3.2031 \(\int \frac{(2+3 x)^5}{\sqrt{1-2 x} (3+5 x)^2} \, dx\)

Optimal. Leaf size=113 \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{55 (5 x+3)}-\frac{78 \sqrt{1-2 x} (3 x+2)^3}{1925}-\frac{1668 \sqrt{1-2 x} (3 x+2)^2}{6875}-\frac{6 \sqrt{1-2 x} (19875 x+59708)}{34375}-\frac{332 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}} \]

[Out]

(-1668*Sqrt[1 - 2*x]*(2 + 3*x)^2)/6875 - (78*Sqrt[1 - 2*x]*(2 + 3*x)^3)/1925 - (
Sqrt[1 - 2*x]*(2 + 3*x)^4)/(55*(3 + 5*x)) - (6*Sqrt[1 - 2*x]*(59708 + 19875*x))/
34375 - (332*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(34375*Sqrt[55])

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Rubi [A]  time = 0.224328, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{\sqrt{1-2 x} (3 x+2)^4}{55 (5 x+3)}-\frac{78 \sqrt{1-2 x} (3 x+2)^3}{1925}-\frac{1668 \sqrt{1-2 x} (3 x+2)^2}{6875}-\frac{6 \sqrt{1-2 x} (19875 x+59708)}{34375}-\frac{332 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{34375 \sqrt{55}} \]

Antiderivative was successfully verified.

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

[Out]

(-1668*Sqrt[1 - 2*x]*(2 + 3*x)^2)/6875 - (78*Sqrt[1 - 2*x]*(2 + 3*x)^3)/1925 - (
Sqrt[1 - 2*x]*(2 + 3*x)^4)/(55*(3 + 5*x)) - (6*Sqrt[1 - 2*x]*(59708 + 19875*x))/
34375 - (332*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(34375*Sqrt[55])

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Rubi in Sympy [A]  time = 25.5487, size = 99, normalized size = 0.88 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}}{55 \left (5 x + 3\right )} - \frac{78 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}}{1925} - \frac{1668 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}}{6875} - \frac{\sqrt{- 2 x + 1} \left (12521250 x + 37616040\right )}{3609375} - \frac{332 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1890625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-2*x + 1)*(3*x + 2)**4/(55*(5*x + 3)) - 78*sqrt(-2*x + 1)*(3*x + 2)**3/192
5 - 1668*sqrt(-2*x + 1)*(3*x + 2)**2/6875 - sqrt(-2*x + 1)*(12521250*x + 3761604
0)/3609375 - 332*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/1890625

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Mathematica [A]  time = 0.112031, size = 68, normalized size = 0.6 \[ \frac{-\frac{55 \sqrt{1-2 x} \left (1670625 x^4+6994350 x^3+13532310 x^2+20175210 x+8527768\right )}{5 x+3}-2324 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{13234375} \]

Antiderivative was successfully verified.

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

[Out]

((-55*Sqrt[1 - 2*x]*(8527768 + 20175210*x + 13532310*x^2 + 6994350*x^3 + 1670625
*x^4))/(3 + 5*x) - 2324*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/13234375

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Maple [A]  time = 0.018, size = 72, normalized size = 0.6 \[{\frac{243}{1400} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{8829}{5000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{35703}{5000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{434043}{25000}\sqrt{1-2\,x}}+{\frac{2}{171875}\sqrt{1-2\,x} \left ( -{\frac{6}{5}}-2\,x \right ) ^{-1}}-{\frac{332\,\sqrt{55}}{1890625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243/1400*(1-2*x)^(7/2)-8829/5000*(1-2*x)^(5/2)+35703/5000*(1-2*x)^(3/2)-434043/2
5000*(1-2*x)^(1/2)+2/171875*(1-2*x)^(1/2)/(-6/5-2*x)-332/1890625*arctanh(1/11*55
^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.51409, size = 120, normalized size = 1.06 \[ \frac{243}{1400} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{8829}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{35703}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{166}{1890625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{434043}{25000} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{34375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243/1400*(-2*x + 1)^(7/2) - 8829/5000*(-2*x + 1)^(5/2) + 35703/5000*(-2*x + 1)^(
3/2) + 166/1890625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqr
t(-2*x + 1))) - 434043/25000*sqrt(-2*x + 1) - 1/34375*sqrt(-2*x + 1)/(5*x + 3)

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Fricas [A]  time = 0.24125, size = 107, normalized size = 0.95 \[ -\frac{\sqrt{55}{\left (\sqrt{55}{\left (1670625 \, x^{4} + 6994350 \, x^{3} + 13532310 \, x^{2} + 20175210 \, x + 8527768\right )} \sqrt{-2 \, x + 1} - 1162 \,{\left (5 \, x + 3\right )} \log \left (\frac{\sqrt{55}{\left (5 \, x - 8\right )} + 55 \, \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )}}{13234375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/13234375*sqrt(55)*(sqrt(55)*(1670625*x^4 + 6994350*x^3 + 13532310*x^2 + 20175
210*x + 8527768)*sqrt(-2*x + 1) - 1162*(5*x + 3)*log((sqrt(55)*(5*x - 8) + 55*sq
rt(-2*x + 1))/(5*x + 3)))/(5*x + 3)

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.213421, size = 143, normalized size = 1.27 \[ -\frac{243}{1400} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{8829}{5000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{35703}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{166}{1890625} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{434043}{25000} \, \sqrt{-2 \, x + 1} - \frac{\sqrt{-2 \, x + 1}}{34375 \,{\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243/1400*(2*x - 1)^3*sqrt(-2*x + 1) - 8829/5000*(2*x - 1)^2*sqrt(-2*x + 1) + 35
703/5000*(-2*x + 1)^(3/2) + 166/1890625*sqrt(55)*ln(1/2*abs(-2*sqrt(55) + 10*sqr
t(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 434043/25000*sqrt(-2*x + 1) - 1/34
375*sqrt(-2*x + 1)/(5*x + 3)