3.2065 \(\int \frac{3+5 x}{(1-2 x)^{3/2} (2+3 x)^6} \, dx\)

Optimal. Leaf size=150 \[ -\frac{369 \sqrt{1-2 x}}{33614 (3 x+2)}-\frac{123 \sqrt{1-2 x}}{4802 (3 x+2)^2}-\frac{123 \sqrt{1-2 x}}{1715 (3 x+2)^3}-\frac{369 \sqrt{1-2 x}}{1715 (3 x+2)^4}+\frac{328}{735 \sqrt{1-2 x} (3 x+2)^4}+\frac{1}{105 \sqrt{1-2 x} (3 x+2)^5}-\frac{123 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{16807} \]

[Out]

1/(105*Sqrt[1 - 2*x]*(2 + 3*x)^5) + 328/(735*Sqrt[1 - 2*x]*(2 + 3*x)^4) - (369*S
qrt[1 - 2*x])/(1715*(2 + 3*x)^4) - (123*Sqrt[1 - 2*x])/(1715*(2 + 3*x)^3) - (123
*Sqrt[1 - 2*x])/(4802*(2 + 3*x)^2) - (369*Sqrt[1 - 2*x])/(33614*(2 + 3*x)) - (12
3*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/16807

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Rubi [A]  time = 0.164618, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{369 \sqrt{1-2 x}}{33614 (3 x+2)}-\frac{123 \sqrt{1-2 x}}{4802 (3 x+2)^2}-\frac{123 \sqrt{1-2 x}}{1715 (3 x+2)^3}-\frac{369 \sqrt{1-2 x}}{1715 (3 x+2)^4}+\frac{328}{735 \sqrt{1-2 x} (3 x+2)^4}+\frac{1}{105 \sqrt{1-2 x} (3 x+2)^5}-\frac{123 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{16807} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 5*x)/((1 - 2*x)^(3/2)*(2 + 3*x)^6),x]

[Out]

1/(105*Sqrt[1 - 2*x]*(2 + 3*x)^5) + 328/(735*Sqrt[1 - 2*x]*(2 + 3*x)^4) - (369*S
qrt[1 - 2*x])/(1715*(2 + 3*x)^4) - (123*Sqrt[1 - 2*x])/(1715*(2 + 3*x)^3) - (123
*Sqrt[1 - 2*x])/(4802*(2 + 3*x)^2) - (369*Sqrt[1 - 2*x])/(33614*(2 + 3*x)) - (12
3*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/16807

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Rubi in Sympy [A]  time = 16.7619, size = 133, normalized size = 0.89 \[ - \frac{369 \sqrt{- 2 x + 1}}{33614 \left (3 x + 2\right )} - \frac{123 \sqrt{- 2 x + 1}}{4802 \left (3 x + 2\right )^{2}} - \frac{123 \sqrt{- 2 x + 1}}{1715 \left (3 x + 2\right )^{3}} - \frac{123 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{117649} + \frac{246}{1715 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} - \frac{41}{735 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{4}} + \frac{1}{105 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)/(1-2*x)**(3/2)/(2+3*x)**6,x)

[Out]

-369*sqrt(-2*x + 1)/(33614*(3*x + 2)) - 123*sqrt(-2*x + 1)/(4802*(3*x + 2)**2) -
123*sqrt(-2*x + 1)/(1715*(3*x + 2)**3) - 123*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x
+ 1)/7)/117649 + 246/(1715*sqrt(-2*x + 1)*(3*x + 2)**3) - 41/(735*sqrt(-2*x + 1)
*(3*x + 2)**4) + 1/(105*sqrt(-2*x + 1)*(3*x + 2)**5)

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Mathematica [A]  time = 0.123837, size = 73, normalized size = 0.49 \[ \frac{\frac{7 \left (298890 x^5+880065 x^4+964197 x^3+430992 x^2+8774 x-32894\right )}{\sqrt{1-2 x} (3 x+2)^5}-1230 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1176490} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 5*x)/((1 - 2*x)^(3/2)*(2 + 3*x)^6),x]

[Out]

((7*(-32894 + 8774*x + 430992*x^2 + 964197*x^3 + 880065*x^4 + 298890*x^5))/(Sqrt
[1 - 2*x]*(2 + 3*x)^5) - 1230*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/1176490

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Maple [A]  time = 0.02, size = 84, normalized size = 0.6 \[{\frac{352}{117649}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{7776}{117649\, \left ( -4-6\,x \right ) ^{5}} \left ({\frac{509}{32} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{7903}{48} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{29302}{45} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{169099}{144} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{6321833}{7776}\sqrt{1-2\,x}} \right ) }-{\frac{123\,\sqrt{21}}{117649}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)/(1-2*x)^(3/2)/(2+3*x)^6,x)

[Out]

352/117649/(1-2*x)^(1/2)+7776/117649*(509/32*(1-2*x)^(9/2)-7903/48*(1-2*x)^(7/2)
+29302/45*(1-2*x)^(5/2)-169099/144*(1-2*x)^(3/2)+6321833/7776*(1-2*x)^(1/2))/(-4
-6*x)^5-123/117649*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50043, size = 185, normalized size = 1.23 \[ \frac{123}{235298} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{149445 \,{\left (2 \, x - 1\right )}^{5} + 1627290 \,{\left (2 \, x - 1\right )}^{4} + 6943104 \,{\left (2 \, x - 1\right )}^{3} + 14283990 \,{\left (2 \, x - 1\right )}^{2} + 27141590 \, x - 9345035}{84035 \,{\left (243 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 2835 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 13230 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 30870 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 36015 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 16807 \, \sqrt{-2 \, x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)/((3*x + 2)^6*(-2*x + 1)^(3/2)),x, algorithm="maxima")

[Out]

123/235298*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) - 1/84035*(149445*(2*x - 1)^5 + 1627290*(2*x - 1)^4 + 6943104*(2*x - 1)^3
+ 14283990*(2*x - 1)^2 + 27141590*x - 9345035)/(243*(-2*x + 1)^(11/2) - 2835*(-2
*x + 1)^(9/2) + 13230*(-2*x + 1)^(7/2) - 30870*(-2*x + 1)^(5/2) + 36015*(-2*x +
1)^(3/2) - 16807*sqrt(-2*x + 1))

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Fricas [A]  time = 0.240928, size = 185, normalized size = 1.23 \[ \frac{\sqrt{7}{\left (615 \, \sqrt{3}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{7}{\left (3 \, x - 5\right )} + 7 \, \sqrt{3} \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{7}{\left (298890 \, x^{5} + 880065 \, x^{4} + 964197 \, x^{3} + 430992 \, x^{2} + 8774 \, x - 32894\right )}\right )}}{1176490 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)/((3*x + 2)^6*(-2*x + 1)^(3/2)),x, algorithm="fricas")

[Out]

1/1176490*sqrt(7)*(615*sqrt(3)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x +
32)*sqrt(-2*x + 1)*log((sqrt(7)*(3*x - 5) + 7*sqrt(3)*sqrt(-2*x + 1))/(3*x + 2)
) + sqrt(7)*(298890*x^5 + 880065*x^4 + 964197*x^3 + 430992*x^2 + 8774*x - 32894)
)/((243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*sqrt(-2*x + 1))

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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((3+5*x)/(1-2*x)**(3/2)/(2+3*x)**6,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.220778, size = 169, normalized size = 1.13 \[ \frac{123}{235298} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{352}{117649 \, \sqrt{-2 \, x + 1}} - \frac{618435 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 6401430 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 25316928 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 45656730 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 31609165 \, \sqrt{-2 \, x + 1}}{18823840 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)/((3*x + 2)^6*(-2*x + 1)^(3/2)),x, algorithm="giac")

[Out]

123/235298*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqr
t(-2*x + 1))) + 352/117649/sqrt(-2*x + 1) - 1/18823840*(618435*(2*x - 1)^4*sqrt(
-2*x + 1) + 6401430*(2*x - 1)^3*sqrt(-2*x + 1) + 25316928*(2*x - 1)^2*sqrt(-2*x
+ 1) - 45656730*(-2*x + 1)^(3/2) + 31609165*sqrt(-2*x + 1))/(3*x + 2)^5