3.1905 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=113 \[ \frac{1138 \sqrt{1-2 x}}{21 (3 x+2)}+\frac{49 \sqrt{1-2 x}}{9 (3 x+2)^2}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3}+\frac{78506 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}-110 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3) + (49*Sqrt[1 - 2*x])/(9*(2 + 3*x)^2) + (1138*Sqrt[1 - 2*x])/(21*(2 + 3*x)) +
(78506*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21]) - 110*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.0463217, antiderivative size = 113, 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, Rules used = {98, 151, 156, 63, 206} \[ \frac{1138 \sqrt{1-2 x}}{21 (3 x+2)}+\frac{49 \sqrt{1-2 x}}{9 (3 x+2)^2}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3}+\frac{78506 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}-110 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)),x]

[Out]

(7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3) + (49*Sqrt[1 - 2*x])/(9*(2 + 3*x)^2) + (1138*Sqrt[1 - 2*x])/(21*(2 + 3*x)) +
(78506*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21]) - 110*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)^4 (3+5 x)} \, dx &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3}+\frac{1}{9} \int \frac{120-163 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3}+\frac{49 \sqrt{1-2 x}}{9 (2+3 x)^2}+\frac{1}{126} \int \frac{9072-10290 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3}+\frac{49 \sqrt{1-2 x}}{9 (2+3 x)^2}+\frac{1138 \sqrt{1-2 x}}{21 (2+3 x)}+\frac{1}{882} \int \frac{390222-238980 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3}+\frac{49 \sqrt{1-2 x}}{9 (2+3 x)^2}+\frac{1138 \sqrt{1-2 x}}{21 (2+3 x)}-\frac{39253}{21} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+3025 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3}+\frac{49 \sqrt{1-2 x}}{9 (2+3 x)^2}+\frac{1138 \sqrt{1-2 x}}{21 (2+3 x)}+\frac{39253}{21} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-3025 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3}+\frac{49 \sqrt{1-2 x}}{9 (2+3 x)^2}+\frac{1138 \sqrt{1-2 x}}{21 (2+3 x)}+\frac{78506 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}-110 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0852701, size = 83, normalized size = 0.73 \[ \frac{\sqrt{1-2 x} \left (10242 x^2+13999 x+4797\right )}{21 (3 x+2)^3}+\frac{78506 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}-110 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)),x]

[Out]

(Sqrt[1 - 2*x]*(4797 + 13999*x + 10242*x^2))/(21*(2 + 3*x)^3) + (78506*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*S
qrt[21]) - 110*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.012, size = 75, normalized size = 0.7 \begin{align*} -54\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{1138\, \left ( 1-2\,x \right ) ^{5/2}}{63}}-{\frac{6926\, \left ( 1-2\,x \right ) ^{3/2}}{81}}+{\frac{8204\,\sqrt{1-2\,x}}{81}} \right ) }+{\frac{78506\,\sqrt{21}}{441}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-110\,{\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)^4/(3+5*x),x)

[Out]

-54*(1138/63*(1-2*x)^(5/2)-6926/81*(1-2*x)^(3/2)+8204/81*(1-2*x)^(1/2))/(-6*x-4)^3+78506/441*arctanh(1/7*21^(1
/2)*(1-2*x)^(1/2))*21^(1/2)-110*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 3.63293, size = 173, normalized size = 1.53 \begin{align*} 55 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{39253}{441} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (5121 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 24241 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 28714 \, \sqrt{-2 \, x + 1}\right )}}{21 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x),x, algorithm="maxima")

[Out]

55*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 39253/441*sqrt(21)*log(-(sqrt(
21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/21*(5121*(-2*x + 1)^(5/2) - 24241*(-2*x + 1)^(3/2)
+ 28714*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 1.38577, size = 377, normalized size = 3.34 \begin{align*} \frac{24255 \, \sqrt{55}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 39253 \, \sqrt{21}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (10242 \, x^{2} + 13999 \, x + 4797\right )} \sqrt{-2 \, x + 1}}{441 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x),x, algorithm="fricas")

[Out]

1/441*(24255*sqrt(55)*(27*x^3 + 54*x^2 + 36*x + 8)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 39253*
sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(10242*x^2 + 139
99*x + 4797)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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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)**4/(3+5*x),x)

[Out]

Timed out

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Giac [A]  time = 1.65629, size = 166, normalized size = 1.47 \begin{align*} 55 \, \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{39253}{441} \, \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{5121 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 24241 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 28714 \, \sqrt{-2 \, x + 1}}{42 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x),x, algorithm="giac")

[Out]

55*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 39253/441*sqrt(21)*l
og(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/42*(5121*(2*x - 1)^2*sqrt(-2*x +
1) - 24241*(-2*x + 1)^(3/2) + 28714*sqrt(-2*x + 1))/(3*x + 2)^3