3.1825 \(\int \frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=127 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{15 (3 x+2)^5}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{630 (3 x+2)^4}-\frac{\sqrt{1-2 x} (59665 x+37224)}{79380 (3 x+2)^3}+\frac{11237 \sqrt{1-2 x}}{111132 (3 x+2)}+\frac{11237 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{55566 \sqrt{21}} \]

[Out]

(11237*Sqrt[1 - 2*x])/(111132*(2 + 3*x)) - (53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(630*(2 + 3*x)^4) - (Sqrt[1 - 2*x]*(
3 + 5*x)^3)/(15*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(37224 + 59665*x))/(79380*(2 + 3*x)^3) + (11237*ArcTanh[Sqrt[3/7
]*Sqrt[1 - 2*x]])/(55566*Sqrt[21])

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Rubi [A]  time = 0.0397103, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {97, 149, 145, 51, 63, 206} \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{15 (3 x+2)^5}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{630 (3 x+2)^4}-\frac{\sqrt{1-2 x} (59665 x+37224)}{79380 (3 x+2)^3}+\frac{11237 \sqrt{1-2 x}}{111132 (3 x+2)}+\frac{11237 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{55566 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^6,x]

[Out]

(11237*Sqrt[1 - 2*x])/(111132*(2 + 3*x)) - (53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(630*(2 + 3*x)^4) - (Sqrt[1 - 2*x]*(
3 + 5*x)^3)/(15*(2 + 3*x)^5) - (Sqrt[1 - 2*x]*(37224 + 59665*x))/(79380*(2 + 3*x)^3) + (11237*ArcTanh[Sqrt[3/7
]*Sqrt[1 - 2*x]])/(55566*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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, 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{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^6} \, dx &=-\frac{\sqrt{1-2 x} (3+5 x)^3}{15 (2+3 x)^5}+\frac{1}{15} \int \frac{(12-35 x) (3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^5} \, dx\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac{\sqrt{1-2 x} (3+5 x)^3}{15 (2+3 x)^5}+\frac{\int \frac{(346-3310 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)^4} \, dx}{1260}\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac{\sqrt{1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac{\sqrt{1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}-\frac{11237 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{15876}\\ &=\frac{11237 \sqrt{1-2 x}}{111132 (2+3 x)}-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac{\sqrt{1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac{\sqrt{1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}-\frac{11237 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{111132}\\ &=\frac{11237 \sqrt{1-2 x}}{111132 (2+3 x)}-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac{\sqrt{1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac{\sqrt{1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}+\frac{11237 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{111132}\\ &=\frac{11237 \sqrt{1-2 x}}{111132 (2+3 x)}-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{630 (2+3 x)^4}-\frac{\sqrt{1-2 x} (3+5 x)^3}{15 (2+3 x)^5}-\frac{\sqrt{1-2 x} (37224+59665 x)}{79380 (2+3 x)^3}+\frac{11237 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{55566 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0267173, size = 52, normalized size = 0.41 \[ \frac{(1-2 x)^{3/2} \left (\frac{4802 \left (78750 x^2+104667 x+34784\right )}{(3 x+2)^5}-1797920 \, _2F_1\left (\frac{3}{2},4;\frac{5}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{27227340} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^6,x]

[Out]

((1 - 2*x)^(3/2)*((4802*(34784 + 104667*x + 78750*x^2))/(2 + 3*x)^5 - 1797920*Hypergeometric2F1[3/2, 4, 5/2, 3
/7 - (6*x)/7]))/27227340

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Maple [A]  time = 0.01, size = 75, normalized size = 0.6 \begin{align*} 1944\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ( -{\frac{11237\, \left ( 1-2\,x \right ) ^{9/2}}{1333584}}+{\frac{4237\, \left ( 1-2\,x \right ) ^{7/2}}{122472}}+{\frac{4954\, \left ( 1-2\,x \right ) ^{5/2}}{229635}}-{\frac{263117\, \left ( 1-2\,x \right ) ^{3/2}}{1102248}}+{\frac{78659\,\sqrt{1-2\,x}}{314928}} \right ) }+{\frac{11237\,\sqrt{21}}{1166886}{\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)^6,x)

[Out]

1944*(-11237/1333584*(1-2*x)^(9/2)+4237/122472*(1-2*x)^(7/2)+4954/229635*(1-2*x)^(5/2)-263117/1102248*(1-2*x)^
(3/2)+78659/314928*(1-2*x)^(1/2))/(-6*x-4)^5+11237/1166886*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 2.11852, size = 173, normalized size = 1.36 \begin{align*} -\frac{11237}{2333772} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4550985 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 18685170 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 11651808 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 128927330 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 134900185 \, \sqrt{-2 \, x + 1}}{277830 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\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)^6,x, algorithm="maxima")

[Out]

-11237/2333772*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/277830*(4550985*
(-2*x + 1)^(9/2) - 18685170*(-2*x + 1)^(7/2) - 11651808*(-2*x + 1)^(5/2) + 128927330*(-2*x + 1)^(3/2) - 134900
185*sqrt(-2*x + 1))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19
208)

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Fricas [A]  time = 1.63715, size = 367, normalized size = 2.89 \begin{align*} \frac{56185 \, \sqrt{21}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (4550985 \, x^{4} + 240615 \, x^{3} - 10100352 \, x^{2} - 8471518 \, x - 1984928\right )} \sqrt{-2 \, x + 1}}{11668860 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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)^6,x, algorithm="fricas")

[Out]

1/11668860*(56185*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x - sqrt(21)*sqrt(-2*x
+ 1) - 5)/(3*x + 2)) + 21*(4550985*x^4 + 240615*x^3 - 10100352*x^2 - 8471518*x - 1984928)*sqrt(-2*x + 1))/(24
3*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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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)**6,x)

[Out]

Timed out

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Giac [A]  time = 1.63224, size = 157, normalized size = 1.24 \begin{align*} -\frac{11237}{2333772} \, \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{4550985 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 18685170 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 11651808 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 128927330 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 134900185 \, \sqrt{-2 \, x + 1}}{8890560 \,{\left (3 \, x + 2\right )}^{5}} \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)^6,x, algorithm="giac")

[Out]

-11237/2333772*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/8890560
*(4550985*(2*x - 1)^4*sqrt(-2*x + 1) + 18685170*(2*x - 1)^3*sqrt(-2*x + 1) - 11651808*(2*x - 1)^2*sqrt(-2*x +
1) + 128927330*(-2*x + 1)^(3/2) - 134900185*sqrt(-2*x + 1))/(3*x + 2)^5