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

Optimal. Leaf size=80 \[ \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^2}+\frac {5 \sqrt {1-2 x} (857 x+541)}{2058 (3 x+2)^2}+\frac {2245 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]

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Rubi [A]  time = 0.02, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {98, 145, 63, 206} \begin {gather*} \frac {11 (5 x+3)^2}{7 \sqrt {1-2 x} (3 x+2)^2}+\frac {5 \sqrt {1-2 x} (857 x+541)}{2058 (3 x+2)^2}+\frac {2245 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(11*(3 + 5*x)^2)/(7*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*Sqrt[1 - 2*x]*(541 + 857*x))/(2058*(2 + 3*x)^2) + (2245*Ar
cTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 145

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b^2*(b*c - a*d)^2*(m + 1)*(m
+ 2)), x] + Dist[(f*h)/b^2 - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] &&  !L
tQ[n, -2]))

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)^{3/2} (2+3 x)^3} \, dx &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^2}-\frac {1}{7} \int \frac {(3+5 x) (25+5 x)}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (541+857 x)}{2058 (2+3 x)^2}-\frac {2245 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2058}\\ &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (541+857 x)}{2058 (2+3 x)^2}+\frac {2245 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2058}\\ &=\frac {11 (3+5 x)^2}{7 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {1-2 x} (541+857 x)}{2058 (2+3 x)^2}+\frac {2245 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 59, normalized size = 0.74 \begin {gather*} \frac {7 \left (36750 x^2+48795 x+16199\right )-4490 (3 x+2)^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )}{6174 \sqrt {1-2 x} (3 x+2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(16199 + 48795*x + 36750*x^2) - 4490*(2 + 3*x)^2*Hypergeometric2F1[-1/2, 1, 1/2, 3/7 - (6*x)/7])/(6174*Sqrt
[1 - 2*x]*(2 + 3*x)^2)

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IntegrateAlgebraic [A]  time = 0.18, size = 70, normalized size = 0.88 \begin {gather*} \frac {36140 (1-2 x)^2-168175 (1-2 x)+195657}{1029 (3 (1-2 x)-7)^2 \sqrt {1-2 x}}+\frac {2245 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(195657 - 168175*(1 - 2*x) + 36140*(1 - 2*x)^2)/(1029*(-7 + 3*(1 - 2*x))^2*Sqrt[1 - 2*x]) + (2245*ArcTanh[Sqrt
[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21])

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fricas [A]  time = 1.28, size = 85, normalized size = 1.06 \begin {gather*} \frac {2245 \, \sqrt {21} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (72280 \, x^{2} + 95895 \, x + 31811\right )} \sqrt {-2 \, x + 1}}{43218 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/43218*(2245*sqrt(21)*(18*x^3 + 15*x^2 - 4*x - 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(72
280*x^2 + 95895*x + 31811)*sqrt(-2*x + 1))/(18*x^3 + 15*x^2 - 4*x - 4)

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giac [A]  time = 1.28, size = 77, normalized size = 0.96 \begin {gather*} -\frac {2245}{43218} \, \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 {1331}{343 \, \sqrt {-2 \, x + 1}} + \frac {29 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 67 \, \sqrt {-2 \, x + 1}}{588 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2245/43218*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1331/343/sqr
t(-2*x + 1) + 1/588*(29*(-2*x + 1)^(3/2) - 67*sqrt(-2*x + 1))/(3*x + 2)^2

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maple [A]  time = 0.01, size = 57, normalized size = 0.71 \begin {gather*} \frac {2245 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{21609}+\frac {1331}{343 \sqrt {-2 x +1}}-\frac {18 \left (-\frac {203 \left (-2 x +1\right )^{\frac {3}{2}}}{54}+\frac {469 \sqrt {-2 x +1}}{54}\right )}{343 \left (-6 x -4\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1331/343/(-2*x+1)^(1/2)-18/343*(-203/54*(-2*x+1)^(3/2)+469/54*(-2*x+1)^(1/2))/(-6*x-4)^2+2245/21609*arctanh(1/
7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.30, size = 83, normalized size = 1.04 \begin {gather*} -\frac {2245}{43218} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (18070 \, {\left (2 \, x - 1\right )}^{2} + 168175 \, x + 13741\right )}}{1029 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 49 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2245/43218*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/1029*(18070*(2*x -
1)^2 + 168175*x + 13741)/(9*(-2*x + 1)^(5/2) - 42*(-2*x + 1)^(3/2) + 49*sqrt(-2*x + 1))

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mupad [B]  time = 0.07, size = 62, normalized size = 0.78 \begin {gather*} \frac {2245\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609}+\frac {\frac {48050\,x}{1323}+\frac {36140\,{\left (2\,x-1\right )}^2}{9261}+\frac {3926}{1323}}{\frac {49\,\sqrt {1-2\,x}}{9}-\frac {14\,{\left (1-2\,x\right )}^{3/2}}{3}+{\left (1-2\,x\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2245*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/21609 + ((48050*x)/1323 + (36140*(2*x - 1)^2)/9261 + 3926/
1323)/((49*(1 - 2*x)^(1/2))/9 - (14*(1 - 2*x)^(3/2))/3 + (1 - 2*x)^(5/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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