3.2892 \(\int \frac{(2+3 x)^{5/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=219 \[ \frac{(3 x+2)^{5/2} (5 x+3)^{5/2}}{\sqrt{1-2 x}}+\frac{5}{3} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac{93}{14} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{4853}{105} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{1284329 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{3780}+\frac{1284329 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18900}+\frac{42696881 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18900} \]

[Out]

(1284329*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3780 + (4853*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*(3 + 5*x)^(3/2))/105 + (93*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5
/2))/14 + (5*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/3 + ((2 + 3*x)^(5/2)
*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x] + (42696881*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7
]*Sqrt[1 - 2*x]], 35/33])/18900 + (1284329*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]
*Sqrt[1 - 2*x]], 35/33])/18900

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Rubi [A]  time = 0.476191, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{(3 x+2)^{5/2} (5 x+3)^{5/2}}{\sqrt{1-2 x}}+\frac{5}{3} \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac{93}{14} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{4853}{105} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{1284329 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{3780}+\frac{1284329 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18900}+\frac{42696881 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{18900} \]

Antiderivative was successfully verified.

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

[Out]

(1284329*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3780 + (4853*Sqrt[1 - 2*x]*S
qrt[2 + 3*x]*(3 + 5*x)^(3/2))/105 + (93*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5
/2))/14 + (5*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/3 + ((2 + 3*x)^(5/2)
*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x] + (42696881*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7
]*Sqrt[1 - 2*x]], 35/33])/18900 + (1284329*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]
*Sqrt[1 - 2*x]], 35/33])/18900

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Rubi in Sympy [A]  time = 46.7852, size = 197, normalized size = 0.9 \[ \frac{5 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{3} + \frac{155 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{14} + \frac{9241 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{126} + \frac{1228883 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{3780} + \frac{42696881 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{56700} + \frac{1284329 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{56700} + \frac{\left (3 x + 2\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{\sqrt{- 2 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*(5*x + 3)**(5/2)/3 + 155*sqrt(-2*x + 1)*(3*x +
2)**(3/2)*(5*x + 3)**(3/2)/14 + 9241*sqrt(-2*x + 1)*(3*x + 2)**(3/2)*sqrt(5*x +
3)/126 + 1228883*sqrt(-2*x + 1)*sqrt(3*x + 2)*sqrt(5*x + 3)/3780 + 42696881*sqr
t(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/56700 + 1284329*sqrt(33
)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/56700 + (3*x + 2)**(5/2)*(5
*x + 3)**(5/2)/sqrt(-2*x + 1)

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Mathematica [A]  time = 0.340544, size = 120, normalized size = 0.55 \[ \frac{-30 \sqrt{3 x+2} \sqrt{5 x+3} \left (94500 x^4+392400 x^3+795150 x^2+1258906 x-2283923\right )[phone] \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-85393762 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{113400 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-2283923 + 1258906*x + 795150*x^2 + 392400*x^3
+ 94500*x^4) - 85393762*Sqrt[2 - 4*x]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
], -33/2] + 43010905*Sqrt[2 - 4*x]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -
33/2])/(113400*Sqrt[1 - 2*x])

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Maple [C]  time = 0.026, size = 179, normalized size = 0.8 \[ -{\frac{1}{3402000\,{x}^{3}+2608200\,{x}^{2}-793800\,x-680400}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -42525000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -85393762\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -230445000\,{x}^{5}-598495500\,{x}^{4}-1090375200\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+411106140 \right ) } \]

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)^(3/2),x)

[Out]

-1/113400*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-42525000*x^6+43010905*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))-85393762*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))-230445000*x^5-598495500*x^4-1090375200*x^3+167061930*x^2+10
75233030*x+411106140)/(30*x^3+23*x^2-7*x-6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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