3.2381 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}+\frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^5}+\frac{2770202075 \sqrt{5 x+3} \sqrt{1-2 x}}{14224896 (3 x+2)}+\frac{26486645 \sqrt{5 x+3} \sqrt{1-2 x}}{1016064 (3 x+2)^2}+\frac{151621 \sqrt{5 x+3} \sqrt{1-2 x}}{36288 (3 x+2)^3}+\frac{647 \sqrt{5 x+3} \sqrt{1-2 x}}{864 (3 x+2)^4}-\frac{391280725 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

[Out]

-((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(18*(2 + 3*x)^6) + ((1 - 2*x)^(3/2)*Sqrt[3 + 5*
x])/(12*(2 + 3*x)^5) + (647*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(864*(2 + 3*x)^4) + (15
1621*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36288*(2 + 3*x)^3) + (26486645*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/(1016064*(2 + 3*x)^2) + (2770202075*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/
(14224896*(2 + 3*x)) - (391280725*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])
/(175616*Sqrt[7])

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Rubi [A]  time = 0.45296, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{18 (3 x+2)^6}+\frac{\sqrt{5 x+3} (1-2 x)^{3/2}}{12 (3 x+2)^5}+\frac{2770202075 \sqrt{5 x+3} \sqrt{1-2 x}}{14224896 (3 x+2)}+\frac{26486645 \sqrt{5 x+3} \sqrt{1-2 x}}{1016064 (3 x+2)^2}+\frac{151621 \sqrt{5 x+3} \sqrt{1-2 x}}{36288 (3 x+2)^3}+\frac{647 \sqrt{5 x+3} \sqrt{1-2 x}}{864 (3 x+2)^4}-\frac{391280725 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^7,x]

[Out]

-((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(18*(2 + 3*x)^6) + ((1 - 2*x)^(3/2)*Sqrt[3 + 5*
x])/(12*(2 + 3*x)^5) + (647*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(864*(2 + 3*x)^4) + (15
1621*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36288*(2 + 3*x)^3) + (26486645*Sqrt[1 - 2*x]*
Sqrt[3 + 5*x])/(1016064*(2 + 3*x)^2) + (2770202075*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/
(14224896*(2 + 3*x)) - (391280725*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])
/(175616*Sqrt[7])

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Rubi in Sympy [A]  time = 43.8989, size = 189, normalized size = 0.9 \[ - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{18 \left (3 x + 2\right )^{6}} + \frac{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{12 \left (3 x + 2\right )^{5}} + \frac{2770202075 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{14224896 \left (3 x + 2\right )} + \frac{26486645 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{1016064 \left (3 x + 2\right )^{2}} + \frac{151621 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{36288 \left (3 x + 2\right )^{3}} + \frac{647 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{864 \left (3 x + 2\right )^{4}} - \frac{391280725 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{1229312} \]

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)**7,x)

[Out]

-(-2*x + 1)**(5/2)*sqrt(5*x + 3)/(18*(3*x + 2)**6) + (-2*x + 1)**(3/2)*sqrt(5*x
+ 3)/(12*(3*x + 2)**5) + 2770202075*sqrt(-2*x + 1)*sqrt(5*x + 3)/(14224896*(3*x
+ 2)) + 26486645*sqrt(-2*x + 1)*sqrt(5*x + 3)/(1016064*(3*x + 2)**2) + 151621*sq
rt(-2*x + 1)*sqrt(5*x + 3)/(36288*(3*x + 2)**3) + 647*sqrt(-2*x + 1)*sqrt(5*x +
3)/(864*(3*x + 2)**4) - 391280725*sqrt(7)*atan(sqrt(7)*sqrt(-2*x + 1)/(7*sqrt(5*
x + 3)))/1229312

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Mathematica [A]  time = 0.155445, size = 92, normalized size = 0.44 \[ \frac{\frac{14 \sqrt{1-2 x} \sqrt{5 x+3} \left (24931818675 x^5+84218501340 x^4+113834022672 x^3+76960600672 x^2+26026519504 x+3522190656\right )}{(3 x+2)^6}-1173842175 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{7375872} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^7,x]

[Out]

((14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3522190656 + 26026519504*x + 76960600672*x^2 +
113834022672*x^3 + 84218501340*x^4 + 24931818675*x^5))/(2 + 3*x)^6 - 1173842175
*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])])/7375872

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Maple [B]  time = 0.018, size = 346, normalized size = 1.7 \[{\frac{1}{7375872\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 855730945575\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1077448409408\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]

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)^7,x)

[Out]

1/7375872*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(855730945575*7^(1/2)*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+3422923782300*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^5+5704872970500*7^(1/2)*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))*x^4+349045461450*x^5*(-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^3+1179059018760*x^4
*(-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^2+1593676317408*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arc
tan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+1077448409408*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))+3
64371273056*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)^6

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Maxima [A]  time = 1.51529, size = 329, normalized size = 1.57 \[ \frac{391280725}{2458624} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{16168625}{131712} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{18 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{19 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{12 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{4673 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{672 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{821945 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{28224 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{9701175 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{87808 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{119647825 \, \sqrt{-10 \, x^{2} - x + 3}}{526848 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="maxima")

[Out]

391280725/2458624*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 16
168625/131712*sqrt(-10*x^2 - x + 3) + 7/18*(-10*x^2 - x + 3)^(3/2)/(729*x^6 + 29
16*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 19/12*(-10*x^2 - x + 3)^
(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 4673/672*(-10*x^2
- x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 821945/28224*(-10*x^2
- x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 9701175/87808*(-10*x^2 - x + 3)^(3
/2)/(9*x^2 + 12*x + 4) - 119647825/526848*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 0.225834, size = 188, normalized size = 0.9 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (24931818675 \, x^{5} + 84218501340 \, x^{4} + 113834022672 \, x^{3} + 76960600672 \, x^{2} + 26026519504 \, x + 3522190656\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1173842175 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{7375872 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="fricas")

[Out]

1/7375872*sqrt(7)*(2*sqrt(7)*(24931818675*x^5 + 84218501340*x^4 + 113834022672*x
^3 + 76960600672*x^2 + 26026519504*x + 3522190656)*sqrt(5*x + 3)*sqrt(-2*x + 1)
+ 1173842175*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*
arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(729*x^6 + 2916
*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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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)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.613194, size = 676, normalized size = 3.23 \[ \frac{78256145}{4917248} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{366025 \,{\left (3207 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{11} - 8960840 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{9} - 4031723136 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} - 929280844800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} - 111701434880000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 5519365017600000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{263424 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^7,x, algorithm="giac")

[Out]

78256145/4917248*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)))) - 366025/263424*(3207*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)))^11 - 89
60840*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)))^9 - 4031723136*sqrt(10)*((sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5
) - sqrt(22)))^7 - 929280844800*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s
qrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 1117014
34880000*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 - 5519365017600000*sqrt(10)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*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)^6