3.19.93 \(\int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^4} \, dx\)

Optimal. Leaf size=80 \[ \frac {\sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}+\frac {5 \sqrt {1-2 x} (1867 x+1205)}{9261 (3 x+2)^2}-\frac {78710 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}} \]

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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 {\sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)^3}+\frac {5 \sqrt {1-2 x} (1867 x+1205)}{9261 (3 x+2)^2}-\frac {78710 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^3/(Sqrt[1 - 2*x]*(2 + 3*x)^4),x]

[Out]

(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(63*(2 + 3*x)^3) + (5*Sqrt[1 - 2*x]*(1205 + 1867*x))/(9261*(2 + 3*x)^2) - (78710*A
rcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9261*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}{\sqrt {1-2 x} (2+3 x)^4} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)^3}-\frac {1}{63} \int \frac {(-290-520 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)^3}+\frac {5 \sqrt {1-2 x} (1205+1867 x)}{9261 (2+3 x)^2}+\frac {39355 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{9261}\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)^3}+\frac {5 \sqrt {1-2 x} (1205+1867 x)}{9261 (2+3 x)^2}-\frac {39355 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{9261}\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)^3}+\frac {5 \sqrt {1-2 x} (1205+1867 x)}{9261 (2+3 x)^2}-\frac {78710 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 67, normalized size = 0.84 \begin {gather*} \frac {\sqrt {1-2 x} \left (\frac {21 \left (31680 x^2+41155 x+13373\right )}{(3 x+2)^3}-\frac {78710 \sqrt {21} \tan ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {2 x-1}\right )}{\sqrt {2 x-1}}\right )}{194481} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^3/(Sqrt[1 - 2*x]*(2 + 3*x)^4),x]

[Out]

(Sqrt[1 - 2*x]*((21*(13373 + 41155*x + 31680*x^2))/(2 + 3*x)^3 - (78710*Sqrt[21]*ArcTan[Sqrt[3/7]*Sqrt[-1 + 2*
x]])/Sqrt[-1 + 2*x]))/194481

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IntegrateAlgebraic [A]  time = 0.23, size = 70, normalized size = 0.88 \begin {gather*} -\frac {4 \sqrt {1-2 x} \left (15840 (1-2 x)^2-72835 (1-2 x)+83741\right )}{9261 (3 (1-2 x)-7)^3}-\frac {78710 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9261 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(3 + 5*x)^3/(Sqrt[1 - 2*x]*(2 + 3*x)^4),x]

[Out]

(-4*(83741 - 72835*(1 - 2*x) + 15840*(1 - 2*x)^2)*Sqrt[1 - 2*x])/(9261*(-7 + 3*(1 - 2*x))^3) - (78710*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/(9261*Sqrt[21])

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fricas [A]  time = 1.40, size = 84, normalized size = 1.05 \begin {gather*} \frac {39355 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (31680 \, x^{2} + 41155 \, x + 13373\right )} \sqrt {-2 \, x + 1}}{194481 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(2+3*x)^4/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/194481*(39355*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*
(31680*x^2 + 41155*x + 13373)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [A]  time = 1.07, size = 84, normalized size = 1.05 \begin {gather*} \frac {39355}{194481} \, \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 {15840 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 72835 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 83741 \, \sqrt {-2 \, x + 1}}{18522 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(2+3*x)^4/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

39355/194481*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/18522*(15
840*(2*x - 1)^2*sqrt(-2*x + 1) - 72835*(-2*x + 1)^(3/2) + 83741*sqrt(-2*x + 1))/(3*x + 2)^3

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maple [A]  time = 0.01, size = 57, normalized size = 0.71 \begin {gather*} -\frac {78710 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{194481}+\frac {-\frac {7040 \left (-2 x +1\right )^{\frac {5}{2}}}{1029}+\frac {41620 \left (-2 x +1\right )^{\frac {3}{2}}}{1323}-\frac {6836 \sqrt {-2 x +1}}{189}}{\left (-6 x -4\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^3/(3*x+2)^4/(-2*x+1)^(1/2),x)

[Out]

54*(-3520/27783*(-2*x+1)^(5/2)+20810/35721*(-2*x+1)^(3/2)-3418/5103*(-2*x+1)^(1/2))/(-6*x-4)^3-78710/194481*ar
ctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.36, size = 92, normalized size = 1.15 \begin {gather*} \frac {39355}{194481} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (15840 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 72835 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 83741 \, \sqrt {-2 \, x + 1}\right )}}{9261 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(2+3*x)^4/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

39355/194481*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/9261*(15840*(-2*x
+ 1)^(5/2) - 72835*(-2*x + 1)^(3/2) + 83741*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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mupad [B]  time = 0.07, size = 71, normalized size = 0.89 \begin {gather*} \frac {\frac {6836\,\sqrt {1-2\,x}}{5103}-\frac {41620\,{\left (1-2\,x\right )}^{3/2}}{35721}+\frac {7040\,{\left (1-2\,x\right )}^{5/2}}{27783}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}}-\frac {78710\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{194481} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^3/((1 - 2*x)^(1/2)*(3*x + 2)^4),x)

[Out]

((6836*(1 - 2*x)^(1/2))/5103 - (41620*(1 - 2*x)^(3/2))/35721 + (7040*(1 - 2*x)^(5/2))/27783)/((98*x)/3 + 7*(2*
x - 1)^2 + (2*x - 1)^3 - 98/27) - (78710*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/194481

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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/(2+3*x)**4/(1-2*x)**(1/2),x)

[Out]

Timed out

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