3.1822 \(\int \frac{\sqrt{1-2 x} (3+5 x)^3}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=100 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{6 (3 x+2)^2}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)}+\frac{5 \sqrt{1-2 x} (2815 x+323)}{1134}+\frac{7559 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]

[Out]

(-53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(6*(2 + 3*x)^2) + (5*Sqrt[1 - 2*x
]*(323 + 2815*x))/1134 + (7559*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*Sqrt[21])

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Rubi [A]  time = 0.0278372, antiderivative size = 100, 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, 147, 63, 206} \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{6 (3 x+2)^2}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{63 (3 x+2)}+\frac{5 \sqrt{1-2 x} (2815 x+323)}{1134}+\frac{7559 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-53*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*x)^3)/(6*(2 + 3*x)^2) + (5*Sqrt[1 - 2*x
]*(323 + 2815*x))/1134 + (7559*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(567*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 147

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

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)^3} \, dx &=-\frac{\sqrt{1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac{1}{6} \int \frac{(12-35 x) (3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac{1}{126} \int \frac{(643-2815 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac{5 \sqrt{1-2 x} (323+2815 x)}{1134}-\frac{7559 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{1134}\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac{5 \sqrt{1-2 x} (323+2815 x)}{1134}+\frac{7559 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1134}\\ &=-\frac{53 \sqrt{1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^3}{6 (2+3 x)^2}+\frac{5 \sqrt{1-2 x} (323+2815 x)}{1134}+\frac{7559 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0381269, size = 63, normalized size = 0.63 \[ \frac{\sqrt{1-2 x} \left (31500 x^3+7350 x^2-32833 x-15815\right )}{1134 (3 x+2)^2}+\frac{7559 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{567 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(-15815 - 32833*x + 7350*x^2 + 31500*x^3))/(1134*(2 + 3*x)^2) + (7559*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/(567*Sqrt[21])

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Maple [A]  time = 0.01, size = 66, normalized size = 0.7 \begin{align*} -{\frac{125}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{50}{27}\sqrt{1-2\,x}}-{\frac{2}{3\, \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{211}{126} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{209}{54}\sqrt{1-2\,x}} \right ) }+{\frac{7559\,\sqrt{21}}{11907}{\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)^3,x)

[Out]

-125/81*(1-2*x)^(3/2)-50/27*(1-2*x)^(1/2)-2/3*(-211/126*(1-2*x)^(3/2)+209/54*(1-2*x)^(1/2))/(-6*x-4)^2+7559/11
907*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.60484, size = 124, normalized size = 1.24 \begin{align*} -\frac{125}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{7559}{23814} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{50}{27} \, \sqrt{-2 \, x + 1} + \frac{633 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1463 \, \sqrt{-2 \, x + 1}}{567 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\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)^3,x, algorithm="maxima")

[Out]

-125/81*(-2*x + 1)^(3/2) - 7559/23814*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)
)) - 50/27*sqrt(-2*x + 1) + 1/567*(633*(-2*x + 1)^(3/2) - 1463*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 1.6353, size = 236, normalized size = 2.36 \begin{align*} \frac{7559 \, \sqrt{21}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (31500 \, x^{3} + 7350 \, x^{2} - 32833 \, x - 15815\right )} \sqrt{-2 \, x + 1}}{23814 \,{\left (9 \, x^{2} + 12 \, x + 4\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)^3,x, algorithm="fricas")

[Out]

1/23814*(7559*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(31500*x^3 +
7350*x^2 - 32833*x - 15815)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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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)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.92169, size = 116, normalized size = 1.16 \begin{align*} -\frac{125}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{7559}{23814} \, \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{50}{27} \, \sqrt{-2 \, x + 1} + \frac{633 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1463 \, \sqrt{-2 \, x + 1}}{2268 \,{\left (3 \, x + 2\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

-125/81*(-2*x + 1)^(3/2) - 7559/23814*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(
-2*x + 1))) - 50/27*sqrt(-2*x + 1) + 1/2268*(633*(-2*x + 1)^(3/2) - 1463*sqrt(-2*x + 1))/(3*x + 2)^2