3.2029 \(\int \frac{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)} \, dx\)

Optimal. Leaf size=67 \[ -\frac{25}{12} (1-2 x)^{5/2}+\frac{400}{27} (1-2 x)^{3/2}-\frac{5135}{108} \sqrt{1-2 x}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

[Out]

(-5135*Sqrt[1 - 2*x])/108 + (400*(1 - 2*x)^(3/2))/27 - (25*(1 - 2*x)^(5/2))/12 + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/(27*Sqrt[21])

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Rubi [A]  time = 0.0209144, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {88, 63, 206} \[ -\frac{25}{12} (1-2 x)^{5/2}+\frac{400}{27} (1-2 x)^{3/2}-\frac{5135}{108} \sqrt{1-2 x}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5135*Sqrt[1 - 2*x])/108 + (400*(1 - 2*x)^(3/2))/27 - (25*(1 - 2*x)^(5/2))/12 + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/(27*Sqrt[21])

Rule 88

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

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{(3+5 x)^3}{\sqrt{1-2 x} (2+3 x)} \, dx &=\int \left (\frac{5135}{108 \sqrt{1-2 x}}-\frac{400}{9} \sqrt{1-2 x}+\frac{125}{12} (1-2 x)^{3/2}-\frac{1}{27 \sqrt{1-2 x} (2+3 x)}\right ) \, dx\\ &=-\frac{5135}{108} \sqrt{1-2 x}+\frac{400}{27} (1-2 x)^{3/2}-\frac{25}{12} (1-2 x)^{5/2}-\frac{1}{27} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{5135}{108} \sqrt{1-2 x}+\frac{400}{27} (1-2 x)^{3/2}-\frac{25}{12} (1-2 x)^{5/2}+\frac{1}{27} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{5135}{108} \sqrt{1-2 x}+\frac{400}{27} (1-2 x)^{3/2}-\frac{25}{12} (1-2 x)^{5/2}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0319022, size = 51, normalized size = 0.76 \[ \frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}}-\frac{5}{27} \sqrt{1-2 x} \left (45 x^2+115 x+188\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Sqrt[1 - 2*x]*(188 + 115*x + 45*x^2))/27 + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(27*Sqrt[21])

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Maple [A]  time = 0.005, size = 47, normalized size = 0.7 \begin{align*}{\frac{400}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{25}{12} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{2\,\sqrt{21}}{567}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{5135}{108}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

400/27*(1-2*x)^(3/2)-25/12*(1-2*x)^(5/2)+2/567*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-5135/108*(1-2*x)^(
1/2)

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Maxima [A]  time = 1.69927, size = 86, normalized size = 1.28 \begin{align*} -\frac{25}{12} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{400}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{567} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{5135}{108} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/12*(-2*x + 1)^(5/2) + 400/27*(-2*x + 1)^(3/2) - 1/567*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21
) + 3*sqrt(-2*x + 1))) - 5135/108*sqrt(-2*x + 1)

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Fricas [A]  time = 1.67621, size = 153, normalized size = 2.28 \begin{align*} -\frac{5}{27} \,{\left (45 \, x^{2} + 115 \, x + 188\right )} \sqrt{-2 \, x + 1} + \frac{1}{567} \, \sqrt{21} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/27*(45*x^2 + 115*x + 188)*sqrt(-2*x + 1) + 1/567*sqrt(21)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)
)

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Sympy [A]  time = 23.4883, size = 102, normalized size = 1.52 \begin{align*} - \frac{25 \left (1 - 2 x\right )^{\frac{5}{2}}}{12} + \frac{400 \left (1 - 2 x\right )^{\frac{3}{2}}}{27} - \frac{5135 \sqrt{1 - 2 x}}{108} - \frac{2 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21}}{3 \sqrt{1 - 2 x}} \right )}}{21} & \text{for}\: \frac{1}{1 - 2 x} > \frac{3}{7} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21}}{3 \sqrt{1 - 2 x}} \right )}}{21} & \text{for}\: \frac{1}{1 - 2 x} < \frac{3}{7} \end{cases}\right )}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25*(1 - 2*x)**(5/2)/12 + 400*(1 - 2*x)**(3/2)/27 - 5135*sqrt(1 - 2*x)/108 - 2*Piecewise((-sqrt(21)*acoth(sqrt
(21)/(3*sqrt(1 - 2*x)))/21, 1/(1 - 2*x) > 3/7), (-sqrt(21)*atanh(sqrt(21)/(3*sqrt(1 - 2*x)))/21, 1/(1 - 2*x) <
3/7))/27

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Giac [A]  time = 1.98613, size = 100, normalized size = 1.49 \begin{align*} -\frac{25}{12} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{400}{27} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{1}{567} \, \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{5135}{108} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/12*(2*x - 1)^2*sqrt(-2*x + 1) + 400/27*(-2*x + 1)^(3/2) - 1/567*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(
-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 5135/108*sqrt(-2*x + 1)