3.1824 \(\int \frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=107 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{12 (3 x+2)^4}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{378 (3 x+2)^3}-\frac{5 \sqrt{1-2 x} (110981 x+70429)}{222264 (3 x+2)^2}+\frac{328715 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{111132 \sqrt{21}} \]

[Out]

(-53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(378*(2 + 3*x)^3) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(12*(2 + 3*x)^4) - (5*Sqrt[1 -
2*x]*(70429 + 110981*x))/(222264*(2 + 3*x)^2) + (328715*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(111132*Sqrt[21])

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Rubi [A]  time = 0.0292461, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {97, 149, 145, 63, 206} \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{12 (3 x+2)^4}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{378 (3 x+2)^3}-\frac{5 \sqrt{1-2 x} (110981 x+70429)}{222264 (3 x+2)^2}+\frac{328715 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{111132 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(378*(2 + 3*x)^3) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(12*(2 + 3*x)^4) - (5*Sqrt[1 -
2*x]*(70429 + 110981*x))/(222264*(2 + 3*x)^2) + (328715*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(111132*Sqrt[21])

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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 145

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m
+ 2)), x] + Dist[(f*h)/b^2 - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] &&  !L
tQ[n, -2]))

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{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^5} \, dx &=-\frac{\sqrt{1-2 x} (3+5 x)^3}{12 (2+3 x)^4}+\frac{1}{12} \int \frac{(12-35 x) (3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac{\sqrt{1-2 x} (3+5 x)^3}{12 (2+3 x)^4}+\frac{1}{756} \int \frac{(445-3145 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac{\sqrt{1-2 x} (3+5 x)^3}{12 (2+3 x)^4}-\frac{5 \sqrt{1-2 x} (70429+110981 x)}{222264 (2+3 x)^2}-\frac{328715 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{222264}\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac{\sqrt{1-2 x} (3+5 x)^3}{12 (2+3 x)^4}-\frac{5 \sqrt{1-2 x} (70429+110981 x)}{222264 (2+3 x)^2}+\frac{328715 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{222264}\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{378 (2+3 x)^3}-\frac{\sqrt{1-2 x} (3+5 x)^3}{12 (2+3 x)^4}-\frac{5 \sqrt{1-2 x} (70429+110981 x)}{222264 (2+3 x)^2}+\frac{328715 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{111132 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0226396, size = 59, normalized size = 0.55 \[ \frac{(1-2 x)^{3/2} \left (343 \left (661500 x^2+880755 x+293191\right )-2629720 (3 x+2)^4 \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{5445468 (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(3/2)*(343*(293191 + 880755*x + 661500*x^2) - 2629720*(2 + 3*x)^4*Hypergeometric2F1[3/2, 3, 5/2, 3/
7 - (6*x)/7]))/(5445468*(2 + 3*x)^4)

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Maple [A]  time = 0.008, size = 66, normalized size = 0.6 \begin{align*} -324\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ( -{\frac{119095\, \left ( 1-2\,x \right ) ^{7/2}}{444528}}+{\frac{3126535\, \left ( 1-2\,x \right ) ^{5/2}}{1714608}}-{\frac{3040873\, \left ( 1-2\,x \right ) ^{3/2}}{734832}}+{\frac{328715\,\sqrt{1-2\,x}}{104976}} \right ) }+{\frac{328715\,\sqrt{21}}{2333772}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-324*(-119095/444528*(1-2*x)^(7/2)+3126535/1714608*(1-2*x)^(5/2)-3040873/734832*(1-2*x)^(3/2)+328715/104976*(1
-2*x)^(1/2))/(-6*x-4)^4+328715/2333772*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.69451, size = 149, normalized size = 1.39 \begin{align*} -\frac{328715}{4667544} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{9646695 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 65657235 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 149002777 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 112749245 \, \sqrt{-2 \, x + 1}}{111132 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-328715/4667544*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/111132*(9646695
*(-2*x + 1)^(7/2) - 65657235*(-2*x + 1)^(5/2) + 149002777*(-2*x + 1)^(3/2) - 112749245*sqrt(-2*x + 1))/(81*(2*
x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 1.62263, size = 316, normalized size = 2.95 \begin{align*} \frac{328715 \, \sqrt{21}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (9646695 \, x^{3} + 18358575 \, x^{2} + 11657098 \, x + 2469626\right )} \sqrt{-2 \, x + 1}}{4667544 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4667544*(328715*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3
*x + 2)) - 21*(9646695*x^3 + 18358575*x^2 + 11657098*x + 2469626)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2
+ 96*x + 16)

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

[Out]

Timed out

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Giac [A]  time = 2.36572, size = 135, normalized size = 1.26 \begin{align*} -\frac{328715}{4667544} \, \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{9646695 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 65657235 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 149002777 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 112749245 \, \sqrt{-2 \, x + 1}}{1778112 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-328715/4667544*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/177811
2*(9646695*(2*x - 1)^3*sqrt(-2*x + 1) + 65657235*(2*x - 1)^2*sqrt(-2*x + 1) - 149002777*(-2*x + 1)^(3/2) + 112
749245*sqrt(-2*x + 1))/(3*x + 2)^4