3.2255 \(\int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ \frac{1466281 \sqrt{1-2 x} \sqrt{5 x+3}}{131712 (3 x+2)}+\frac{14023 \sqrt{1-2 x} \sqrt{5 x+3}}{9408 (3 x+2)^2}+\frac{403 \sqrt{1-2 x} \sqrt{5 x+3}}{1680 (3 x+2)^3}+\frac{37 \sqrt{1-2 x} \sqrt{5 x+3}}{840 (3 x+2)^4}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{15 (3 x+2)^5}-\frac{5591773 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(840*(2 + 3*x)^4) + (403*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1680*(2 + 3*x)^3) + (1
4023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(9408*(2 + 3*x)^2) + (1466281*Sqrt[1 - 2*x]*Sq
rt[3 + 5*x])/(131712*(2 + 3*x)) - (5591773*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3
+ 5*x])])/(43904*Sqrt[7])

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Rubi [A]  time = 0.37009, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{1466281 \sqrt{1-2 x} \sqrt{5 x+3}}{131712 (3 x+2)}+\frac{14023 \sqrt{1-2 x} \sqrt{5 x+3}}{9408 (3 x+2)^2}+\frac{403 \sqrt{1-2 x} \sqrt{5 x+3}}{1680 (3 x+2)^3}+\frac{37 \sqrt{1-2 x} \sqrt{5 x+3}}{840 (3 x+2)^4}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{15 (3 x+2)^5}-\frac{5591773 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x
])/(840*(2 + 3*x)^4) + (403*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1680*(2 + 3*x)^3) + (1
4023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(9408*(2 + 3*x)^2) + (1466281*Sqrt[1 - 2*x]*Sq
rt[3 + 5*x])/(131712*(2 + 3*x)) - (5591773*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3
+ 5*x])])/(43904*Sqrt[7])

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Rubi in Sympy [A]  time = 35.1247, size = 163, normalized size = 0.91 \[ \frac{1466281 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{131712 \left (3 x + 2\right )} + \frac{14023 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{9408 \left (3 x + 2\right )^{2}} + \frac{403 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1680 \left (3 x + 2\right )^{3}} + \frac{37 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{840 \left (3 x + 2\right )^{4}} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{15 \left (3 x + 2\right )^{5}} - \frac{5591773 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{307328} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1466281*sqrt(-2*x + 1)*sqrt(5*x + 3)/(131712*(3*x + 2)) + 14023*sqrt(-2*x + 1)*s
qrt(5*x + 3)/(9408*(3*x + 2)**2) + 403*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1680*(3*x +
2)**3) + 37*sqrt(-2*x + 1)*sqrt(5*x + 3)/(840*(3*x + 2)**4) - sqrt(-2*x + 1)*sq
rt(5*x + 3)/(15*(3*x + 2)**5) - 5591773*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*s
qrt(5*x + 3)))/307328

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Mathematica [A]  time = 0.119201, size = 87, normalized size = 0.48 \[ \frac{\frac{42 \sqrt{1-2 x} \sqrt{5 x+3} \left (197947935 x^4+536695650 x^3+546004068 x^2+247045192 x+41933792\right )}{(3 x+2)^5}-83876595 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{9219840} \]

Antiderivative was successfully verified.

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

[Out]

((42*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(41933792 + 247045192*x + 546004068*x^2 + 53669
5650*x^3 + 197947935*x^4))/(2 + 3*x)^5 - 83876595*Sqrt[7]*ArcTan[(-20 - 37*x)/(2
*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/9219840

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Maple [B]  time = 0.02, size = 298, normalized size = 1.7 \[{\frac{1}{3073280\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 6794004195\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+7644056952\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3073280*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(6794004195*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+22646680650*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/
2)/(-10*x^2-x+3)^(1/2))*x^4+30195574200*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))*x^3+2771271090*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+7513739100*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+
7644056952*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 = 1.52604, size = 267, normalized size = 1.48 \[ \frac{5591773}{614656} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{231065}{32928} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{35 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{111 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{280 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{1305 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{784 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{138639 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{21952 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{1709881 \, \sqrt{-10 \, x^{2} - x + 3}}{131712 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5591773/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 23106
5/32928*sqrt(-10*x^2 - x + 3) + 3/35*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4
+ 1080*x^3 + 720*x^2 + 240*x + 32) + 111/280*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 2
16*x^3 + 216*x^2 + 96*x + 16) + 1305/784*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^
2 + 36*x + 8) + 138639/21952*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 170988
1/131712*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.233808, size = 167, normalized size = 0.93 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (197947935 \, x^{4} + 536695650 \, x^{3} + 546004068 \, x^{2} + 247045192 \, x + 41933792\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 27958865 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{3073280 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3073280*sqrt(7)*(2*sqrt(7)*(197947935*x^4 + 536695650*x^3 + 546004068*x^2 + 24
7045192*x + 41933792)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 27958865*(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]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3}}{\left (3 x + 2\right )^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.501274, size = 582, normalized size = 3.23 \[ \frac{121}{6146560} \, \sqrt{5}{\left (46213 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{280 \, \sqrt{2}{\left (46213 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} - 85961680 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 30665564160 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 4732042112000 \,{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - \frac{284050977280000 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} + \frac{1136203909120000 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{5}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

121/6146560*sqrt(5)*(46213*sqrt(70)*sqrt(2)*(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(-1
0*x + 5) - sqrt(22)))) - 280*sqrt(2)*(46213*((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)))^9 - 8596
1680*((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)))^7 - 30665564160*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 -
4732042112000*((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 - 284050977280000*(sqrt(2)*sqrt(-10
*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1136203909120000*sqrt(5*x + 3)/(sqrt(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)