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

Optimal. Leaf size=67 \[ -\frac {125}{12} \sqrt {1-2 x}-\frac {2178}{49 \sqrt {1-2 x}}+\frac {1331}{84 (1-2 x)^{3/2}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \]

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Rubi [A]  time = 0.03, antiderivative size = 67, 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 = {87, 63, 206} \begin {gather*} -\frac {125}{12} \sqrt {1-2 x}-\frac {2178}{49 \sqrt {1-2 x}}+\frac {1331}{84 (1-2 x)^{3/2}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1331/(84*(1 - 2*x)^(3/2)) - 2178/(49*Sqrt[1 - 2*x]) - (125*Sqrt[1 - 2*x])/12 + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/(147*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 87

Int[(((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegr
and[(e + f*x)^FractionalPart[p], ((c + d*x)^n*(e + f*x)^IntegerPart[p])/(a + b*x), x], x] /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]

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}{(1-2 x)^{5/2} (2+3 x)} \, dx &=\int \left (\frac {1331}{28 (1-2 x)^{5/2}}-\frac {2178}{49 (1-2 x)^{3/2}}+\frac {125}{12 \sqrt {1-2 x}}-\frac {1}{147 \sqrt {1-2 x} (2+3 x)}\right ) \, dx\\ &=\frac {1331}{84 (1-2 x)^{3/2}}-\frac {2178}{49 \sqrt {1-2 x}}-\frac {125}{12} \sqrt {1-2 x}-\frac {1}{147} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {1331}{84 (1-2 x)^{3/2}}-\frac {2178}{49 \sqrt {1-2 x}}-\frac {125}{12} \sqrt {1-2 x}+\frac {1}{147} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {1331}{84 (1-2 x)^{3/2}}-\frac {2178}{49 \sqrt {1-2 x}}-\frac {125}{12} \sqrt {1-2 x}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 45, normalized size = 0.67 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )+35 \left (675 x^2-2115 x+632\right )}{567 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/567*(35*(632 - 2115*x + 675*x^2) + 2*Hypergeometric2F1[-3/2, 1, -1/2, 3/7 - (6*x)/7])/(1 - 2*x)^(3/2)

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IntegrateAlgebraic [A]  time = 0.08, size = 59, normalized size = 0.88 \begin {gather*} \frac {-6125 (1-2 x)^2-26136 (1-2 x)+9317}{588 (1-2 x)^{3/2}}+\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{147 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9317 - 26136*(1 - 2*x) - 6125*(1 - 2*x)^2)/(588*(1 - 2*x)^(3/2)) + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(147*
Sqrt[21])

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fricas [A]  time = 1.23, size = 74, normalized size = 1.10 \begin {gather*} \frac {\sqrt {21} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (6125 \, x^{2} - 19193 \, x + 5736\right )} \sqrt {-2 \, x + 1}}{3087 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3087*(sqrt(21)*(4*x^2 - 4*x + 1)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(6125*x^2 - 19193*x
+ 5736)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.34, size = 70, normalized size = 1.04 \begin {gather*} -\frac {1}{3087} \, \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 {125}{12} \, \sqrt {-2 \, x + 1} - \frac {121 \, {\left (432 \, x - 139\right )}}{588 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3087*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125/12*sqrt(-2*x
+ 1) - 121/588*(432*x - 139)/((2*x - 1)*sqrt(-2*x + 1))

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maple [A]  time = 0.01, size = 47, normalized size = 0.70 \begin {gather*} \frac {2 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{3087}+\frac {1331}{84 \left (-2 x +1\right )^{\frac {3}{2}}}-\frac {2178}{49 \sqrt {-2 x +1}}-\frac {125 \sqrt {-2 x +1}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1331/84/(-2*x+1)^(3/2)+2/3087*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)-2178/49/(-2*x+1)^(1/2)-125/12*(-2*
x+1)^(1/2)

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maxima [A]  time = 1.15, size = 60, normalized size = 0.90 \begin {gather*} -\frac {1}{3087} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {125}{12} \, \sqrt {-2 \, x + 1} + \frac {121 \, {\left (432 \, x - 139\right )}}{588 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3087*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125/12*sqrt(-2*x + 1) + 1
21/588*(432*x - 139)/(-2*x + 1)^(3/2)

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mupad [B]  time = 0.06, size = 41, normalized size = 0.61 \begin {gather*} \frac {\frac {4356\,x}{49}-\frac {16819}{588}}{{\left (1-2\,x\right )}^{3/2}}+\frac {2\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{3087}-\frac {125\,\sqrt {1-2\,x}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((4356*x)/49 - 16819/588)/(1 - 2*x)^(3/2) + (2*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/3087 - (125*(1 -
2*x)^(1/2))/12

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sympy [A]  time = 58.15, size = 102, normalized size = 1.52 \begin {gather*} - \frac {125 \sqrt {1 - 2 x}}{12} - \frac {2 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 < - \frac {7}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 > - \frac {7}{3} \end {cases}\right )}{147} - \frac {2178}{49 \sqrt {1 - 2 x}} + \frac {1331}{84 \left (1 - 2 x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125*sqrt(1 - 2*x)/12 - 2*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 < -7/3), (-sqrt(21)
*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 > -7/3))/147 - 2178/(49*sqrt(1 - 2*x)) + 1331/(84*(1 - 2*x)**(3/2
))

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