3.2313 \(\int \frac{(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^3} \, dx\)

Optimal. Leaf size=300 \[ -\frac{(5-4 x) (2 x+1)^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}-\frac{(957-592 x) \sqrt{2 x+1}}{9610 \left (5 x^2+3 x+2\right )}-\frac{\sqrt{\frac{1}{310} \left (1806875 \sqrt{35}-9651062\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{9610}+\frac{\sqrt{\frac{1}{310} \left (1806875 \sqrt{35}-9651062\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{9610}-\frac{\sqrt{\frac{1}{310} \left (9651062+1806875 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{4805}+\frac{\sqrt{\frac{1}{310} \left (9651062+1806875 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{4805} \]

[Out]

-((5 - 4*x)*(1 + 2*x)^(5/2))/(62*(2 + 3*x + 5*x^2)^2) - ((957 - 592*x)*Sqrt[1 +
2*x])/(9610*(2 + 3*x + 5*x^2)) - (Sqrt[(9651062 + 1806875*Sqrt[35])/310]*ArcTan[
(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/4805 + (
Sqrt[(9651062 + 1806875*Sqrt[35])/310]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt
[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/4805 - (Sqrt[(-9651062 + 1806875*Sqrt[35])
/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/9610
+ (Sqrt[(-9651062 + 1806875*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])
]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/9610

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Rubi [A]  time = 1.2995, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{(5-4 x) (2 x+1)^{5/2}}{62 \left (5 x^2+3 x+2\right )^2}-\frac{(957-592 x) \sqrt{2 x+1}}{9610 \left (5 x^2+3 x+2\right )}-\frac{\sqrt{\frac{1}{310} \left (1806875 \sqrt{35}-9651062\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{9610}+\frac{\sqrt{\frac{1}{310} \left (1806875 \sqrt{35}-9651062\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )}{9610}-\frac{\sqrt{\frac{1}{310} \left (9651062+1806875 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{4805}+\frac{\sqrt{\frac{1}{310} \left (9651062+1806875 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )}{4805} \]

Antiderivative was successfully verified.

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

[Out]

-((5 - 4*x)*(1 + 2*x)^(5/2))/(62*(2 + 3*x + 5*x^2)^2) - ((957 - 592*x)*Sqrt[1 +
2*x])/(9610*(2 + 3*x + 5*x^2)) - (Sqrt[(9651062 + 1806875*Sqrt[35])/310]*ArcTan[
(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/4805 + (
Sqrt[(9651062 + 1806875*Sqrt[35])/310]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt
[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/4805 - (Sqrt[(-9651062 + 1806875*Sqrt[35])
/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/9610
+ (Sqrt[(-9651062 + 1806875*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])
]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/9610

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Rubi in Sympy [A]  time = 78.9642, size = 406, normalized size = 1.35 \[ - \frac{\left (- 592 x + 957\right ) \sqrt{2 x + 1}}{9610 \left (5 x^{2} + 3 x + 2\right )} - \frac{\left (- 4 x + 5\right ) \left (2 x + 1\right )^{\frac{5}{2}}}{62 \left (5 x^{2} + 3 x + 2\right )^{2}} - \frac{\sqrt{14} \left (- \frac{544 \sqrt{35}}{5} + 1253\right ) \log{\left (2 x - \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{134540 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{14} \left (- \frac{544 \sqrt{35}}{5} + 1253\right ) \log{\left (2 x + \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \sqrt{2 x + 1}}{5} + 1 + \frac{\sqrt{35}}{5} \right )}}{134540 \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- \frac{1088 \sqrt{35}}{5} + 2506\right )}{10} + \frac{2506 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} - \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{67270 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} + \frac{\sqrt{35} \left (- \frac{\sqrt{10} \sqrt{2 + \sqrt{35}} \left (- \frac{1088 \sqrt{35}}{5} + 2506\right )}{10} + \frac{2506 \sqrt{10} \sqrt{2 + \sqrt{35}}}{5}\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (\sqrt{2 x + 1} + \frac{\sqrt{20 + 10 \sqrt{35}}}{10}\right )}{\sqrt{-2 + \sqrt{35}}} \right )}}{67270 \sqrt{-2 + \sqrt{35}} \sqrt{2 + \sqrt{35}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*x)**(7/2)/(5*x**2+3*x+2)**3,x)

[Out]

-(-592*x + 957)*sqrt(2*x + 1)/(9610*(5*x**2 + 3*x + 2)) - (-4*x + 5)*(2*x + 1)**
(5/2)/(62*(5*x**2 + 3*x + 2)**2) - sqrt(14)*(-544*sqrt(35)/5 + 1253)*log(2*x - s
qrt(10)*sqrt(2 + sqrt(35))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(134540*sqrt(2 + sq
rt(35))) + sqrt(14)*(-544*sqrt(35)/5 + 1253)*log(2*x + sqrt(10)*sqrt(2 + sqrt(35
))*sqrt(2*x + 1)/5 + 1 + sqrt(35)/5)/(134540*sqrt(2 + sqrt(35))) + sqrt(35)*(-sq
rt(10)*sqrt(2 + sqrt(35))*(-1088*sqrt(35)/5 + 2506)/10 + 2506*sqrt(10)*sqrt(2 +
sqrt(35))/5)*atan(sqrt(10)*(sqrt(2*x + 1) - sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 +
sqrt(35)))/(67270*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35))) + sqrt(35)*(-sqrt(10)
*sqrt(2 + sqrt(35))*(-1088*sqrt(35)/5 + 2506)/10 + 2506*sqrt(10)*sqrt(2 + sqrt(3
5))/5)*atan(sqrt(10)*(sqrt(2*x + 1) + sqrt(20 + 10*sqrt(35))/10)/sqrt(-2 + sqrt(
35)))/(67270*sqrt(-2 + sqrt(35))*sqrt(2 + sqrt(35)))

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Mathematica [C]  time = 1.25534, size = 159, normalized size = 0.53 \[ \frac{\frac{155 \sqrt{2 x+1} \left (5440 x^3-3629 x^2-4167 x-2689\right )}{2 \left (5 x^2+3 x+2\right )^2}+\frac{\left (16864-7353 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2-i \sqrt{31}}}\right )}{\sqrt{-\frac{1}{5} i \left (\sqrt{31}-2 i\right )}}+\frac{\left (16864+7353 i \sqrt{31}\right ) \tan ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{-2+i \sqrt{31}}}\right )}{\sqrt{\frac{1}{5} i \left (\sqrt{31}+2 i\right )}}}{744775} \]

Antiderivative was successfully verified.

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

[Out]

((155*Sqrt[1 + 2*x]*(-2689 - 4167*x - 3629*x^2 + 5440*x^3))/(2*(2 + 3*x + 5*x^2)
^2) + ((16864 - (7353*I)*Sqrt[31])*ArcTan[Sqrt[5 + 10*x]/Sqrt[-2 - I*Sqrt[31]]])
/Sqrt[(-I/5)*(-2*I + Sqrt[31])] + ((16864 + (7353*I)*Sqrt[31])*ArcTan[Sqrt[5 + 1
0*x]/Sqrt[-2 + I*Sqrt[31]]])/Sqrt[(I/5)*(2*I + Sqrt[31])])/744775

_______________________________________________________________________________________

Maple [B]  time = 0.053, size = 662, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1600*(17/24025*(1+2*x)^(7/2)-11789/3844000*(1+2*x)^(5/2)+1771/961000*(1+2*x)^(3/
2)-8771/3844000*(1+2*x)^(1/2))/(5*(1+2*x)^2-8*x+3)^2-7353/2979100*ln(-(2*5^(1/2)
*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5^(1/2)*7^(1/2)+10*x+5)*(2*5^(1/2)*7^(1/
2)+4)^(1/2)*5^(1/2)+451/297910*ln(-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(
1/2)+5^(1/2)*7^(1/2)+10*x+5)*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-7353/297910/(10
*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)+10*(1+2*
x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)+451/148955/(10*5^
(1/2)*7^(1/2)-20)^(1/2)*arctan((-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)+10*(1+2*x)^
(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)*5^(1/2)*7^(1/2)+358/
4805/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)+
10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*5^(1/2)*7^(1/2)+7353/2979100*ln
(5^(1/2)*7^(1/2)+10*x+5+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))*(2*5^
(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)-451/297910*ln(5^(1/2)*7^(1/2)+10*x+5+(2*5^(1/2)*7
^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2))*7^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-7353/
297910/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)+(2*5^(1/2)*7^(1/2)
+4)^(1/2)*5^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)+451/1489
55/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)+(2*5^(1/2)*7^(1/2)+4)^
(1/2)*5^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*(2*5^(1/2)*7^(1/2)+4)*5^(1/2)*7^(1
/2)+358/4805/(10*5^(1/2)*7^(1/2)-20)^(1/2)*arctan((10*(1+2*x)^(1/2)+(2*5^(1/2)*7
^(1/2)+4)^(1/2)*5^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))*5^(1/2)*7^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{7}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^3, x)

_______________________________________________________________________________________

Fricas [A]  time = 0.266566, size = 1183, normalized size = 3.94 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/175766900*sqrt(118)*(155*sqrt(118)*(52501777280*x^3 - 35023703998*x^2 - 180687
5*sqrt(35)*(5440*x^3 - 3629*x^2 - 4167*x - 2689) - 40215975354*x - 25951705718)*
sqrt(2*x + 1)*sqrt((9651062*sqrt(35) - 63240625)/(34876525302500*sqrt(35) - 2074
10902024719)) - 102361876*3045875^(1/4)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4)*ar
ctan(59*3045875^(1/4)*(902*sqrt(35)*sqrt(31) - 7353*sqrt(31))/(sqrt(9145)*sqrt(1
18)*(1806875*sqrt(35) - 9651062)*sqrt((3045875^(1/4)*sqrt(118)*(1140095851593998
833449962*sqrt(35)*sqrt(31) - 6745474887126660693511557*sqrt(31))*sqrt(2*x + 1)*
sqrt((9651062*sqrt(35) - 63240625)/(34876525302500*sqrt(35) - 207410902024719))
+ 1241167876198096285903010*sqrt(35)*(2*x + 1) + 59*sqrt(35)*(420734873287490266
4078*sqrt(35) - 24897637857221863934375) - 14689606335760899721281250*x - 734480
3167880449860640625)/(4207348732874902664078*sqrt(35) - 24897637857221863934375)
)*sqrt((9651062*sqrt(35) - 63240625)/(34876525302500*sqrt(35) - 207410902024719)
) + 295*sqrt(118)*(1806875*sqrt(35)*sqrt(31) - 9651062*sqrt(31))*sqrt(2*x + 1)*s
qrt((9651062*sqrt(35) - 63240625)/(34876525302500*sqrt(35) - 207410902024719)) +
1829*3045875^(1/4)*(179*sqrt(35) - 544))) - 102361876*3045875^(1/4)*(25*x^4 + 3
0*x^3 + 29*x^2 + 12*x + 4)*arctan(59*3045875^(1/4)*(902*sqrt(35)*sqrt(31) - 7353
*sqrt(31))/(sqrt(9145)*sqrt(118)*(1806875*sqrt(35) - 9651062)*sqrt(-(3045875^(1/
4)*sqrt(118)*(1140095851593998833449962*sqrt(35)*sqrt(31) - 67454748871266606935
11557*sqrt(31))*sqrt(2*x + 1)*sqrt((9651062*sqrt(35) - 63240625)/(34876525302500
*sqrt(35) - 207410902024719)) - 1241167876198096285903010*sqrt(35)*(2*x + 1) - 5
9*sqrt(35)*(4207348732874902664078*sqrt(35) - 24897637857221863934375) + 1468960
6335760899721281250*x + 7344803167880449860640625)/(4207348732874902664078*sqrt(
35) - 24897637857221863934375))*sqrt((9651062*sqrt(35) - 63240625)/(348765253025
00*sqrt(35) - 207410902024719)) + 295*sqrt(118)*(1806875*sqrt(35)*sqrt(31) - 965
1062*sqrt(31))*sqrt(2*x + 1)*sqrt((9651062*sqrt(35) - 63240625)/(34876525302500*
sqrt(35) - 207410902024719)) - 1829*3045875^(1/4)*(179*sqrt(35) - 544))) - 30458
75^(1/4)*(1806875*sqrt(35)*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 9651
062*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4))*log(13723215625*(3045875^(1/
4)*sqrt(118)*(1140095851593998833449962*sqrt(35)*sqrt(31) - 67454748871266606935
11557*sqrt(31))*sqrt(2*x + 1)*sqrt((9651062*sqrt(35) - 63240625)/(34876525302500
*sqrt(35) - 207410902024719)) + 1241167876198096285903010*sqrt(35)*(2*x + 1) + 5
9*sqrt(35)*(4207348732874902664078*sqrt(35) - 24897637857221863934375) - 1468960
6335760899721281250*x - 7344803167880449860640625)/(4207348732874902664078*sqrt(
35) - 24897637857221863934375)) + 3045875^(1/4)*(1806875*sqrt(35)*sqrt(31)*(25*x
^4 + 30*x^3 + 29*x^2 + 12*x + 4) - 9651062*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 +
12*x + 4))*log(-13723215625*(3045875^(1/4)*sqrt(118)*(1140095851593998833449962*
sqrt(35)*sqrt(31) - 6745474887126660693511557*sqrt(31))*sqrt(2*x + 1)*sqrt((9651
062*sqrt(35) - 63240625)/(34876525302500*sqrt(35) - 207410902024719)) - 12411678
76198096285903010*sqrt(35)*(2*x + 1) - 59*sqrt(35)*(4207348732874902664078*sqrt(
35) - 24897637857221863934375) + 14689606335760899721281250*x + 7344803167880449
860640625)/(4207348732874902664078*sqrt(35) - 24897637857221863934375)))/((24127
6550*x^4 + 289531860*x^3 + 279880798*x^2 - 1806875*sqrt(35)*(25*x^4 + 30*x^3 + 2
9*x^2 + 12*x + 4) + 115812744*x + 38604248)*sqrt((9651062*sqrt(35) - 63240625)/(
34876525302500*sqrt(35) - 207410902024719)))

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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)**(7/2)/(5*x**2+3*x+2)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, x + 1\right )}^{\frac{7}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^3, x)