3.2945 \(\int \frac{(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=253 \[ \frac{(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac{203 (5 x+3)^{5/2} (3 x+2)^{5/2}}{33 \sqrt{1-2 x}}-\frac{225}{22} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}-\frac{6277}{154} \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}-\frac{1310203 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{4620}-\frac{1313411}{630} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{1313411 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3150}-\frac{174654791 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12600} \]

[Out]

(-1313411*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/630 - (1310203*Sqrt[1 - 2*x
]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/4620 - (6277*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5
*x)^(5/2))/154 - (225*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/22 - (203*(
2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/(33*Sqrt[1 - 2*x]) + ((2 + 3*x)^(7/2)*(3 + 5*x)^
(5/2))/(3*(1 - 2*x)^(3/2)) - (174654791*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sq
rt[1 - 2*x]], 35/33])/12600 - (1313411*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqr
t[1 - 2*x]], 35/33])/3150

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Rubi [A]  time = 0.569912, 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 \[ \frac{(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac{203 (5 x+3)^{5/2} (3 x+2)^{5/2}}{33 \sqrt{1-2 x}}-\frac{225}{22} \sqrt{1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}-\frac{6277}{154} \sqrt{1-2 x} (5 x+3)^{5/2} \sqrt{3 x+2}-\frac{1310203 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{4620}-\frac{1313411}{630} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{1313411 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3150}-\frac{174654791 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{12600} \]

Antiderivative was successfully verified.

[In]  Int[((2 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/(1 - 2*x)^(5/2),x]

[Out]

(-1313411*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/630 - (1310203*Sqrt[1 - 2*x
]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/4620 - (6277*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5
*x)^(5/2))/154 - (225*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/22 - (203*(
2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/(33*Sqrt[1 - 2*x]) + ((2 + 3*x)^(7/2)*(3 + 5*x)^
(5/2))/(3*(1 - 2*x)^(3/2)) - (174654791*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sq
rt[1 - 2*x]], 35/33])/12600 - (1313411*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqr
t[1 - 2*x]], 35/33])/3150

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Rubi in Sympy [A]  time = 54.22, size = 228, normalized size = 0.9 \[ - \frac{485 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{18} - \frac{12871 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{126} - \frac{107983 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{252} - \frac{2513419 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{1260} - \frac{174654791 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{37800} - \frac{1313411 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{9450} - \frac{29 \left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3 \sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**(7/2)*(3+5*x)**(5/2)/(1-2*x)**(5/2),x)

[Out]

-485*sqrt(-2*x + 1)*(3*x + 2)**(7/2)*sqrt(5*x + 3)/18 - 12871*sqrt(-2*x + 1)*(3*
x + 2)**(5/2)*sqrt(5*x + 3)/126 - 107983*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*
x + 3)/252 - 2513419*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/1260 - 174654791
*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/37800 - 1313411*sqr
t(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/9450 - 29*(3*x + 2)**(7
/2)*(5*x + 3)**(3/2)/(3*sqrt(-2*x + 1)) + (3*x + 2)**(7/2)*(5*x + 3)**(5/2)/(3*(
-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.341317, size = 135, normalized size = 0.53 \[ -\frac{30 \sqrt{3 x+2} \sqrt{5 x+3} \left (94500 x^5+486900 x^4+1279350 x^3+2783146 x^2-12151171 x+4641769\right )-87969665 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{37800 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/(1 - 2*x)^(5/2),x]

[Out]

-(30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(4641769 - 12151171*x + 2783146*x^2 + 1279350*x
^3 + 486900*x^4 + 94500*x^5) + 174654791*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticE[ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 87969665*Sqrt[2 - 4*x]*(-1 + 2*x)*Ellipti
cF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(37800*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.03, size = 296, normalized size = 1.2 \[{\frac{1}{37800\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( -42525000\,{x}^{7}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-349309582\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-272970000\,{x}^{6}-87969665\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -870250500\,{x}^{5}-2069287200\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-458597550\,x-835518420 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(5/2),x)

[Out]

1/37800*(-42525000*x^7+175939330*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)-349309582*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)-272970000*x^6-879
69665*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))-870250500*x^5-2069287200*x^4+3651350730*x^3+4336
405140*x^2-458597550*x-835518420)*(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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2),x, algorithm="fricas")

[Out]

integral((675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*sqrt(5*x + 3)*s
qrt(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)**(7/2)*(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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2), x)