3.2554 \(\int \frac{(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^3}{1815 (5 x+3)^{3/2}}-\frac{4487 \sqrt{1-2 x} (3 x+2)^2}{99825 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (1078860 x+2571547)}{5324000}-\frac{111321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4000 \sqrt{10}} \]

[Out]

(-107*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^4)/(11*Sq
rt[1 - 2*x]*(3 + 5*x)^(3/2)) - (4487*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(99825*Sqrt[3 +
5*x]) + (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2571547 + 1078860*x))/5324000 - (111321*
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4000*Sqrt[10])

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Rubi [A]  time = 0.270838, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} (5 x+3)^{3/2}}-\frac{107 \sqrt{1-2 x} (3 x+2)^3}{1815 (5 x+3)^{3/2}}-\frac{4487 \sqrt{1-2 x} (3 x+2)^2}{99825 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (1078860 x+2571547)}{5324000}-\frac{111321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4000 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-107*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(1815*(3 + 5*x)^(3/2)) + (7*(2 + 3*x)^4)/(11*Sq
rt[1 - 2*x]*(3 + 5*x)^(3/2)) - (4487*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(99825*Sqrt[3 +
5*x]) + (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2571547 + 1078860*x))/5324000 - (111321*
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4000*Sqrt[10])

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Rubi in Sympy [A]  time = 26.091, size = 133, normalized size = 0.94 \[ - \frac{107 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{1815 \left (5 x + 3\right )^{\frac{3}{2}}} - \frac{4487 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{99825 \sqrt{5 x + 3}} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{28320075 x}{4} + \frac{270012435}{16}\right )}{4991250} - \frac{111321 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{40000} + \frac{7 \left (3 x + 2\right )^{4}}{11 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-107*sqrt(-2*x + 1)*(3*x + 2)**3/(1815*(5*x + 3)**(3/2)) - 4487*sqrt(-2*x + 1)*(
3*x + 2)**2/(99825*sqrt(5*x + 3)) + sqrt(-2*x + 1)*sqrt(5*x + 3)*(28320075*x/4 +
270012435/16)/4991250 - 111321*sqrt(10)*asin(sqrt(22)*sqrt(5*x + 3)/11)/40000 +
7*(3*x + 2)**4/(11*sqrt(-2*x + 1)*(5*x + 3)**(3/2))

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Mathematica [A]  time = 0.20713, size = 83, normalized size = 0.58 \[ \frac{10 \left (-194059800 x^4-1128781170 x^3+612106475 x^2+1785872944 x+632498543\right )[phone] \sqrt{10-20 x} (5 x+3)^{3/2} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{159720000 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(10*(632498543 + 1785872944*x + 612106475*x^2 - 1128781170*x^3 - 194059800*x^4)
+ 444504753*Sqrt[10 - 20*x]*(3 + 5*x)^(3/2)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1
59720000*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.023, size = 168, normalized size = 1.2 \[ -{\frac{1}{-319440000+638880000\,x}\sqrt{1-2\,x} \left ( 22225237650\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-3881196000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-22575623400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-5334057036\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+12242129500\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-4000542777\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/319440000*(1-2*x)^(1/2)*(22225237650*10^(1/2)*arcsin(20/11*x+1/11)*x^3-388119
6000*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x^2-22575
623400*x^3*(-10*x^2-x+3)^(1/2)-5334057036*10^(1/2)*arcsin(20/11*x+1/11)*x+122421
29500*x^2*(-10*x^2-x+3)^(1/2)-4000542777*10^(1/2)*arcsin(20/11*x+1/11)[phone]
80*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-1+2*x)/(-10*x^2-x+3)
^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.51371, size = 151, normalized size = 1.06 \[ -\frac{243 \, x^{3}}{100 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{111321}{80000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{25353 \, x^{2}}{2000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1219513649 \, x}{79860000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{5270823773}{399300000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2}{103125 \,{\left (5 \, \sqrt{-10 \, x^{2} - x + 3} x + 3 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243/100*x^3/sqrt(-10*x^2 - x + 3) - 111321/80000*sqrt(5)*sqrt(2)*arcsin(20/11*x
+ 1/11) - 25353/2000*x^2/sqrt(-10*x^2 - x + 3) + 1219513649/79860000*x/sqrt(-10
*x^2 - x + 3) + 5270823773/399300000/sqrt(-10*x^2 - x + 3) - 2/103125/(5*sqrt(-1
0*x^2 - x + 3)*x + 3*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 0.233619, size = 140, normalized size = 0.99 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (194059800 \, x^{4} + 1128781170 \, x^{3} - 612106475 \, x^{2} - 1785872944 \, x - 632498543\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 444504753 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{319440000 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/319440000*sqrt(10)*(2*sqrt(10)*(194059800*x^4 + 1128781170*x^3 - 612106475*x^2
- 1785872944*x - 632498543)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 444504753*(50*x^3 +
35*x^2 - 12*x - 9)*arctan(1/20*sqrt(10)*(20*x + 1)/(sqrt(5*x + 3)*sqrt(-2*x + 1)
)))/(50*x^3 + 35*x^2 - 12*x - 9)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272929, size = 265, normalized size = 1.87 \[ -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{199650000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{111321}{40000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (215622 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 205 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 741559591 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{665500000 \,{\left (2 \, x - 1\right )}} - \frac{337 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{16637500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{1011 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{12478125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/199650000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 1
11321/40000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/665500000*(215622*(
12*sqrt(5)*(5*x + 3) + 205*sqrt(5))*(5*x + 3) - 741559591*sqrt(5))*sqrt(5*x + 3)
*sqrt(-10*x + 5)/(2*x - 1) - 337/16637500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) + 1/12478125*(1011*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt
(22))^3