3.1864 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=148 \[ \frac{59 (1-2 x)^{5/2}}{1890 (3 x+2)^5}-\frac{(1-2 x)^{5/2}}{378 (3 x+2)^6}-\frac{991 (1-2 x)^{3/2}}{4536 (3 x+2)^4}-\frac{991 \sqrt{1-2 x}}{444528 (3 x+2)}-\frac{991 \sqrt{1-2 x}}{190512 (3 x+2)^2}+\frac{991 \sqrt{1-2 x}}{13608 (3 x+2)^3}-\frac{991 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}} \]

[Out]

-(1 - 2*x)^(5/2)/(378*(2 + 3*x)^6) + (59*(1 - 2*x)^(5/2))/(1890*(2 + 3*x)^5) - (
991*(1 - 2*x)^(3/2))/(4536*(2 + 3*x)^4) + (991*Sqrt[1 - 2*x])/(13608*(2 + 3*x)^3
) - (991*Sqrt[1 - 2*x])/(190512*(2 + 3*x)^2) - (991*Sqrt[1 - 2*x])/(444528*(2 +
3*x)) - (991*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(222264*Sqrt[21])

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Rubi [A]  time = 0.165024, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{59 (1-2 x)^{5/2}}{1890 (3 x+2)^5}-\frac{(1-2 x)^{5/2}}{378 (3 x+2)^6}-\frac{991 (1-2 x)^{3/2}}{4536 (3 x+2)^4}-\frac{991 \sqrt{1-2 x}}{444528 (3 x+2)}-\frac{991 \sqrt{1-2 x}}{190512 (3 x+2)^2}+\frac{991 \sqrt{1-2 x}}{13608 (3 x+2)^3}-\frac{991 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{222264 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(3/2)*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

-(1 - 2*x)^(5/2)/(378*(2 + 3*x)^6) + (59*(1 - 2*x)^(5/2))/(1890*(2 + 3*x)^5) - (
991*(1 - 2*x)^(3/2))/(4536*(2 + 3*x)^4) + (991*Sqrt[1 - 2*x])/(13608*(2 + 3*x)^3
) - (991*Sqrt[1 - 2*x])/(190512*(2 + 3*x)^2) - (991*Sqrt[1 - 2*x])/(444528*(2 +
3*x)) - (991*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(222264*Sqrt[21])

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Rubi in Sympy [A]  time = 17.8758, size = 131, normalized size = 0.89 \[ \frac{59 \left (- 2 x + 1\right )^{\frac{5}{2}}}{1890 \left (3 x + 2\right )^{5}} - \frac{\left (- 2 x + 1\right )^{\frac{5}{2}}}{378 \left (3 x + 2\right )^{6}} - \frac{991 \left (- 2 x + 1\right )^{\frac{3}{2}}}{4536 \left (3 x + 2\right )^{4}} - \frac{991 \sqrt{- 2 x + 1}}{444528 \left (3 x + 2\right )} - \frac{991 \sqrt{- 2 x + 1}}{190512 \left (3 x + 2\right )^{2}} + \frac{991 \sqrt{- 2 x + 1}}{13608 \left (3 x + 2\right )^{3}} - \frac{991 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{4667544} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)*(3+5*x)**2/(2+3*x)**7,x)

[Out]

59*(-2*x + 1)**(5/2)/(1890*(3*x + 2)**5) - (-2*x + 1)**(5/2)/(378*(3*x + 2)**6)
- 991*(-2*x + 1)**(3/2)/(4536*(3*x + 2)**4) - 991*sqrt(-2*x + 1)/(444528*(3*x +
2)) - 991*sqrt(-2*x + 1)/(190512*(3*x + 2)**2) + 991*sqrt(-2*x + 1)/(13608*(3*x
+ 2)**3) - 991*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/4667544

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Mathematica [A]  time = 0.123164, size = 73, normalized size = 0.49 \[ \frac{-\frac{21 \sqrt{1-2 x} \left (1204065 x^5+4950045 x^4-6094818 x^3-9658494 x^2-1262200 x+858112\right )}{(3 x+2)^6}-9910 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{46675440} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

((-21*Sqrt[1 - 2*x]*(858112 - 1262200*x - 9658494*x^2 - 6094818*x^3 + 4950045*x^
4 + 1204065*x^5))/(2 + 3*x)^6 - 9910*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/
46675440

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Maple [A]  time = 0.02, size = 84, normalized size = 0.6 \[ 23328\,{\frac{1}{ \left ( -4-6\,x \right ) ^{6}} \left ({\frac{991\, \left ( 1-2\,x \right ) ^{11/2}}{21337344}}-{\frac{16847\, \left ( 1-2\,x \right ) ^{9/2}}{27433728}}+{\frac{10303\, \left ( 1-2\,x \right ) ^{7/2}}{9797760}}+{\frac{29843\, \left ( 1-2\,x \right ) ^{5/2}}{9797760}}-{\frac{117929\, \left ( 1-2\,x \right ) ^{3/2}}{15116544}}+{\frac{48559\,\sqrt{1-2\,x}}{15116544}} \right ) }-{\frac{991\,\sqrt{21}}{4667544}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)*(3+5*x)^2/(2+3*x)^7,x)

[Out]

23328*(991/21337344*(1-2*x)^(11/2)-16847/27433728*(1-2*x)^(9/2)+10303/9797760*(1
-2*x)^(7/2)+29843/9797760*(1-2*x)^(5/2)-117929/15116544*(1-2*x)^(3/2)+48559/1511
6544*(1-2*x)^(1/2))/(-4-6*x)^6-991/4667544*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*2
1^(1/2)

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Maxima [A]  time = 1.50967, size = 197, normalized size = 1.33 \[ \frac{991}{9335088} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1204065 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 15920415 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 27261738 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 78964578 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 202248235 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 83278685 \, \sqrt{-2 \, x + 1}}{1111320 \,{\left (729 \,{\left (2 \, x - 1\right )}^{6} + 10206 \,{\left (2 \, x - 1\right )}^{5} + 59535 \,{\left (2 \, x - 1\right )}^{4} + 185220 \,{\left (2 \, x - 1\right )}^{3} + 324135 \,{\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(3/2)/(3*x + 2)^7,x, algorithm="maxima")

[Out]

991/9335088*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x
+ 1))) + 1/1111320*(1204065*(-2*x + 1)^(11/2) - 15920415*(-2*x + 1)^(9/2) + 2726
1738*(-2*x + 1)^(7/2) + 78964578*(-2*x + 1)^(5/2) - 202248235*(-2*x + 1)^(3/2) +
83278685*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59535*(2*x - 1)
^4 + 185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)

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Fricas [A]  time = 0.217875, size = 181, normalized size = 1.22 \[ -\frac{\sqrt{21}{\left (\sqrt{21}{\left (1204065 \, x^{5} + 4950045 \, x^{4} - 6094818 \, x^{3} - 9658494 \, x^{2} - 1262200 \, x + 858112\right )} \sqrt{-2 \, x + 1} - 4955 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{46675440 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(3/2)/(3*x + 2)^7,x, algorithm="fricas")

[Out]

-1/46675440*sqrt(21)*(sqrt(21)*(1204065*x^5 + 4950045*x^4 - 6094818*x^3 - 965849
4*x^2 - 1262200*x + 858112)*sqrt(-2*x + 1) - 4955*(729*x^6 + 2916*x^5 + 4860*x^4
+ 4320*x^3 + 2160*x^2 + 576*x + 64)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1)
)/(3*x + 2)))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**(3/2)*(3+5*x)**2/(2+3*x)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213485, size = 178, normalized size = 1.2 \[ \frac{991}{9335088} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{1204065 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 15920415 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 27261738 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 78964578 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 202248235 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 83278685 \, \sqrt{-2 \, x + 1}}{71124480 \,{\left (3 \, x + 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(-2*x + 1)^(3/2)/(3*x + 2)^7,x, algorithm="giac")

[Out]

991/9335088*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sq
rt(-2*x + 1))) - 1/71124480*(1204065*(2*x - 1)^5*sqrt(-2*x + 1) + 15920415*(2*x
- 1)^4*sqrt(-2*x + 1) + 27261738*(2*x - 1)^3*sqrt(-2*x + 1) - 78964578*(2*x - 1)
^2*sqrt(-2*x + 1) + 202248235*(-2*x + 1)^(3/2) - 83278685*sqrt(-2*x + 1))/(3*x +
2)^6