3.2430 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=193 \[ -\frac{508 (1-2 x)^{3/2} (3 x+2)^4}{75 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}+\frac{2514}{625} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^3+\frac{23991 (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2}{25000}+\frac{21 (1-2 x)^{3/2} \sqrt{5 x+3} (118392 x+64435)}{4000000}+\frac{8026963 \sqrt{1-2 x} \sqrt{5 x+3}}{40000000}+\frac{88296593 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40000000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^4)/(15*(3 + 5*x)^(3/2)) - (508*(1 - 2*x)^(3/2)*(2
+ 3*x)^4)/(75*Sqrt[3 + 5*x]) + (8026963*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/40000000 +
(23991*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/25000 + (2514*(1 - 2*x)^(3/2)*
(2 + 3*x)^3*Sqrt[3 + 5*x])/625 + (21*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]*(64435 + 1183
92*x))/4000000 + (88296593*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(40000000*Sqrt[10])

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Rubi [A]  time = 0.345857, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{508 (1-2 x)^{3/2} (3 x+2)^4}{75 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}+\frac{2514}{625} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^3+\frac{23991 (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2}{25000}+\frac{21 (1-2 x)^{3/2} \sqrt{5 x+3} (118392 x+64435)}{4000000}+\frac{8026963 \sqrt{1-2 x} \sqrt{5 x+3}}{40000000}+\frac{88296593 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40000000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*(2 + 3*x)^4)/(3 + 5*x)^(5/2),x]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^4)/(15*(3 + 5*x)^(3/2)) - (508*(1 - 2*x)^(3/2)*(2
+ 3*x)^4)/(75*Sqrt[3 + 5*x]) + (8026963*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/40000000 +
(23991*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/25000 + (2514*(1 - 2*x)^(3/2)*
(2 + 3*x)^3*Sqrt[3 + 5*x])/625 + (21*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]*(64435 + 1183
92*x))/4000000 + (88296593*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(40000000*Sqrt[10])

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Rubi in Sympy [A]  time = 31.6952, size = 173, normalized size = 0.9 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{4}}{15 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{508 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{3}}{825 \sqrt{5 x + 3}} + \frac{2969 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{6875} + \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3} \left (\frac{5348295 x}{2} + \frac{8453709}{8}\right )}{12375000} + \frac{8026963 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{132000000} + \frac{8026963 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{40000000} + \frac{88296593 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{400000000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(2+3*x)**4/(3+5*x)**(5/2),x)

[Out]

-2*(-2*x + 1)**(5/2)*(3*x + 2)**4/(15*(5*x + 3)**(3/2)) - 508*(-2*x + 1)**(5/2)*
(3*x + 2)**3/(825*sqrt(5*x + 3)) + 2969*(-2*x + 1)**(5/2)*(3*x + 2)**2*sqrt(5*x
+ 3)/6875 + (-2*x + 1)**(5/2)*sqrt(5*x + 3)*(5348295*x/2 + 8453709/8)/12375000 +
8026963*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/132000000 + 8026963*sqrt(-2*x + 1)*sqrt
(5*x + 3)/40000000 + 88296593*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/400000000

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Mathematica [A]  time = 0.241855, size = 80, normalized size = 0.41 \[ \frac{\frac{10 \sqrt{1-2 x} \left (1555200000 x^6+1626480000 x^5-1419228000 x^4-1405199700 x^3+865945995 x^2+980658710 x+210855251\right )}{(5 x+3)^{3/2}}-264889779 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1200000000} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*(2 + 3*x)^4)/(3 + 5*x)^(5/2),x]

[Out]

((10*Sqrt[1 - 2*x]*(210855251 + 980658710*x + 865945995*x^2 - 1405199700*x^3 - 1
419228000*x^4 + 1626480000*x^5 + 1555200000*x^6))/(3 + 5*x)^(3/2) - 264889779*Sq
rt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/1200000000

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Maple [A]  time = 0.02, size = 181, normalized size = 0.9 \[{\frac{1}{2400000000} \left ( 31104000000\,{x}^{6}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-28384560000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-28103994000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+17318919900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x)^(5/2),x)

[Out]

1/2400000000*(31104000000*x^6*(-10*x^2-x+3)^(1/2)[phone]*x^5*(-10*x^2-x+3)^
(1/2)-28384560000*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/1
1)*x^2-28103994000*x^3*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/
11)*x+17318919900*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/1
1)[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/
2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.54239, size = 478, normalized size = 2.48 \[ \frac{81}{15625} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{891}{25000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{70759953}{800000000} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{23}{11}\right ) + \frac{27401}{1250000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{8811}{500000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3125 \,{\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} + \frac{6 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3125 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{18 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3125 \,{\left (5 \, x + 3\right )}} + \frac{584793}{2000000} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} x + \frac{13450239}{40000000} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} + \frac{3267}{62500} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{11 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{18750 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{3125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{99 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{6250 \,{\left (5 \, x + 3\right )}} - \frac{121 \, \sqrt{-10 \, x^{2} - x + 3}}{93750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{638 \, \sqrt{-10 \, x^{2} - x + 3}}{9375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2),x, algorithm="maxima")

[Out]

81/15625*(-10*x^2 - x + 3)^(5/2) + 891/25000*(-10*x^2 - x + 3)^(3/2)*x - 7075995
3/800000000*I*sqrt(5)*sqrt(2)*arcsin(20/11*x + 23/11) + 27401/1250000*sqrt(5)*sq
rt(2)*arcsin(20/11*x + 1/11) + 8811/500000*(-10*x^2 - x + 3)^(3/2) + 1/3125*(-10
*x^2 - x + 3)^(5/2)/(625*x^4 + 1500*x^3 + 1350*x^2 + 540*x + 81) + 6/3125*(-10*x
^2 - x + 3)^(5/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 18/3125*(-10*x^2 - x + 3)^(
5/2)/(25*x^2 + 30*x + 9) + 27/3125*(-10*x^2 - x + 3)^(5/2)/(5*x + 3) + 584793/20
00000*sqrt(10*x^2 + 23*x + 51/5)*x + 13450239/40000000*sqrt(10*x^2 + 23*x + 51/5
) + 3267/62500*sqrt(-10*x^2 - x + 3) - 11/18750*(-10*x^2 - x + 3)^(3/2)/(125*x^3
+ 225*x^2 + 135*x + 27) + 33/3125*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) +
99/6250*(-10*x^2 - x + 3)^(3/2)/(5*x + 3) - 121/93750*sqrt(-10*x^2 - x + 3)/(25
*x^2 + 30*x + 9) - 638/9375*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 0.227222, size = 140, normalized size = 0.73 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (1555200000 \, x^{6} + 1626480000 \, x^{5} - 1419228000 \, x^{4} - 1405199700 \, x^{3} + 865945995 \, x^{2} + 980658710 \, x + 210855251\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 264889779 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{2400000000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2),x, algorithm="fricas")

[Out]

1/2400000000*sqrt(10)*(2*sqrt(10)*(1555200000*x^6 + 1626480000*x^5 - 1419228000*
x^4 - 1405199700*x^3 + 865945995*x^2 + 980658710*x + 210855251)*sqrt(5*x + 3)*sq
rt(-2*x + 1) + 264889779*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)/(sq
rt(5*x + 3)*sqrt(-2*x + 1))))/(25*x^2 + 30*x + 9)

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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((1-2*x)**(5/2)*(2+3*x)**4/(3+5*x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.367993, size = 290, normalized size = 1.5 \[ \frac{1}{1000000000} \,{\left (12 \,{\left (24 \,{\left (12 \,{\left (48 \, \sqrt{5}{\left (5 \, x + 3\right )} - 613 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 19439 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1264235 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 10674335 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{18750000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{88296593}{400000000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{561 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{312500 \, \sqrt{5 \, x + 3}} + \frac{11 \,{\left (\frac{765 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{1171875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 2)^4*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2),x, algorithm="giac")

[Out]

1/1000000000*(12*(24*(12*(48*sqrt(5)*(5*x + 3) - 613*sqrt(5))*(5*x + 3) + 19439*
sqrt(5))*(5*x + 3) + 1264235*sqrt(5))*(5*x + 3) - 10674335*sqrt(5))*sqrt(5*x + 3
)*sqrt(-10*x + 5) - 11/18750000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/
(5*x + 3)^(3/2) + 88296593/400000000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)
) - 561/312500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 11/
1171875*(765*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(
10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3