)/(21*sqrt(-2*x + 1)) + (3*x + 2)**(5/2)*(5*x + 3)**(5
/2)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.342712, size = 130, normalized size = 0.59 \[ -\frac{10 \sqrt{3 x+2} \sqrt{5 x+3} \left (2700 x^4+12960 x^3+36606 x^2-175958 x+66663\right )-421995 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+837828 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{840 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(66663 - 175958*x + 36606*x^2 + 12960*x^3 + 270
0*x^4) + 837828*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*
x]], -33/2] - 421995*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2])/(840*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.03, size = 291, normalized size = 1.3 \[{\frac{1}{840\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 843990\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1675656\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-405000\,{x}^{6}-421995\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +837828\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2457000\,{x}^{5}-8115300\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-2108490\,x-3999780 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/840*(843990*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)-1675656*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)-405000*x^6-421995*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))+837828*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
))-2457000*x^5-8115300*x^4+18660960*x^3+21236210*x^2-2108490*x-3999780)*(1-2*x)^
(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)/(-1+2*x)^2/(15*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*sqrt(5*x + 3)*sqrt(3*x + 2)/
((4*x^2 - 4*x + 1)*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)**(5/2)*(3+5*x)**(5/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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