3.2380 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ \frac{3 (5 x+3)^{3/2} (1-2 x)^{7/2}}{35 (3 x+2)^5}+\frac{251 (5 x+3)^{3/2} (1-2 x)^{5/2}}{280 (3 x+2)^4}+\frac{2761 (5 x+3)^{3/2} (1-2 x)^{3/2}}{336 (3 x+2)^3}+\frac{30371 (5 x+3)^{3/2} \sqrt{1-2 x}}{448 (3 x+2)^2}-\frac{334081 \sqrt{5 x+3} \sqrt{1-2 x}}{6272 (3 x+2)}-\frac{3674891 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

[Out]

(-334081*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6272*(2 + 3*x)) + (3*(1 - 2*x)^(7/2)*(3 +
5*x)^(3/2))/(35*(2 + 3*x)^5) + (251*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(280*(2 +
3*x)^4) + (2761*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(336*(2 + 3*x)^3) + (30371*Sqrt
[1 - 2*x]*(3 + 5*x)^(3/2))/(448*(2 + 3*x)^2) - (3674891*ArcTan[Sqrt[1 - 2*x]/(Sq
rt[7]*Sqrt[3 + 5*x])])/(6272*Sqrt[7])

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Rubi [A]  time = 0.264144, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3 (5 x+3)^{3/2} (1-2 x)^{7/2}}{35 (3 x+2)^5}+\frac{251 (5 x+3)^{3/2} (1-2 x)^{5/2}}{280 (3 x+2)^4}+\frac{2761 (5 x+3)^{3/2} (1-2 x)^{3/2}}{336 (3 x+2)^3}+\frac{30371 (5 x+3)^{3/2} \sqrt{1-2 x}}{448 (3 x+2)^2}-\frac{334081 \sqrt{5 x+3} \sqrt{1-2 x}}{6272 (3 x+2)}-\frac{3674891 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^6,x]

[Out]

(-334081*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6272*(2 + 3*x)) + (3*(1 - 2*x)^(7/2)*(3 +
5*x)^(3/2))/(35*(2 + 3*x)^5) + (251*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(280*(2 +
3*x)^4) + (2761*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(336*(2 + 3*x)^3) + (30371*Sqrt
[1 - 2*x]*(3 + 5*x)^(3/2))/(448*(2 + 3*x)^2) - (3674891*ArcTan[Sqrt[1 - 2*x]/(Sq
rt[7]*Sqrt[3 + 5*x])])/(6272*Sqrt[7])

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Rubi in Sympy [A]  time = 20.3696, size = 165, normalized size = 0.92 \[ - \frac{251 \left (- 2 x + 1\right )^{\frac{7}{2}} \sqrt{5 x + 3}}{1960 \left (3 x + 2\right )^{4}} + \frac{3 \left (- 2 x + 1\right )^{\frac{7}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{35 \left (3 x + 2\right )^{5}} + \frac{2761 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{11760 \left (3 x + 2\right )^{3}} + \frac{30371 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{9408 \left (3 x + 2\right )^{2}} + \frac{334081 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6272 \left (3 x + 2\right )} - \frac{3674891 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{43904} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(1/2)/(2+3*x)**6,x)

[Out]

-251*(-2*x + 1)**(7/2)*sqrt(5*x + 3)/(1960*(3*x + 2)**4) + 3*(-2*x + 1)**(7/2)*(
5*x + 3)**(3/2)/(35*(3*x + 2)**5) + 2761*(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(11760*
(3*x + 2)**3) + 30371*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(9408*(3*x + 2)**2) + 3340
81*sqrt(-2*x + 1)*sqrt(5*x + 3)/(6272*(3*x + 2)) - 3674891*sqrt(7)*atan(sqrt(7)*
sqrt(-2*x + 1)/(7*sqrt(5*x + 3)))/43904

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Mathematica [A]  time = 0.112539, size = 87, normalized size = 0.48 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (390269835 x^4+1058136330 x^3+1076423732 x^2+487066088 x+82697568\right )}{(3 x+2)^5}-55123365 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{1317120} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^6,x]

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(82697568 + 487066088*x + 1076423732*x^2 + 1058
136330*x^3 + 390269835*x^4))/(2 + 3*x)^5 - 55123365*Sqrt[7]*ArcTan[(-20 - 37*x)/
(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/1317120

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Maple [B]  time = 0.018, size = 298, normalized size = 1.7 \[{\frac{1}{1317120\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 13394977695\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+15069932248\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^6,x)

[Out]

1/1317120*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(13394977695*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+44649925650*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^4+59533234200*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))*x^3+5463777690*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+14813908620*x^3*(-10*x^2-x
+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
*x+15069932248*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^
2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^5

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Maxima [A]  time = 1.63515, size = 267, normalized size = 1.48 \[ \frac{3674891}{87808} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{151855}{4704} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{15 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{73 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{40 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{2573 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{336 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{91113 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{3136 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{1123727 \, \sqrt{-10 \, x^{2} - x + 3}}{18816 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^6,x, algorithm="maxima")

[Out]

3674891/87808*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 151855
/4704*sqrt(-10*x^2 - x + 3) + 7/15*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 +
1080*x^3 + 720*x^2 + 240*x + 32) + 73/40*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x
^3 + 216*x^2 + 96*x + 16) + 2573/336*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 +
36*x + 8) + 91113/3136*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 1123727/1881
6*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.226937, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (390269835 \, x^{4} + 1058136330 \, x^{3} + 1076423732 \, x^{2} + 487066088 \, x + 82697568\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 55123365 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{1317120 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^6,x, algorithm="fricas")

[Out]

1/1317120*sqrt(7)*(2*sqrt(7)*(390269835*x^4 + 1058136330*x^3 + 1076423732*x^2 +
487066088*x + 82697568)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 55123365*(243*x^5 + 810*x
^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x
+ 3)*sqrt(-2*x + 1))))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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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)*(3+5*x)**(1/2)/(2+3*x)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.498039, size = 594, normalized size = 3.3 \[ \frac{3674891}{878080} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{14641 \,{\left (753 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} - 1524880 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 503767040 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 77139328000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 4628359680000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{9408 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^6,x, algorithm="giac")

[Out]

3674891/878080*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22)))) - 14641/9408*(753*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr
t(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 - 1524880*s
qrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(s
qrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 - 503767040*sqrt(10)*((sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt
(22)))^5 - 77139328000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x +
3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 4628359680000*sq
rt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(
5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^5