3.1878 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^6}{3+5 x} \, dx\)

Optimal. Leaf size=134 \[ \frac{243}{800} (1-2 x)^{15/2}-\frac{43011 (1-2 x)^{13/2}}{10400}+\frac{507627 (1-2 x)^{11/2}}{22000}-\frac{665817 (1-2 x)^{9/2}}{10000}+\frac{70752609 (1-2 x)^{7/2}}{700000}-\frac{167115051 (1-2 x)^{5/2}}{2500000}+\frac{2 (1-2 x)^{3/2}}{234375}+\frac{22 \sqrt{1-2 x}}{390625}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{390625} \]

[Out]

(22*Sqrt[1 - 2*x])/390625 + (2*(1 - 2*x)^(3/2))/234375 - (167115051*(1 - 2*x)^(5
/2))/2500000 + (70752609*(1 - 2*x)^(7/2))/700000 - (665817*(1 - 2*x)^(9/2))/1000
0 + (507627*(1 - 2*x)^(11/2))/22000 - (43011*(1 - 2*x)^(13/2))/10400 + (243*(1 -
2*x)^(15/2))/800 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/390625

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Rubi [A]  time = 0.119211, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{243}{800} (1-2 x)^{15/2}-\frac{43011 (1-2 x)^{13/2}}{10400}+\frac{507627 (1-2 x)^{11/2}}{22000}-\frac{665817 (1-2 x)^{9/2}}{10000}+\frac{70752609 (1-2 x)^{7/2}}{700000}-\frac{167115051 (1-2 x)^{5/2}}{2500000}+\frac{2 (1-2 x)^{3/2}}{234375}+\frac{22 \sqrt{1-2 x}}{390625}-\frac{22 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{390625} \]

Antiderivative was successfully verified.

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

[Out]

(22*Sqrt[1 - 2*x])/390625 + (2*(1 - 2*x)^(3/2))/234375 - (167115051*(1 - 2*x)^(5
/2))/2500000 + (70752609*(1 - 2*x)^(7/2))/700000 - (665817*(1 - 2*x)^(9/2))/1000
0 + (507627*(1 - 2*x)^(11/2))/22000 - (43011*(1 - 2*x)^(13/2))/10400 + (243*(1 -
2*x)^(15/2))/800 - (22*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/390625

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Rubi in Sympy [A]  time = 14.0578, size = 119, normalized size = 0.89 \[ \frac{243 \left (- 2 x + 1\right )^{\frac{15}{2}}}{800} - \frac{43011 \left (- 2 x + 1\right )^{\frac{13}{2}}}{10400} + \frac{507627 \left (- 2 x + 1\right )^{\frac{11}{2}}}{22000} - \frac{665817 \left (- 2 x + 1\right )^{\frac{9}{2}}}{10000} + \frac{70752609 \left (- 2 x + 1\right )^{\frac{7}{2}}}{700000} - \frac{167115051 \left (- 2 x + 1\right )^{\frac{5}{2}}}{2500000} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{234375} + \frac{22 \sqrt{- 2 x + 1}}{390625} - \frac{22 \sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{1953125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243*(-2*x + 1)**(15/2)/800 - 43011*(-2*x + 1)**(13/2)/10400 + 507627*(-2*x + 1)*
*(11/2)/22000 - 665817*(-2*x + 1)**(9/2)/10000 + 70752609*(-2*x + 1)**(7/2)/7000
00 - 167115051*(-2*x + 1)**(5/2)/2500000 + 2*(-2*x + 1)**(3/2)/234375 + 22*sqrt(
-2*x + 1)/390625 - 22*sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/1953125

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Mathematica [A]  time = 0.127264, size = 76, normalized size = 0.57 \[ \frac{-5 \sqrt{1-2 x} \left (45608062500 x^7+150857437500 x^6+174123928125 x^5+49094797500 x^4-61883481375 x^3-56176961670 x^2-9645684935 x+15379193944\right )-66066 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5865234375} \]

Antiderivative was successfully verified.

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

[Out]

(-5*Sqrt[1 - 2*x]*(15379193944 - 9645684935*x - 56176961670*x^2 - 61883481375*x^
3 + 49094797500*x^4 + 174123928125*x^5 + 150857437500*x^6 + 45608062500*x^7) - 6
6066*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5865234375

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Maple [A]  time = 0.013, size = 92, normalized size = 0.7 \[{\frac{2}{234375} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{167115051}{2500000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{70752609}{700000} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{665817}{10000} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}+{\frac{507627}{22000} \left ( 1-2\,x \right ) ^{{\frac{11}{2}}}}-{\frac{43011}{10400} \left ( 1-2\,x \right ) ^{{\frac{13}{2}}}}+{\frac{243}{800} \left ( 1-2\,x \right ) ^{{\frac{15}{2}}}}-{\frac{22\,\sqrt{55}}{1953125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{22}{390625}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/234375*(1-2*x)^(3/2)-167115051/2500000*(1-2*x)^(5/2)[phone]/700000*(1-2*x)^(
7/2)-665817/10000*(1-2*x)^(9/2)+507627/22000*(1-2*x)^(11/2)-43011/10400*(1-2*x)^
(13/2)+243/800*(1-2*x)^(15/2)-22/1953125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55
^(1/2)+22/390625*(1-2*x)^(1/2)

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Maxima [A]  time = 1.49478, size = 147, normalized size = 1.1 \[ \frac{243}{800} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} - \frac{43011}{10400} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{507627}{22000} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{665817}{10000} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{70752609}{700000} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{167115051}{2500000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{2}{234375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{1953125} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{22}{390625} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243/800*(-2*x + 1)^(15/2) - 43011/10400*(-2*x + 1)^(13/2) + 507627/22000*(-2*x +
1)^(11/2) - 665817/10000*(-2*x + 1)^(9/2) + 70752609/700000*(-2*x + 1)^(7/2) -
167115051/2500000*(-2*x + 1)^(5/2) + 2/234375*(-2*x + 1)^(3/2) + 11/1953125*sqrt
(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/3906
25*sqrt(-2*x + 1)

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Fricas [A]  time = 0.211562, size = 119, normalized size = 0.89 \[ -\frac{1}{5865234375} \, \sqrt{5}{\left (\sqrt{5}{\left (45608062500 \, x^{7} + 150857437500 \, x^{6} + 174123928125 \, x^{5} + 49094797500 \, x^{4} - 61883481375 \, x^{3} - 56176961670 \, x^{2} - 9645684935 \, x + 15379193944\right )} \sqrt{-2 \, x + 1} - 33033 \, \sqrt{11} \log \left (\frac{\sqrt{5}{\left (5 \, x - 8\right )} + 5 \, \sqrt{11} \sqrt{-2 \, x + 1}}{5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5865234375*sqrt(5)*(sqrt(5)*(45608062500*x^7 + 150857437500*x^6 + 17412392812
5*x^5 + 49094797500*x^4 - 61883481375*x^3 - 56176961670*x^2 - 9645684935*x + 153
79193944)*sqrt(-2*x + 1) - 33033*sqrt(11)*log((sqrt(5)*(5*x - 8) + 5*sqrt(11)*sq
rt(-2*x + 1))/(5*x + 3)))

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Sympy [A]  time = 25.4479, size = 158, normalized size = 1.18 \[ \frac{243 \left (- 2 x + 1\right )^{\frac{15}{2}}}{800} - \frac{43011 \left (- 2 x + 1\right )^{\frac{13}{2}}}{10400} + \frac{507627 \left (- 2 x + 1\right )^{\frac{11}{2}}}{22000} - \frac{665817 \left (- 2 x + 1\right )^{\frac{9}{2}}}{10000} + \frac{70752609 \left (- 2 x + 1\right )^{\frac{7}{2}}}{700000} - \frac{167115051 \left (- 2 x + 1\right )^{\frac{5}{2}}}{2500000} + \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}}}{234375} + \frac{22 \sqrt{- 2 x + 1}}{390625} + \frac{242 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 > \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}}{55} & \text{for}\: - 2 x + 1 < \frac{11}{5} \end{cases}\right )}{390625} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

243*(-2*x + 1)**(15/2)/800 - 43011*(-2*x + 1)**(13/2)/10400 + 507627*(-2*x + 1)*
*(11/2)/22000 - 665817*(-2*x + 1)**(9/2)/10000 + 70752609*(-2*x + 1)**(7/2)/7000
00 - 167115051*(-2*x + 1)**(5/2)/2500000 + 2*(-2*x + 1)**(3/2)/234375 + 22*sqrt(
-2*x + 1)/390625 + 242*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(-2*x + 1)/11)/55
, -2*x + 1 > 11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(-2*x + 1)/11)/55, -2*x + 1 <
11/5))/390625

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GIAC/XCAS [A]  time = 0.216279, size = 208, normalized size = 1.55 \[ -\frac{243}{800} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} - \frac{43011}{10400} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{507627}{22000} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{665817}{10000} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{70752609}{700000} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{167115051}{2500000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{2}{234375} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{11}{1953125} \, \sqrt{55}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{22}{390625} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-243/800*(2*x - 1)^7*sqrt(-2*x + 1) - 43011/10400*(2*x - 1)^6*sqrt(-2*x + 1) - 5
07627/22000*(2*x - 1)^5*sqrt(-2*x + 1) - 665817/10000*(2*x - 1)^4*sqrt(-2*x + 1)
- 70752609/700000*(2*x - 1)^3*sqrt(-2*x + 1) - 167115051/2500000*(2*x - 1)^2*sq
rt(-2*x + 1) + 2/234375*(-2*x + 1)^(3/2) + 11/1953125*sqrt(55)*ln(1/2*abs(-2*sqr
t(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 22/390625*sqrt(-2*x
+ 1)