3.2446 \(\int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2} \, dx\)

Optimal. Leaf size=229 \[ -\frac{1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}+\frac{439 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}}{1404}+\frac{205}{351} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{(389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}}{227448}+\frac{74167 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{186624}-\frac{519169 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{13436928}+\frac{2595845 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{644972544}-\frac{2595845 (6 x+5) \sqrt{3 x^2+5 x+2}}{5159780352}+\frac{2595845 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{10319560704 \sqrt{3}} \]

[Out]

(-2595845*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/5159780352 + (2595845*(5 + 6*x)*(2 +
5*x + 3*x^2)^(3/2))/644972544 - (519169*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/13436
928 + (74167*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/186624 + (205*(3 + 2*x)^2*(2 + 5
*x + 3*x^2)^(9/2))/351 + (439*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(9/2))/1404 - ((3 +
2*x)^4*(2 + 5*x + 3*x^2)^(9/2))/39 + ((852175 + 389394*x)*(2 + 5*x + 3*x^2)^(9/2
))/227448 + (2595845*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1031
9560704*Sqrt[3])

_______________________________________________________________________________________

Rubi [A]  time = 0.38124, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}+\frac{439 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}}{1404}+\frac{205}{351} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{(389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}}{227448}+\frac{74167 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{186624}-\frac{519169 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{13436928}+\frac{2595845 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{644972544}-\frac{2595845 (6 x+5) \sqrt{3 x^2+5 x+2}}{5159780352}+\frac{2595845 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{10319560704 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(5 - x)*(3 + 2*x)^4*(2 + 5*x + 3*x^2)^(7/2),x]

[Out]

(-2595845*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/5159780352 + (2595845*(5 + 6*x)*(2 +
5*x + 3*x^2)^(3/2))/644972544 - (519169*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/13436
928 + (74167*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/186624 + (205*(3 + 2*x)^2*(2 + 5
*x + 3*x^2)^(9/2))/351 + (439*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(9/2))/1404 - ((3 +
2*x)^4*(2 + 5*x + 3*x^2)^(9/2))/39 + ((852175 + 389394*x)*(2 + 5*x + 3*x^2)^(9/2
))/227448 + (2595845*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(1031
9560704*Sqrt[3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 39.0641, size = 212, normalized size = 0.93 \[ - \frac{\left (2 x + 3\right )^{4} \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{39} + \frac{439 \left (2 x + 3\right )^{3} \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{1404} + \frac{205 \left (2 x + 3\right )^{2} \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{351} + \frac{74167 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{186624} - \frac{519169 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{13436928} + \frac{2595845 \left (6 x + 5\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{644972544} - \frac{2595845 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{5159780352} + \frac{\left (64250010 x + 140608875\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{9}{2}}}{37528920} + \frac{2595845 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{30958682112} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**4*(3*x**2+5*x+2)**(7/2),x)

[Out]

-(2*x + 3)**4*(3*x**2 + 5*x + 2)**(9/2)/39 + 439*(2*x + 3)**3*(3*x**2 + 5*x + 2)
**(9/2)/1404 + 205*(2*x + 3)**2*(3*x**2 + 5*x + 2)**(9/2)/351 + 74167*(6*x + 5)*
(3*x**2 + 5*x + 2)**(7/2)/186624 - 519169*(6*x + 5)*(3*x**2 + 5*x + 2)**(5/2)/13
436928 + 2595845*(6*x + 5)*(3*x**2 + 5*x + 2)**(3/2)/644972544 - 2595845*(6*x +
5)*sqrt(3*x**2 + 5*x + 2)/5159780352 + (64250010*x + 140608875)*(3*x**2 + 5*x +
2)**(9/2)/37528920 + 2595845*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*
x + 2)))/30958682112

_______________________________________________________________________________________

Mathematica [A]  time = 0.161238, size = 110, normalized size = 0.48 \[ \frac{33745985 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-6 \sqrt{3 x^2+5 x+2} \left (2229025112064 x^{12}[phone] x^{11}-2110350163968 x^{10}-333952593887232 x^9-1590604366381056 x^8-4022427759003648 x^7-6524509131334656 x^6-7203650864723712 x^5-5499074981552256 x^4-2865856228323984 x^3-975104480077800 x^2-195441229635490 x-17510968283403\right )}{402462867456} \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - x)*(3 + 2*x)^4*(2 + 5*x + 3*x^2)^(7/2),x]

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-17510968283403 - 195441229635490*x - 975104480077800
*x^2 - 2865856228323984*x^3 - 5499074981552256*x^4 - 7203650864723712*x^5 - 6524
509131334656*x^6 - 4022427759003648*x^7 - 1590604366381056*x^8 - 333952593887232
*x^9 - 2110350163968*x^10 + 14643456638976*x^11 + 2229025112064*x^12) + 33745985
*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/402462867456

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 187, normalized size = 0.8 \[{\frac{370835+445002\,x}{186624} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{2595845+3115014\,x}{13436928} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{12979225+15575070\,x}{644972544} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{12979225+15575070\,x}{5159780352}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{2595845\,\sqrt{3}}{30958682112}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }+{\frac{3495529}{227448} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{84521\,x}{4212} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{2827\,{x}^{2}}{351} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{14\,{x}^{3}}{351} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}-{\frac{16\,{x}^{4}}{39} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3+2*x)^4*(3*x^2+5*x+2)^(7/2),x)

[Out]

74167/186624*(5+6*x)*(3*x^2+5*x+2)^(7/2)-519169/13436928*(5+6*x)*(3*x^2+5*x+2)^(
5/2)+2595845/644972544*(5+6*x)*(3*x^2+5*x+2)^(3/2)-2595845/5159780352*(5+6*x)*(3
*x^2+5*x+2)^(1/2)+2595845/30958682112*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/
2))*3^(1/2)+3495529/227448*(3*x^2+5*x+2)^(9/2)+84521/4212*x*(3*x^2+5*x+2)^(9/2)+
2827/351*x^2*(3*x^2+5*x+2)^(9/2)+14/351*x^3*(3*x^2+5*x+2)^(9/2)-16/39*x^4*(3*x^2
+5*x+2)^(9/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.801536, size = 304, normalized size = 1.33 \[ -\frac{16}{39} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x^{4} + \frac{14}{351} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x^{3} + \frac{2827}{351} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x^{2} + \frac{84521}{4212} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x + \frac{3495529}{227448} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} + \frac{74167}{31104} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{370835}{186624} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{519169}{2239488} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{2595845}{13436928} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{2595845}{107495424} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{12979225}{644972544} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{2595845}{859963392} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{2595845}{30958682112} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{12979225}{5159780352} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(2*x + 3)^4*(x - 5),x, algorithm="maxima")

[Out]

-16/39*(3*x^2 + 5*x + 2)^(9/2)*x^4 + 14/351*(3*x^2 + 5*x + 2)^(9/2)*x^3 + 2827/3
51*(3*x^2 + 5*x + 2)^(9/2)*x^2 + 84521/4212*(3*x^2 + 5*x + 2)^(9/2)*x + 3495529/
227448*(3*x^2 + 5*x + 2)^(9/2) + 74167/31104*(3*x^2 + 5*x + 2)^(7/2)*x + 370835/
186624*(3*x^2 + 5*x + 2)^(7/2) - 519169/2239488*(3*x^2 + 5*x + 2)^(5/2)*x - 2595
845/13436928*(3*x^2 + 5*x + 2)^(5/2) + 2595845/107495424*(3*x^2 + 5*x + 2)^(3/2)
*x + 12979225/644972544*(3*x^2 + 5*x + 2)^(3/2) - 2595845/859963392*sqrt(3*x^2 +
5*x + 2)*x + 2595845/30958682112*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) +
6*x + 5) - 12979225/5159780352*sqrt(3*x^2 + 5*x + 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.281704, size = 162, normalized size = 0.71 \[ -\frac{1}{804925734912} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (2229025112064 \, x^{12} + 14643456638976 \, x^{11} - 2110350163968 \, x^{10} - 333952593887232 \, x^{9} - 1590604366381056 \, x^{8} - 4022427759003648 \, x^{7} - 6524509131334656 \, x^{6} - 7203650864723712 \, x^{5} - 5499074981552256 \, x^{4} - 2865856228323984 \, x^{3} - 975104480077800 \, x^{2} - 195441229635490 \, x - 17510968283403\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 33745985 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(2*x + 3)^4*(x - 5),x, algorithm="fricas")

[Out]

-1/804925734912*sqrt(3)*(4*sqrt(3)*(2229025112064*x^12 + 14643456638976*x^11 - 2
110350163968*x^10 - 333952593887232*x^9 - 1590604366381056*x^8 - 402242775900364
8*x^7 - 6524509131334656*x^6 - 7203650864723712*x^5 - 5499074981552256*x^4 - 286
5856228323984*x^3 - 975104480077800*x^2 - 195441229635490*x - 17510968283403)*sq
rt(3*x^2 + 5*x + 2) - 33745985*log(sqrt(3)*(72*x^2 + 120*x + 49) + 12*sqrt(3*x^2
+ 5*x + 2)*(6*x + 5)))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- 32292 x \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 142182 x^{2} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 363291 x^{3} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 594106 x^{4} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 644932 x^{5} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 463440 x^{6} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 209413 x^{7} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 49624 x^{8} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 504 x^{9} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 2592 x^{10} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 432 x^{11} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 3240 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**4*(3*x**2+5*x+2)**(7/2),x)

[Out]

-Integral(-32292*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-142182*x**2*sqrt(3*x**
2 + 5*x + 2), x) - Integral(-363291*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-
594106*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-644932*x**5*sqrt(3*x**2 + 5*x
+ 2), x) - Integral(-463440*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(-209413*
x**7*sqrt(3*x**2 + 5*x + 2), x) - Integral(-49624*x**8*sqrt(3*x**2 + 5*x + 2), x
) - Integral(-504*x**9*sqrt(3*x**2 + 5*x + 2), x) - Integral(2592*x**10*sqrt(3*x
**2 + 5*x + 2), x) - Integral(432*x**11*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3
240*sqrt(3*x**2 + 5*x + 2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.29388, size = 147, normalized size = 0.64 \[ -\frac{1}{67077144576} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (2 \,{\left (48 \,{\left (54 \,{\left (4 \,{\left (6 \,{\left (72 \, x + 473\right )} x - 409\right )} x - 258889\right )} x - 66586273\right )} x - 8082617507\right )} x - 26220538883\right )} x - 1042194858901\right )} x - 4773502588153\right )} x - 19901779363361\right )} x - 40629353336575\right )} x - 97720614817745\right )} x - 17510968283403\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{2595845}{30958682112} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(2*x + 3)^4*(x - 5),x, algorithm="giac")

[Out]

-1/67077144576*(2*(12*(6*(8*(6*(36*(2*(48*(54*(4*(6*(72*x + 473)*x - 409)*x - 25
8889)*x - 66586273)*x - 8082617507)*x - 26220538883)*x - 1042194858901)*x - 4773
502588153)*x - 19901779363361)*x - 40629353336575)*x - 97720614817745)*x - 17510
968283403)*sqrt(3*x^2 + 5*x + 2) - 2595845/30958682112*sqrt(3)*ln(abs(-2*sqrt(3)
*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))