3.2937 \(\int \frac{(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=222 \[ \frac{(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac{56 (5 x+3)^{3/2} (3 x+2)^{5/2}}{11 \sqrt{1-2 x}}-\frac{1341}{154} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac{140289 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{3850}-\frac{2166399 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{7700}-\frac{722133 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500}-\frac{6547351 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500} \]

[Out]

(-2166399*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/7700 - (140289*Sqrt[1 - 2*x
]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/3850 - (1341*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 +
5*x)^(3/2))/154 - (56*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/(11*Sqrt[1 - 2*x]) + ((2
+ 3*x)^(7/2)*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) - (6547351*Sqrt[11/3]*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3500 - (722133*Sqrt[3/11]*EllipticF[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3500

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Rubi [A]  time = 0.468229, 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 \[ \frac{(5 x+3)^{3/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac{56 (5 x+3)^{3/2} (3 x+2)^{5/2}}{11 \sqrt{1-2 x}}-\frac{1341}{154} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac{140289 \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}}{3850}-\frac{2166399 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{7700}-\frac{722133 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500}-\frac{6547351 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500} \]

Antiderivative was successfully verified.

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

[Out]

(-2166399*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/7700 - (140289*Sqrt[1 - 2*x
]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/3850 - (1341*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 +
5*x)^(3/2))/154 - (56*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/(11*Sqrt[1 - 2*x]) + ((2
+ 3*x)^(7/2)*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) - (6547351*Sqrt[11/3]*Ellipti
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3500 - (722133*Sqrt[3/11]*EllipticF[
ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3500

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Rubi in Sympy [A]  time = 45.7676, size = 197, normalized size = 0.89 \[ - \frac{193 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{14} - \frac{2024 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{35} - \frac{188443 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{700} - \frac{6547351 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{10500} - \frac{2166399 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{122500} - \frac{8 \left (3 x + 2\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{\sqrt{- 2 x + 1}} + \frac{\left (3 x + 2\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-193*sqrt(-2*x + 1)*(3*x + 2)**(5/2)*sqrt(5*x + 3)/14 - 2024*sqrt(-2*x + 1)*(3*x
+ 2)**(3/2)*sqrt(5*x + 3)/35 - 188443*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3
)/700 - 6547351*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/1050
0 - 2166399*sqrt(35)*elliptic_f(asin(sqrt(55)*sqrt(-2*x + 1)/11), 33/35)/122500
- 8*(3*x + 2)**(7/2)*sqrt(5*x + 3)/sqrt(-2*x + 1) + (3*x + 2)**(7/2)*(5*x + 3)**
(3/2)/(3*(-2*x + 1)**(3/2))

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Mathematica [A]  time = 0.361144, size = 130, normalized size = 0.59 \[ -\frac{10 \sqrt{3 x+2} \sqrt{5 x+3} \left (40500 x^4+198180 x^3+567906 x^2-2751916 x+1041609\right )-6595505 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )[phone] \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{21000 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(1041609 - 2751916*x + 567906*x^2 + 198180*x^3
+ 40500*x^4) + 13094702*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]], -33/2] - 6595505*Sqrt[2 - 4*x]*(-1 + 2*x)*EllipticF[ArcSin[Sqrt[2/1
1]*Sqrt[3 + 5*x]], -33/2])/(21000*(1 - 2*x)^(3/2))

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Maple [C]  time = 0.03, size = 291, normalized size = 1.3 \[{\frac{1}{21000\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 13191010\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-26189404\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-6075000\,{x}^{6}-6595505\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -37422000\,{x}^{5}-125270100\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-32790750\,x-62496540 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21000*(13191010*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)-26189404*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)-6075000*x^6-6595505*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))-37422000*x^5-125270100*x^4+292994460*x^3+332548330*x^2-32790750*x-62
496540)*(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. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*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)**(7/2)*(3+5*x)**(3/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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