3.1918 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^2} \, dx\)

Optimal. Leaf size=181 \[ -\frac{8110915 \sqrt{1-2 x}}{1176 (5 x+3)}+\frac{302668 \sqrt{1-2 x}}{441 (3 x+2) (5 x+3)}+\frac{23173 \sqrt{1-2 x}}{504 (3 x+2)^2 (5 x+3)}+\frac{83 \sqrt{1-2 x}}{18 (3 x+2)^3 (5 x+3)}+\frac{7 \sqrt{1-2 x}}{12 (3 x+2)^4 (5 x+3)}-\frac{55953383 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{196 \sqrt{21}}+8400 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-8110915*Sqrt[1 - 2*x])/(1176*(3 + 5*x)) + (7*Sqrt[1 - 2*x])/(12*(2 + 3*x)^4*(3 + 5*x)) + (83*Sqrt[1 - 2*x])/
(18*(2 + 3*x)^3*(3 + 5*x)) + (23173*Sqrt[1 - 2*x])/(504*(2 + 3*x)^2*(3 + 5*x)) + (302668*Sqrt[1 - 2*x])/(441*(
2 + 3*x)*(3 + 5*x)) - (55953383*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(196*Sqrt[21]) + 8400*Sqrt[55]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.0732897, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 151, 156, 63, 206} \[ -\frac{8110915 \sqrt{1-2 x}}{1176 (5 x+3)}+\frac{302668 \sqrt{1-2 x}}{441 (3 x+2) (5 x+3)}+\frac{23173 \sqrt{1-2 x}}{504 (3 x+2)^2 (5 x+3)}+\frac{83 \sqrt{1-2 x}}{18 (3 x+2)^3 (5 x+3)}+\frac{7 \sqrt{1-2 x}}{12 (3 x+2)^4 (5 x+3)}-\frac{55953383 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{196 \sqrt{21}}+8400 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8110915*Sqrt[1 - 2*x])/(1176*(3 + 5*x)) + (7*Sqrt[1 - 2*x])/(12*(2 + 3*x)^4*(3 + 5*x)) + (83*Sqrt[1 - 2*x])/
(18*(2 + 3*x)^3*(3 + 5*x)) + (23173*Sqrt[1 - 2*x])/(504*(2 + 3*x)^2*(3 + 5*x)) + (302668*Sqrt[1 - 2*x])/(441*(
2 + 3*x)*(3 + 5*x)) - (55953383*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(196*Sqrt[21]) + 8400*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 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)^{3/2}}{(2+3 x)^5 (3+5 x)^2} \, dx &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac{1}{12} \int \frac{188-299 x}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^2} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac{83 \sqrt{1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac{1}{252} \int \frac{26957-40670 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac{83 \sqrt{1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac{23173 \sqrt{1-2 x}}{504 (2+3 x)^2 (3+5 x)}+\frac{\int \frac{2946286-4055275 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx}{3528}\\ &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac{83 \sqrt{1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac{23173 \sqrt{1-2 x}}{504 (2+3 x)^2 (3+5 x)}+\frac{302668 \sqrt{1-2 x}}{441 (2+3 x) (3+5 x)}+\frac{\int \frac{222179601-254241120 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx}{24696}\\ &=-\frac{8110915 \sqrt{1-2 x}}{1176 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac{83 \sqrt{1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac{23173 \sqrt{1-2 x}}{504 (2+3 x)^2 (3+5 x)}+\frac{302668 \sqrt{1-2 x}}{441 (2+3 x) (3+5 x)}-\frac{\int \frac{9177988743-5620864095 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{271656}\\ &=-\frac{8110915 \sqrt{1-2 x}}{1176 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac{83 \sqrt{1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac{23173 \sqrt{1-2 x}}{504 (2+3 x)^2 (3+5 x)}+\frac{302668 \sqrt{1-2 x}}{441 (2+3 x) (3+5 x)}+\frac{55953383}{392} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-231000 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{8110915 \sqrt{1-2 x}}{1176 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac{83 \sqrt{1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac{23173 \sqrt{1-2 x}}{504 (2+3 x)^2 (3+5 x)}+\frac{302668 \sqrt{1-2 x}}{441 (2+3 x) (3+5 x)}-\frac{55953383}{392} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+231000 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{8110915 \sqrt{1-2 x}}{1176 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 (3+5 x)}+\frac{83 \sqrt{1-2 x}}{18 (2+3 x)^3 (3+5 x)}+\frac{23173 \sqrt{1-2 x}}{504 (2+3 x)^2 (3+5 x)}+\frac{302668 \sqrt{1-2 x}}{441 (2+3 x) (3+5 x)}-\frac{55953383 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{196 \sqrt{21}}+8400 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.138688, size = 100, normalized size = 0.55 \[ -\frac{\sqrt{1-2 x} \left (218994705 x^4+576721848 x^3+569295605 x^2+249642200 x+41029970\right )}{392 (3 x+2)^4 (5 x+3)}-\frac{55953383 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{196 \sqrt{21}}+8400 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(41029970 + 249642200*x + 569295605*x^2 + 576721848*x^3 + 218994705*x^4))/(392*(2 + 3*x)^4*(3
+ 5*x)) - (55953383*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(196*Sqrt[21]) + 8400*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1
- 2*x]]

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Maple [A]  time = 0.013, size = 100, normalized size = 0.6 \begin{align*} 162\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{1298783\, \left ( 1-2\,x \right ) ^{7/2}}{1176}}-{\frac{11773333\, \left ( 1-2\,x \right ) ^{5/2}}{1512}}+{\frac{11859787\, \left ( 1-2\,x \right ) ^{3/2}}{648}}-{\frac{344197\,\sqrt{1-2\,x}}{24}} \right ) }-{\frac{55953383\,\sqrt{21}}{4116}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+550\,{\frac{\sqrt{1-2\,x}}{-2\,x-6/5}}+8400\,{\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)^(3/2)/(2+3*x)^5/(3+5*x)^2,x)

[Out]

162*(1298783/1176*(1-2*x)^(7/2)-11773333/1512*(1-2*x)^(5/2)[phone]/648*(1-2*x)^(3/2)-344197/24*(1-2*x)^(1/2)
)/(-6*x-4)^4-55953383/4116*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+550*(1-2*x)^(1/2)/(-2*x-6/5)+8400*arct
anh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.5574, size = 221, normalized size = 1.22 \begin{align*} -4200 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{55953383}{8232} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{218994705 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 2029422516 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 7051481738 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 10887812348 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 6303237941 \, \sqrt{-2 \, x + 1}}{196 \,{\left (405 \,{\left (2 \, x - 1\right )}^{5} + 4671 \,{\left (2 \, x - 1\right )}^{4} + 21546 \,{\left (2 \, x - 1\right )}^{3} + 49686 \,{\left (2 \, x - 1\right )}^{2} + 114562 \, x - 30870\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4200*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 55953383/8232*sqrt(21)*log(
-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/196*(218994705*(-2*x + 1)^(9/2) - 2029422516
*(-2*x + 1)^(7/2) + 7051481738*(-2*x + 1)^(5/2) - 10887812348*(-2*x + 1)^(3/2) + 6303237941*sqrt(-2*x + 1))/(4
05*(2*x - 1)^5 + 4671*(2*x - 1)^4 + 21546*(2*x - 1)^3 + 49686*(2*x - 1)^2 + 114562*x - 30870)

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Fricas [A]  time = 1.60655, size = 555, normalized size = 3.07 \begin{align*} \frac{34574400 \, \sqrt{55}{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55953383 \, \sqrt{21}{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (218994705 \, x^{4} + 576721848 \, x^{3} + 569295605 \, x^{2} + 249642200 \, x + 41029970\right )} \sqrt{-2 \, x + 1}}{8232 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8232*(34574400*sqrt(55)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*log((5*x - sqrt(55)*sqrt(-2*
x + 1) - 8)/(5*x + 3)) + 55953383*sqrt(21)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*log((3*x +
sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(218994705*x^4 + 576721848*x^3 + 569295605*x^2 + 249642200*x + 41
029970)*sqrt(-2*x + 1))/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)

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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)**(3/2)/(2+3*x)**5/(3+5*x)**2,x)

[Out]

Timed out

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Giac [A]  time = 2.29161, size = 209, normalized size = 1.15 \begin{align*} -4200 \, \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{55953383}{8232} \, \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{1375 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} - \frac{35067141 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 247239993 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 581129563 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 455372631 \, \sqrt{-2 \, x + 1}}{3136 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4200*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 55953383/8232*sqr
t(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1375*sqrt(-2*x + 1)/(5*x +
3) - 1/3136*(35067141*(2*x - 1)^3*sqrt(-2*x + 1) + 247239993*(2*x - 1)^2*sqrt(-2*x + 1) - 581129563*(-2*x + 1)
^(3/2) + 455372631*sqrt(-2*x + 1))/(3*x + 2)^4