3.2883 \(\int \frac{(2+3 x)^{5/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=188 \[ \frac{(5 x+3)^{3/2} (3 x+2)^{5/2}}{\sqrt{1-2 x}}+\frac{12}{7} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}+\frac{2511}{350} \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}+\frac{9694}{175} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{9694}{875} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1289089 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500} \]

[Out]

(9694*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/175 + (2511*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*(3 + 5*x)^(3/2))/350 + (12*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2
))/7 + ((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (1289089*Sqrt[11/3]*Ell
ipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3500 + (9694*Sqrt[11/3]*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/875

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Rubi [A]  time = 0.380547, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{(5 x+3)^{3/2} (3 x+2)^{5/2}}{\sqrt{1-2 x}}+\frac{12}{7} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}+\frac{2511}{350} \sqrt{1-2 x} (5 x+3)^{3/2} \sqrt{3 x+2}+\frac{9694}{175} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{9694}{875} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1289089 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3500} \]

Antiderivative was successfully verified.

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

[Out]

(9694*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/175 + (2511*Sqrt[1 - 2*x]*Sqrt[
2 + 3*x]*(3 + 5*x)^(3/2))/350 + (12*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2
))/7 + ((2 + 3*x)^(5/2)*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (1289089*Sqrt[11/3]*Ell
ipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3500 + (9694*Sqrt[11/3]*Elliptic
F[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/875

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Rubi in Sympy [A]  time = 39.3929, size = 168, normalized size = 0.89 \[ \frac{12 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{7} + \frac{837 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{70} + \frac{18551 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{350} + \frac{1289089 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{10500} + \frac{9694 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2625} + \frac{\left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*(5*x + 3)**(3/2)/7 + 837*sqrt(-2*x + 1)*(3*x
+ 2)**(3/2)*sqrt(5*x + 3)/70 + 18551*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/
350 + 1289089*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/10500
+ 9694*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/2625 + (3*x +
2)**(5/2)*(5*x + 3)**(3/2)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.285382, size = 115, normalized size = 0.61 \[ \frac{-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (2250 x^3+8460 x^2+17487 x-34721\right )+649285 \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-1289089 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{10500 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-34721 + 17487*x + 8460*x^2 + 2250*x^3) - 1289
089*Sqrt[2 - 4*x]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 649285*Sq
rt[2 - 4*x]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(10500*Sqrt[1 -
2*x])

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Maple [C]  time = 0.025, size = 174, normalized size = 0.9 \[ -{\frac{1}{315000\,{x}^{3}+241500\,{x}^{2}-73500\,x-63000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 649285\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1289089\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1012500\,{x}^{5}-5089500\,{x}^{4}-13096350\,{x}^{3}+4134060\,{x}^{2}[phone]\,x+6249780 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(649285*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))-1289089*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
))-1012500*x^5-5089500*x^4-13096350*x^3+4134060*x^2+16643310*x+6249780)/(30*x^3+
23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}{{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(45*x^3 + 87*x^2 + 56*x + 12)*sqrt(5*x + 3)*sqrt(3*x + 2)/((2*x - 1)*s
qrt(-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)**(3/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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