3.1996 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=97 \[ -\frac{11 (1-2 x)^{3/2}}{10 (5 x+3)^2}+\frac{803 \sqrt{1-2 x}}{50 (5 x+3)}+98 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2523}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-11*(1 - 2*x)^(3/2))/(10*(3 + 5*x)^2) + (803*Sqrt[1 - 2*x])/(50*(3 + 5*x)) + 98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]] - (2523*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/25

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Rubi [A]  time = 0.0334687, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 149, 156, 63, 206} \[ -\frac{11 (1-2 x)^{3/2}}{10 (5 x+3)^2}+\frac{803 \sqrt{1-2 x}}{50 (5 x+3)}+98 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2523}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*(1 - 2*x)^(3/2))/(10*(3 + 5*x)^2) + (803*Sqrt[1 - 2*x])/(50*(3 + 5*x)) + 98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*S
qrt[1 - 2*x]] - (2523*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/25

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 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) (3+5 x)^3} \, dx &=-\frac{11 (1-2 x)^{3/2}}{10 (3+5 x)^2}-\frac{1}{10} \int \frac{(136-41 x) \sqrt{1-2 x}}{(2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{11 (1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac{803 \sqrt{1-2 x}}{50 (3+5 x)}-\frac{1}{50} \int \frac{-4056+2491 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{11 (1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac{803 \sqrt{1-2 x}}{50 (3+5 x)}-343 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{27753}{50} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{11 (1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac{803 \sqrt{1-2 x}}{50 (3+5 x)}+343 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{27753}{50} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{11 (1-2 x)^{3/2}}{10 (3+5 x)^2}+\frac{803 \sqrt{1-2 x}}{50 (3+5 x)}+98 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2523}{25} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0962945, size = 81, normalized size = 0.84 \[ 98 \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{1}{250} \left (\frac{55 \sqrt{1-2 x} (375 x+214)}{(5 x+3)^2}-5046 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

98*Sqrt[7/3]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + ((55*Sqrt[1 - 2*x]*(214 + 375*x))/(3 + 5*x)^2 - 5046*Sqrt[55]*
ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/250

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Maple [A]  time = 0.01, size = 66, normalized size = 0.7 \begin{align*}{\frac{98\,\sqrt{21}}{3}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+550\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -3/10\, \left ( 1-2\,x \right ) ^{3/2}+{\frac{803\,\sqrt{1-2\,x}}{1250}} \right ) }-{\frac{2523\,\sqrt{55}}{125}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

98/3*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+550*(-3/10*(1-2*x)^(3/2)+803/1250*(1-2*x)^(1/2))/(-10*x-6)^2
-2523/125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.83706, size = 149, normalized size = 1.54 \begin{align*} \frac{2523}{250} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{49}{3} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{11 \,{\left (375 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 803 \, \sqrt{-2 \, x + 1}\right )}}{25 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2523/250*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 49/3*sqrt(21)*log(-(sqrt
(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 11/25*(375*(-2*x + 1)^(3/2) - 803*sqrt(-2*x + 1))/(2
5*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 1.42232, size = 362, normalized size = 3.73 \begin{align*} \frac{7569 \, \sqrt{11} \sqrt{5}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 12250 \, \sqrt{7} \sqrt{3}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 165 \,{\left (375 \, x + 214\right )} \sqrt{-2 \, x + 1}}{750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/750*(7569*sqrt(11)*sqrt(5)*(25*x^2 + 30*x + 9)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) +
12250*sqrt(7)*sqrt(3)*(25*x^2 + 30*x + 9)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 165*(37
5*x + 214)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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

[Out]

Timed out

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Giac [A]  time = 2.76646, size = 144, normalized size = 1.48 \begin{align*} \frac{2523}{250} \, \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{49}{3} \, \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{11 \,{\left (375 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 803 \, \sqrt{-2 \, x + 1}\right )}}{100 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2523/250*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 49/3*sqrt(21)*
log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 11/100*(375*(-2*x + 1)^(3/2) - 80
3*sqrt(-2*x + 1))/(5*x + 3)^2