3.2018 \(\int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)} \, dx\)

Optimal. Leaf size=54 \[ \frac {25}{18} (1-2 x)^{3/2}-\frac {155}{18} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}} \]

[Out]

25/18*(1-2*x)^(3/2)-2/189*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-155/18*(1-2*x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, 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} \[ \frac {25}{18} (1-2 x)^{3/2}-\frac {155}{18} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-155*Sqrt[1 - 2*x])/18 + (25*(1 - 2*x)^(3/2))/18 - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9*Sqrt[21])

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 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 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)^2}{\sqrt {1-2 x} (2+3 x)} \, dx &=\int \left (\frac {155}{18 \sqrt {1-2 x}}-\frac {25}{6} \sqrt {1-2 x}+\frac {1}{9 \sqrt {1-2 x} (2+3 x)}\right ) \, dx\\ &=-\frac {155}{18} \sqrt {1-2 x}+\frac {25}{18} (1-2 x)^{3/2}+\frac {1}{9} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {155}{18} \sqrt {1-2 x}+\frac {25}{18} (1-2 x)^{3/2}-\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {155}{18} \sqrt {1-2 x}+\frac {25}{18} (1-2 x)^{3/2}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 46, normalized size = 0.85 \[ -\frac {5}{9} \sqrt {1-2 x} (5 x+13)-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Sqrt[1 - 2*x]*(13 + 5*x))/9 - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9*Sqrt[21])

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fricas [A]  time = 0.97, size = 45, normalized size = 0.83 \[ -\frac {5}{9} \, {\left (5 \, x + 13\right )} \sqrt {-2 \, x + 1} + \frac {1}{189} \, \sqrt {21} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/9*(5*x + 13)*sqrt(-2*x + 1) + 1/189*sqrt(21)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2))

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giac [A]  time = 1.12, size = 58, normalized size = 1.07 \[ \frac {25}{18} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{189} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {155}{18} \, \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/18*(-2*x + 1)^(3/2) + 1/189*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x +
1))) - 155/18*sqrt(-2*x + 1)

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maple [A]  time = 0.01, size = 38, normalized size = 0.70 \[ -\frac {2 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{189}+\frac {25 \left (-2 x +1\right )^{\frac {3}{2}}}{18}-\frac {155 \sqrt {-2 x +1}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/18*(-2*x+1)^(3/2)-2/189*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)-155/18*(-2*x+1)^(1/2)

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maxima [A]  time = 1.58, size = 55, normalized size = 1.02 \[ \frac {25}{18} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{189} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {155}{18} \, \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/18*(-2*x + 1)^(3/2) + 1/189*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 15
5/18*sqrt(-2*x + 1)

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mupad [B]  time = 0.06, size = 37, normalized size = 0.69 \[ \frac {25\,{\left (1-2\,x\right )}^{3/2}}{18}-\frac {155\,\sqrt {1-2\,x}}{18}-\frac {2\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{189} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(25*(1 - 2*x)^(3/2))/18 - (155*(1 - 2*x)^(1/2))/18 - (2*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/189

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sympy [A]  time = 22.53, size = 90, normalized size = 1.67 \[ \frac {25 \left (1 - 2 x\right )^{\frac {3}{2}}}{18} - \frac {155 \sqrt {1 - 2 x}}{18} + \frac {2 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21}}{3 \sqrt {1 - 2 x}} \right )}}{21} & \text {for}\: \frac {1}{1 - 2 x} > \frac {3}{7} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21}}{3 \sqrt {1 - 2 x}} \right )}}{21} & \text {for}\: \frac {1}{1 - 2 x} < \frac {3}{7} \end {cases}\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25*(1 - 2*x)**(3/2)/18 - 155*sqrt(1 - 2*x)/18 + 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))/9

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