3.2489 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=144 \[ -\frac{7396875 \sqrt{1-2 x}}{30184 \sqrt{5 x+3}}+\frac{44475 \sqrt{1-2 x}}{2744 (3 x+2) \sqrt{5 x+3}}+\frac{255 \sqrt{1-2 x}}{196 (3 x+2)^2 \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 \sqrt{5 x+3}}+\frac{4616025 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

[Out]

(-7396875*Sqrt[1 - 2*x])/(30184*Sqrt[3 + 5*x]) + Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*Sq
rt[3 + 5*x]) + (255*Sqrt[1 - 2*x])/(196*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (44475*Sqrt
[1 - 2*x])/(2744*(2 + 3*x)*Sqrt[3 + 5*x]) + (4616025*ArcTan[Sqrt[1 - 2*x]/(Sqrt[
7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Rubi [A]  time = 0.322053, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{7396875 \sqrt{1-2 x}}{30184 \sqrt{5 x+3}}+\frac{44475 \sqrt{1-2 x}}{2744 (3 x+2) \sqrt{5 x+3}}+\frac{255 \sqrt{1-2 x}}{196 (3 x+2)^2 \sqrt{5 x+3}}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 \sqrt{5 x+3}}+\frac{4616025 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

(-7396875*Sqrt[1 - 2*x])/(30184*Sqrt[3 + 5*x]) + Sqrt[1 - 2*x]/(7*(2 + 3*x)^3*Sq
rt[3 + 5*x]) + (255*Sqrt[1 - 2*x])/(196*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (44475*Sqrt
[1 - 2*x])/(2744*(2 + 3*x)*Sqrt[3 + 5*x]) + (4616025*ArcTan[Sqrt[1 - 2*x]/(Sqrt[
7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Rubi in Sympy [A]  time = 28.078, size = 131, normalized size = 0.91 \[ - \frac{7396875 \sqrt{- 2 x + 1}}{30184 \sqrt{5 x + 3}} + \frac{44475 \sqrt{- 2 x + 1}}{2744 \left (3 x + 2\right ) \sqrt{5 x + 3}} + \frac{255 \sqrt{- 2 x + 1}}{196 \left (3 x + 2\right )^{2} \sqrt{5 x + 3}} + \frac{\sqrt{- 2 x + 1}}{7 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}} + \frac{4616025 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{19208} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2+3*x)**4/(3+5*x)**(3/2)/(1-2*x)**(1/2),x)

[Out]

-7396875*sqrt(-2*x + 1)/(30184*sqrt(5*x + 3)) + 44475*sqrt(-2*x + 1)/(2744*(3*x
+ 2)*sqrt(5*x + 3)) + 255*sqrt(-2*x + 1)/(196*(3*x + 2)**2*sqrt(5*x + 3)) + sqrt
(-2*x + 1)/(7*(3*x + 2)**3*sqrt(5*x + 3)) + 4616025*sqrt(7)*atan(sqrt(7)*sqrt(-2
*x + 1)/(7*sqrt(5*x + 3)))/19208

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Mathematica [A]  time = 0.106817, size = 82, normalized size = 0.57 \[ \frac{50776275 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )-\frac{14 \sqrt{1-2 x} \left (199715625 x^3+395028225 x^2+260298990 x+57135248\right )}{(3 x+2)^3 \sqrt{5 x+3}}}{422576} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

((-14*Sqrt[1 - 2*x]*(57135248 + 260298990*x + 395028225*x^2 + 199715625*x^3))/((
2 + 3*x)^3*Sqrt[3 + 5*x]) + 50776275*Sqrt[7]*ArcTan[(-20 - 37*x)/(2*Sqrt[7 - 14*
x]*Sqrt[3 + 5*x])])/422576

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Maple [B]  time = 0.025, size = 250, normalized size = 1.7 \[ -{\frac{1}{422576\, \left ( 2+3\,x \right ) ^{3}} \left ( 6854797125\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+5530395150\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2+3*x)^4/(3+5*x)^(3/2)/(1-2*x)^(1/2),x)

[Out]

-1/422576*(6854797125*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
*x^4+17822472525*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+
17365486050*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+27960
18750*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+5530395150*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arct
an(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)+
799893472*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^3/(-10*x^2-x+3)^(1/2)/(3+5*
x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{4} \sqrt{-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^4*sqrt(-2*x + 1)),x, algorithm="maxima")

[Out]

integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^4*sqrt(-2*x + 1)), x)

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Fricas [A]  time = 0.237866, size = 147, normalized size = 1.02 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (199715625 \, x^{3} + 395028225 \, x^{2} + 260298990 \, x + 57135248\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 50776275 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{422576 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^4*sqrt(-2*x + 1)),x, algorithm="fricas")

[Out]

-1/422576*sqrt(7)*(2*sqrt(7)*(199715625*x^3 + 395028225*x^2 + 260298990*x + 5713
5248)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 50776275*(135*x^4 + 351*x^3 + 342*x^2 + 148
*x + 24)*arctan(1/14*sqrt(7)*(37*x + 20)/(sqrt(5*x + 3)*sqrt(-2*x + 1))))/(135*x
^4 + 351*x^3 + 342*x^2 + 148*x + 24)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2+3*x)**4/(3+5*x)**(3/2)/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.344673, size = 509, normalized size = 3.53 \[ -\frac{923205}{76832} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{125}{22} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{7425 \,{\left (487 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 217280 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 25693248 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{1372 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^4*sqrt(-2*x + 1)),x, algorithm="giac")

[Out]

-923205/76832*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((s
qrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22)))) - 125/22*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))) - 7425/1372*(487*sqrt(1
0)*((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)))^5 + 217280*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3
+ 25693248*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2
+ 280)^3