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

Optimal. Leaf size=54 \[ \frac {25}{6} \sqrt {1-2 x}+\frac {121}{14 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \]

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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 = {87, 63, 206} \begin {gather*} \frac {25}{6} \sqrt {1-2 x}+\frac {121}{14 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

121/(14*Sqrt[1 - 2*x]) + (25*Sqrt[1 - 2*x])/6 - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*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)^2}{(1-2 x)^{3/2} (2+3 x)} \, dx &=\int \left (\frac {121}{14 (1-2 x)^{3/2}}-\frac {25}{6 \sqrt {1-2 x}}+\frac {1}{21 \sqrt {1-2 x} (2+3 x)}\right ) \, dx\\ &=\frac {121}{14 \sqrt {1-2 x}}+\frac {25}{6} \sqrt {1-2 x}+\frac {1}{21} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {121}{14 \sqrt {1-2 x}}+\frac {25}{6} \sqrt {1-2 x}-\frac {1}{21} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {121}{14 \sqrt {1-2 x}}+\frac {25}{6} \sqrt {1-2 x}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 37, normalized size = 0.69 \begin {gather*} \frac {2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )-525 x+805}{63 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(805 - 525*x + 2*Hypergeometric2F1[-1/2, 1, 1/2, 3/7 - (6*x)/7])/(63*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.07, size = 50, normalized size = 0.93 \begin {gather*} \frac {175 (1-2 x)+363}{42 \sqrt {1-2 x}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(363 + 175*(1 - 2*x))/(42*Sqrt[1 - 2*x]) - (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21])

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fricas [A]  time = 1.81, size = 58, normalized size = 1.07 \begin {gather*} \frac {\sqrt {21} {\left (2 \, x - 1\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (175 \, x - 269\right )} \sqrt {-2 \, x + 1}}{441 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/441*(sqrt(21)*(2*x - 1)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(175*x - 269)*sqrt(-2*x + 1)
)/(2*x - 1)

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giac [A]  time = 1.26, size = 58, normalized size = 1.07 \begin {gather*} \frac {1}{441} \, \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 {25}{6} \, \sqrt {-2 \, x + 1} + \frac {121}{14 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/441*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 25/6*sqrt(-2*x + 1
) + 121/14/sqrt(-2*x + 1)

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maple [A]  time = 0.01, size = 38, normalized size = 0.70 \begin {gather*} -\frac {2 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{441}+\frac {121}{14 \sqrt {-2 x +1}}+\frac {25 \sqrt {-2 x +1}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/441*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)+121/14/(-2*x+1)^(1/2)+25/6*(-2*x+1)^(1/2)

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maxima [A]  time = 1.23, size = 55, normalized size = 1.02 \begin {gather*} \frac {1}{441} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {25}{6} \, \sqrt {-2 \, x + 1} + \frac {121}{14 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/441*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 25/6*sqrt(-2*x + 1) + 121/1
4/sqrt(-2*x + 1)

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mupad [B]  time = 0.06, size = 37, normalized size = 0.69 \begin {gather*} \frac {121}{14\,\sqrt {1-2\,x}}-\frac {2\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{441}+\frac {25\,\sqrt {1-2\,x}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

121/(14*(1 - 2*x)^(1/2)) - (2*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/441 + (25*(1 - 2*x)^(1/2))/6

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sympy [A]  time = 38.90, size = 90, normalized size = 1.67 \begin {gather*} \frac {25 \sqrt {1 - 2 x}}{6} + \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 )}{21} + \frac {121}{14 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25*sqrt(1 - 2*x)/6 + 2*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 < -7/3), (-sqrt(21)*at
anh(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 > -7/3))/21 + 121/(14*sqrt(1 - 2*x))

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