3.1979 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^7 (3+5 x)} \, dx\)

Optimal. Leaf size=173 \[ \frac{7 (1-2 x)^{3/2}}{18 (3 x+2)^6}+\frac{736065535 \sqrt{1-2 x}}{49392 (3 x+2)}+\frac{31700335 \sqrt{1-2 x}}{21168 (3 x+2)^2}+\frac{302651 \sqrt{1-2 x}}{1512 (3 x+2)^3}+\frac{2165 \sqrt{1-2 x}}{72 (3 x+2)^4}+\frac{91 \sqrt{1-2 x}}{18 (3 x+2)^5}+\frac{25388847535 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{24696 \sqrt{21}}-30250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*(1 - 2*x)^(3/2))/(18*(2 + 3*x)^6) + (91*Sqrt[1 - 2*x])/(18*(2 + 3*x)^5) + (2165*Sqrt[1 - 2*x])/(72*(2 + 3*x
)^4) + (302651*Sqrt[1 - 2*x])/(1512*(2 + 3*x)^3) + (31700335*Sqrt[1 - 2*x])/(21168*(2 + 3*x)^2) + (736065535*S
qrt[1 - 2*x])/(49392*(2 + 3*x)) + (25388847535*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(24696*Sqrt[21]) - 30250*Sqrt
[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.0843072, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 151, 156, 63, 206} \[ \frac{7 (1-2 x)^{3/2}}{18 (3 x+2)^6}+\frac{736065535 \sqrt{1-2 x}}{49392 (3 x+2)}+\frac{31700335 \sqrt{1-2 x}}{21168 (3 x+2)^2}+\frac{302651 \sqrt{1-2 x}}{1512 (3 x+2)^3}+\frac{2165 \sqrt{1-2 x}}{72 (3 x+2)^4}+\frac{91 \sqrt{1-2 x}}{18 (3 x+2)^5}+\frac{25388847535 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{24696 \sqrt{21}}-30250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(1 - 2*x)^(3/2))/(18*(2 + 3*x)^6) + (91*Sqrt[1 - 2*x])/(18*(2 + 3*x)^5) + (2165*Sqrt[1 - 2*x])/(72*(2 + 3*x
)^4) + (302651*Sqrt[1 - 2*x])/(1512*(2 + 3*x)^3) + (31700335*Sqrt[1 - 2*x])/(21168*(2 + 3*x)^2) + (736065535*S
qrt[1 - 2*x])/(49392*(2 + 3*x)) + (25388847535*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(24696*Sqrt[21]) - 30250*Sqrt
[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^7 (3+5 x)} \, dx &=\frac{7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac{1}{18} \int \frac{(261-291 x) \sqrt{1-2 x}}{(2+3 x)^6 (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac{91 \sqrt{1-2 x}}{18 (2+3 x)^5}-\frac{1}{270} \int \frac{-36765+58515 x}{\sqrt{1-2 x} (2+3 x)^5 (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac{91 \sqrt{1-2 x}}{18 (2+3 x)^5}+\frac{2165 \sqrt{1-2 x}}{72 (2+3 x)^4}-\frac{\int \frac{-5288535+7956375 x}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)} \, dx}{7560}\\ &=\frac{7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac{91 \sqrt{1-2 x}}{18 (2+3 x)^5}+\frac{2165 \sqrt{1-2 x}}{72 (2+3 x)^4}+\frac{302651 \sqrt{1-2 x}}{1512 (2+3 x)^3}-\frac{\int \frac{-579872475+794458875 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)} \, dx}{158760}\\ &=\frac{7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac{91 \sqrt{1-2 x}}{18 (2+3 x)^5}+\frac{2165 \sqrt{1-2 x}}{72 (2+3 x)^4}+\frac{302651 \sqrt{1-2 x}}{1512 (2+3 x)^3}+\frac{31700335 \sqrt{1-2 x}}{21168 (2+3 x)^2}-\frac{\int \frac{-44001529425+49928027625 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{2222640}\\ &=\frac{7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac{91 \sqrt{1-2 x}}{18 (2+3 x)^5}+\frac{2165 \sqrt{1-2 x}}{72 (2+3 x)^4}+\frac{302651 \sqrt{1-2 x}}{1512 (2+3 x)^3}+\frac{31700335 \sqrt{1-2 x}}{21168 (2+3 x)^2}+\frac{736065535 \sqrt{1-2 x}}{49392 (2+3 x)}-\frac{\int \frac{-1892960179425+1159303217625 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{15558480}\\ &=\frac{7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac{91 \sqrt{1-2 x}}{18 (2+3 x)^5}+\frac{2165 \sqrt{1-2 x}}{72 (2+3 x)^4}+\frac{302651 \sqrt{1-2 x}}{1512 (2+3 x)^3}+\frac{31700335 \sqrt{1-2 x}}{21168 (2+3 x)^2}+\frac{736065535 \sqrt{1-2 x}}{49392 (2+3 x)}-\frac{25388847535 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{49392}+831875 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac{91 \sqrt{1-2 x}}{18 (2+3 x)^5}+\frac{2165 \sqrt{1-2 x}}{72 (2+3 x)^4}+\frac{302651 \sqrt{1-2 x}}{1512 (2+3 x)^3}+\frac{31700335 \sqrt{1-2 x}}{21168 (2+3 x)^2}+\frac{736065535 \sqrt{1-2 x}}{49392 (2+3 x)}+\frac{25388847535 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{49392}-831875 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{7 (1-2 x)^{3/2}}{18 (2+3 x)^6}+\frac{91 \sqrt{1-2 x}}{18 (2+3 x)^5}+\frac{2165 \sqrt{1-2 x}}{72 (2+3 x)^4}+\frac{302651 \sqrt{1-2 x}}{1512 (2+3 x)^3}+\frac{31700335 \sqrt{1-2 x}}{21168 (2+3 x)^2}+\frac{736065535 \sqrt{1-2 x}}{49392 (2+3 x)}+\frac{25388847535 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{24696 \sqrt{21}}-30250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.132238, size = 98, normalized size = 0.57 \[ \frac{\sqrt{1-2 x} \left (178863925005 x^5+602204446665 x^4+811194684822 x^3+546491397114 x^2+184131053992 x+24823128464\right )}{49392 (3 x+2)^6}+\frac{25388847535 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{24696 \sqrt{21}}-30250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(24823128464 + 184131053992*x + 546491397114*x^2 + 811194684822*x^3 + 602204446665*x^4 + 178863
925005*x^5))/(49392*(2 + 3*x)^6) + (25388847535*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(24696*Sqrt[21]) - 30250*Sqr
t[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.01, size = 102, normalized size = 0.6 \begin{align*} -1458\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{6}} \left ({\frac{736065535\, \left ( 1-2\,x \right ) ^{11/2}}{148176}}-{\frac{11104383695\, \left ( 1-2\,x \right ) ^{9/2}}{190512}}+{\frac{1240999441\, \left ( 1-2\,x \right ) ^{7/2}}{4536}}-{\frac{3744956269\, \left ( 1-2\,x \right ) ^{5/2}}{5832}}+{\frac{79114433335\, \left ( 1-2\,x \right ) ^{3/2}}{104976}}-{\frac{37144080785\,\sqrt{1-2\,x}}{104976}} \right ) }+{\frac{25388847535\,\sqrt{21}}{518616}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-30250\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1458*(736065535/148176*(1-2*x)^(11/2)-11104383695/190512*(1-2*x)^(9/2)[phone]/4536*(1-2*x)^(7/2)-37449562
69/5832*(1-2*x)^(5/2)[phone]/104976*(1-2*x)^(3/2)-37144080785/104976*(1-2*x)^(1/2))/(-6*x-4)^6+2538884753
5/518616*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-30250*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 4.34986, size = 246, normalized size = 1.42 \begin{align*} 15125 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{25388847535}{1037232} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{178863925005 \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - 2098728518355 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + 9851053562658 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 23121360004806 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 27136250633905 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 12740419709255 \, \sqrt{-2 \, x + 1}}{24696 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 25388847535/1037232*sqrt(21
)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/24696*(178863925005*(-2*x + 1)^(11/2)
- 2098728518355*(-2*x + 1)^(9/2) + 9851053562658*(-2*x + 1)^(7/2) - 23121360004806*(-2*x + 1)^(5/2) + 27136250
633905*(-2*x + 1)^(3/2) - 12740419709255*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 = 1.39735, size = 657, normalized size = 3.8 \begin{align*} \frac{15688134000 \, \sqrt{55}{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 25388847535 \, \sqrt{21}{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (178863925005 \, x^{5} + 602204446665 \, x^{4} + 811194684822 \, x^{3} + 546491397114 \, x^{2} + 184131053992 \, x + 24823128464\right )} \sqrt{-2 \, x + 1}}{1037232 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1037232*(15688134000*sqrt(55)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log((5*x +
sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 25388847535*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 216
0*x^2 + 576*x + 64)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(178863925005*x^5 + 602204446665*x
^4 + 811194684822*x^3 + 546491397114*x^2 + 184131053992*x + 24823128464)*sqrt(-2*x + 1))/(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. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 2.54283, size = 231, normalized size = 1.34 \begin{align*} 15125 \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{25388847535}{1037232} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{178863925005 \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + 2098728518355 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 9851053562658 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 23121360004806 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 27136250633905 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 12740419709255 \, \sqrt{-2 \, x + 1}}{1580544 \,{\left (3 \, x + 2\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 25388847535/10372
32*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1580544*(1788639250
05*(2*x - 1)^5*sqrt(-2*x + 1) + 2098728518355*(2*x - 1)^4*sqrt(-2*x + 1) + 9851053562658*(2*x - 1)^3*sqrt(-2*x
+ 1) + 23121360004806*(2*x - 1)^2*sqrt(-2*x + 1) - 27136250633905*(-2*x + 1)^(3/2) + 12740419709255*sqrt(-2*x
+ 1))/(3*x + 2)^6