3.2151 \(\int \frac{(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^6} \, dx\)

Optimal. Leaf size=160 \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^5}+\frac{2 (83544 x+55633)}{5145 \sqrt{1-2 x} (3 x+2)^5}-\frac{81737 \sqrt{1-2 x}}{352947 (3 x+2)}-\frac{81737 \sqrt{1-2 x}}{151263 (3 x+2)^2}-\frac{163474 \sqrt{1-2 x}}{108045 (3 x+2)^3}-\frac{163474 \sqrt{1-2 x}}{36015 (3 x+2)^4}-\frac{163474 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{352947 \sqrt{21}} \]

[Out]

(-163474*Sqrt[1 - 2*x])/(36015*(2 + 3*x)^4) - (163474*Sqrt[1 - 2*x])/(108045*(2
+ 3*x)^3) - (81737*Sqrt[1 - 2*x])/(151263*(2 + 3*x)^2) - (81737*Sqrt[1 - 2*x])/(
352947*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^5) + (2*(5563
3 + 83544*x))/(5145*Sqrt[1 - 2*x]*(2 + 3*x)^5) - (163474*ArcTanh[Sqrt[3/7]*Sqrt[
1 - 2*x]])/(352947*Sqrt[21])

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Rubi [A]  time = 0.21142, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^5}+\frac{2 (83544 x+55633)}{5145 \sqrt{1-2 x} (3 x+2)^5}-\frac{81737 \sqrt{1-2 x}}{352947 (3 x+2)}-\frac{81737 \sqrt{1-2 x}}{151263 (3 x+2)^2}-\frac{163474 \sqrt{1-2 x}}{108045 (3 x+2)^3}-\frac{163474 \sqrt{1-2 x}}{36015 (3 x+2)^4}-\frac{163474 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{352947 \sqrt{21}} \]

Antiderivative was successfully verified.

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

[Out]

(-163474*Sqrt[1 - 2*x])/(36015*(2 + 3*x)^4) - (163474*Sqrt[1 - 2*x])/(108045*(2
+ 3*x)^3) - (81737*Sqrt[1 - 2*x])/(151263*(2 + 3*x)^2) - (81737*Sqrt[1 - 2*x])/(
352947*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^5) + (2*(5563
3 + 83544*x))/(5145*Sqrt[1 - 2*x]*(2 + 3*x)^5) - (163474*ArcTanh[Sqrt[3/7]*Sqrt[
1 - 2*x]])/(352947*Sqrt[21])

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Rubi in Sympy [A]  time = 20.4585, size = 143, normalized size = 0.89 \[ - \frac{81737 \sqrt{- 2 x + 1}}{352947 \left (3 x + 2\right )} - \frac{81737 \sqrt{- 2 x + 1}}{151263 \left (3 x + 2\right )^{2}} - \frac{163474 \sqrt{- 2 x + 1}}{108045 \left (3 x + 2\right )^{3}} - \frac{163474 \sqrt{- 2 x + 1}}{36015 \left (3 x + 2\right )^{4}} - \frac{163474 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{7411887} + \frac{501264 x + 333798}{15435 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{5}} + \frac{11 \left (5 x + 3\right )^{2}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-81737*sqrt(-2*x + 1)/(352947*(3*x + 2)) - 81737*sqrt(-2*x + 1)/(151263*(3*x + 2
)**2) - 163474*sqrt(-2*x + 1)/(108045*(3*x + 2)**3) - 163474*sqrt(-2*x + 1)/(360
15*(3*x + 2)**4) - 163474*sqrt(21)*atanh(sqrt(21)*sqrt(-2*x + 1)/7)/7411887 + (5
01264*x + 333798)/(15435*sqrt(-2*x + 1)*(3*x + 2)**5) + 11*(5*x + 3)**2/(21*(-2*
x + 1)**(3/2)*(3*x + 2)**5)

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Mathematica [A]  time = 0.174617, size = 78, normalized size = 0.49 \[ \frac{\frac{21 \left (-132413940 x^6-323678520 x^5-232214817 x^4+22641149 x^3+99751837 x^2+42553376 x+5615203\right )}{(1-2 x)^{3/2} (3 x+2)^5}-817370 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{37059435} \]

Antiderivative was successfully verified.

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

[Out]

((21*(5615203 + 42553376*x + 99751837*x^2 + 22641149*x^3 - 232214817*x^4 - 32367
8520*x^5 - 132413940*x^6))/((1 - 2*x)^(3/2)*(2 + 3*x)^5) - 817370*Sqrt[21]*ArcTa
nh[Sqrt[3/7]*Sqrt[1 - 2*x]])/37059435

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Maple [A]  time = 0.024, size = 93, normalized size = 0.6 \[{\frac{10648}{352947} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{90024}{823543}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{1944}{823543\, \left ( -4-6\,x \right ) ^{5}} \left ({\frac{167051}{36} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{7270739}{162} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{196782187}{1215} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{377074649}{1458} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{449872969}{2916}\sqrt{1-2\,x}} \right ) }-{\frac{163474\,\sqrt{21}}{7411887}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10648/352947/(1-2*x)^(3/2)+90024/823543/(1-2*x)^(1/2)+1944/823543*(167051/36*(1-
2*x)^(9/2)-7270739/162*(1-2*x)^(7/2)[phone]/1215*(1-2*x)^(5/2)-377074649/1458
*(1-2*x)^(3/2)[phone]/2916*(1-2*x)^(1/2))/(-4-6*x)^5-163474/7411887*arctanh(1
/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.50531, size = 197, normalized size = 1.23 \[ \frac{81737}{7411887} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (33103485 \,{\left (2 \, x - 1\right )}^{6} + 360460170 \,{\left (2 \, x - 1\right )}^{5} + 1537963392 \,{\left (2 \, x - 1\right )}^{4} + 3164039270 \,{\left (2 \, x - 1\right )}^{3} + 2973379535 \,{\left (2 \, x - 1\right )}^{2} + 1324775760 \, x - 1109790220\right )}}{1764735 \,{\left (243 \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - 2835 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + 13230 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 30870 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 36015 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 16807 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81737/7411887*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) + 2/1764735*(33103485*(2*x - 1)^6 + 360460170*(2*x - 1)^5 + 1537963392*
(2*x - 1)^4 + 3164039270*(2*x - 1)^3 + 2973379535*(2*x - 1)^2 + 1324775760*x - 1
109790220)/(243*(-2*x + 1)^(13/2) - 2835*(-2*x + 1)^(11/2) + 13230*(-2*x + 1)^(9
/2) - 30870*(-2*x + 1)^(7/2) + 36015*(-2*x + 1)^(5/2) - 16807*(-2*x + 1)^(3/2))

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Fricas [A]  time = 0.222523, size = 197, normalized size = 1.23 \[ \frac{\sqrt{21}{\left (408685 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \sqrt{-2 \, x + 1} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} + 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right ) + \sqrt{21}{\left (132413940 \, x^{6} + 323678520 \, x^{5} + 232214817 \, x^{4} - 22641149 \, x^{3} - 99751837 \, x^{2} - 42553376 \, x - 5615203\right )}\right )}}{37059435 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \sqrt{-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/37059435*sqrt(21)*(408685*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 -
176*x - 32)*sqrt(-2*x + 1)*log((sqrt(21)*(3*x - 5) + 21*sqrt(-2*x + 1))/(3*x +
2)) + sqrt(21)*(132413940*x^6 + 323678520*x^5 + 232214817*x^4 - 22641149*x^3 - 9
9751837*x^2 - 42553376*x - 5615203))/((486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 -
240*x^2 - 176*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)**3/(1-2*x)**(5/2)/(2+3*x)**6,x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.225394, size = 185, normalized size = 1.16 \[ \frac{81737}{7411887} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{1936 \,{\left (279 \, x - 178\right )}}{2470629 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{67655655 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 654366510 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 2361386244 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 3770746490 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2249364845 \, \sqrt{-2 \, x + 1}}{197650320 \,{\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

81737/7411887*sqrt(21)*ln(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*
sqrt(-2*x + 1))) + 1936/2470629*(279*x - 178)/((2*x - 1)*sqrt(-2*x + 1)) - 1/197
650320*(67655655*(2*x - 1)^4*sqrt(-2*x + 1) + 654366510*(2*x - 1)^3*sqrt(-2*x +
1) + 2361386244*(2*x - 1)^2*sqrt(-2*x + 1) - 3770746490*(-2*x + 1)^(3/2) + 22493
64845*sqrt(-2*x + 1))/(3*x + 2)^5