3.20.43 \(\int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^6} \, dx\)

Optimal. Leaf size=143 \[ \frac {123}{16807 \sqrt {1-2 x}}-\frac {41}{4802 \sqrt {1-2 x} (3 x+2)}-\frac {41}{3430 \sqrt {1-2 x} (3 x+2)^2}-\frac {41}{1715 \sqrt {1-2 x} (3 x+2)^3}-\frac {41}{735 \sqrt {1-2 x} (3 x+2)^4}+\frac {1}{105 \sqrt {1-2 x} (3 x+2)^5}-\frac {123 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{16807} \]

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Rubi [A]  time = 0.05, antiderivative size = 150, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \begin {gather*} -\frac {369 \sqrt {1-2 x}}{33614 (3 x+2)}-\frac {123 \sqrt {1-2 x}}{4802 (3 x+2)^2}-\frac {123 \sqrt {1-2 x}}{1715 (3 x+2)^3}-\frac {369 \sqrt {1-2 x}}{1715 (3 x+2)^4}+\frac {328}{735 \sqrt {1-2 x} (3 x+2)^4}+\frac {1}{105 \sqrt {1-2 x} (3 x+2)^5}-\frac {123 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{16807} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/((1 - 2*x)^(3/2)*(2 + 3*x)^6),x]

[Out]

1/(105*Sqrt[1 - 2*x]*(2 + 3*x)^5) + 328/(735*Sqrt[1 - 2*x]*(2 + 3*x)^4) - (369*Sqrt[1 - 2*x])/(1715*(2 + 3*x)^
4) - (123*Sqrt[1 - 2*x])/(1715*(2 + 3*x)^3) - (123*Sqrt[1 - 2*x])/(4802*(2 + 3*x)^2) - (369*Sqrt[1 - 2*x])/(33
614*(2 + 3*x)) - (123*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/16807

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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}{(1-2 x)^{3/2} (2+3 x)^6} \, dx &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {164}{105} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}+\frac {1476}{245} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}-\frac {369 \sqrt {1-2 x}}{1715 (2+3 x)^4}+\frac {369}{245} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}-\frac {369 \sqrt {1-2 x}}{1715 (2+3 x)^4}-\frac {123 \sqrt {1-2 x}}{1715 (2+3 x)^3}+\frac {123}{343} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}-\frac {369 \sqrt {1-2 x}}{1715 (2+3 x)^4}-\frac {123 \sqrt {1-2 x}}{1715 (2+3 x)^3}-\frac {123 \sqrt {1-2 x}}{4802 (2+3 x)^2}+\frac {369 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{4802}\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}-\frac {369 \sqrt {1-2 x}}{1715 (2+3 x)^4}-\frac {123 \sqrt {1-2 x}}{1715 (2+3 x)^3}-\frac {123 \sqrt {1-2 x}}{4802 (2+3 x)^2}-\frac {369 \sqrt {1-2 x}}{33614 (2+3 x)}+\frac {369 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{33614}\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}-\frac {369 \sqrt {1-2 x}}{1715 (2+3 x)^4}-\frac {123 \sqrt {1-2 x}}{1715 (2+3 x)^3}-\frac {123 \sqrt {1-2 x}}{4802 (2+3 x)^2}-\frac {369 \sqrt {1-2 x}}{33614 (2+3 x)}-\frac {369 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{33614}\\ &=\frac {1}{105 \sqrt {1-2 x} (2+3 x)^5}+\frac {328}{735 \sqrt {1-2 x} (2+3 x)^4}-\frac {369 \sqrt {1-2 x}}{1715 (2+3 x)^4}-\frac {123 \sqrt {1-2 x}}{1715 (2+3 x)^3}-\frac {123 \sqrt {1-2 x}}{4802 (2+3 x)^2}-\frac {369 \sqrt {1-2 x}}{33614 (2+3 x)}-\frac {123 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{16807}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 42, normalized size = 0.29 \begin {gather*} \frac {5248 \, _2F_1\left (-\frac {1}{2},5;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )+\frac {16807}{(3 x+2)^5}}{1764735 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)/((1 - 2*x)^(3/2)*(2 + 3*x)^6),x]

[Out]

(16807/(2 + 3*x)^5 + 5248*Hypergeometric2F1[-1/2, 5, 1/2, 3/7 - (6*x)/7])/(1764735*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.38, size = 99, normalized size = 0.69 \begin {gather*} \frac {149445 (1-2 x)^5-1627290 (1-2 x)^4+6943104 (1-2 x)^3-14283990 (1-2 x)^2+13570795 (1-2 x)-4225760}{84035 (3 (1-2 x)-7)^5 \sqrt {1-2 x}}-\frac {123 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{16807} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(3 + 5*x)/((1 - 2*x)^(3/2)*(2 + 3*x)^6),x]

[Out]

(-4225760 + 13570795*(1 - 2*x) - 14283990*(1 - 2*x)^2 + 6943104*(1 - 2*x)^3 - 1627290*(1 - 2*x)^4 + 149445*(1
- 2*x)^5)/(84035*(-7 + 3*(1 - 2*x))^5*Sqrt[1 - 2*x]) - (123*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/16807

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fricas [A]  time = 1.38, size = 135, normalized size = 0.94 \begin {gather*} \frac {615 \, \sqrt {7} \sqrt {3} {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (298890 \, x^{5} + 880065 \, x^{4} + 964197 \, x^{3} + 430992 \, x^{2} + 8774 \, x - 32894\right )} \sqrt {-2 \, x + 1}}{1176490 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)^(3/2)/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/1176490*(615*sqrt(7)*sqrt(3)*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*log((sqrt(7)*s
qrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 7*(298890*x^5 + 880065*x^4 + 964197*x^3 + 430992*x^2 + 8774*x -
32894)*sqrt(-2*x + 1))/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)

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giac [A]  time = 1.23, size = 125, normalized size = 0.87 \begin {gather*} \frac {123}{235298} \, \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 {352}{117649 \, \sqrt {-2 \, x + 1}} - \frac {618435 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 6401430 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 25316928 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 45656730 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 31609165 \, \sqrt {-2 \, x + 1}}{18823840 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)^(3/2)/(2+3*x)^6,x, algorithm="giac")

[Out]

123/235298*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 352/117649/sq
rt(-2*x + 1) - 1/18823840*(618435*(2*x - 1)^4*sqrt(-2*x + 1) + 6401430*(2*x - 1)^3*sqrt(-2*x + 1) + 25316928*(
2*x - 1)^2*sqrt(-2*x + 1) - 45656730*(-2*x + 1)^(3/2) + 31609165*sqrt(-2*x + 1))/(3*x + 2)^5

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maple [A]  time = 0.02, size = 84, normalized size = 0.59 \begin {gather*} -\frac {123 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{117649}+\frac {352}{117649 \sqrt {-2 x +1}}+\frac {\frac {123687 \left (-2 x +1\right )^{\frac {9}{2}}}{117649}-\frac {182898 \left (-2 x +1\right )^{\frac {7}{2}}}{16807}+\frac {516672 \left (-2 x +1\right )^{\frac {5}{2}}}{12005}-\frac {26622 \left (-2 x +1\right )^{\frac {3}{2}}}{343}+\frac {2633 \sqrt {-2 x +1}}{49}}{\left (-6 x -4\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)/(-2*x+1)^(3/2)/(3*x+2)^6,x)

[Out]

352/117649/(-2*x+1)^(1/2)+7776/117649*(509/32*(-2*x+1)^(9/2)-7903/48*(-2*x+1)^(7/2)+29302/45*(-2*x+1)^(5/2)-16
9099/144*(-2*x+1)^(3/2)+6321833/7776*(-2*x+1)^(1/2))/(-6*x-4)^5-123/117649*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2)
)*21^(1/2)

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maxima [A]  time = 1.19, size = 137, normalized size = 0.96 \begin {gather*} \frac {123}{235298} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {149445 \, {\left (2 \, x - 1\right )}^{5} + 1627290 \, {\left (2 \, x - 1\right )}^{4} + 6943104 \, {\left (2 \, x - 1\right )}^{3} + 14283990 \, {\left (2 \, x - 1\right )}^{2} + 27141590 \, x - 9345035}{84035 \, {\left (243 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 2835 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 13230 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 30870 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 36015 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 16807 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)^(3/2)/(2+3*x)^6,x, algorithm="maxima")

[Out]

123/235298*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/84035*(149445*(2*x -
1)^5 + 1627290*(2*x - 1)^4 + 6943104*(2*x - 1)^3 + 14283990*(2*x - 1)^2 + 27141590*x - 9345035)/(243*(-2*x +
1)^(11/2) - 2835*(-2*x + 1)^(9/2) + 13230*(-2*x + 1)^(7/2) - 30870*(-2*x + 1)^(5/2) + 36015*(-2*x + 1)^(3/2) -
16807*sqrt(-2*x + 1))

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mupad [B]  time = 0.09, size = 118, normalized size = 0.83 \begin {gather*} \frac {\frac {15826\,x}{11907}+\frac {6478\,{\left (2\,x-1\right )}^2}{9261}+\frac {5248\,{\left (2\,x-1\right )}^3}{15435}+\frac {82\,{\left (2\,x-1\right )}^4}{1029}+\frac {123\,{\left (2\,x-1\right )}^5}{16807}-\frac {5449}{11907}}{\frac {16807\,\sqrt {1-2\,x}}{243}-\frac {12005\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {3430\,{\left (1-2\,x\right )}^{5/2}}{27}-\frac {490\,{\left (1-2\,x\right )}^{7/2}}{9}+\frac {35\,{\left (1-2\,x\right )}^{9/2}}{3}-{\left (1-2\,x\right )}^{11/2}}-\frac {123\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{117649} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)/((1 - 2*x)^(3/2)*(3*x + 2)^6),x)

[Out]

((15826*x)/11907 + (6478*(2*x - 1)^2)/9261 + (5248*(2*x - 1)^3)/15435 + (82*(2*x - 1)^4)/1029 + (123*(2*x - 1)
^5)/16807 - 5449/11907)/((16807*(1 - 2*x)^(1/2))/243 - (12005*(1 - 2*x)^(3/2))/81 + (3430*(1 - 2*x)^(5/2))/27
- (490*(1 - 2*x)^(7/2))/9 + (35*(1 - 2*x)^(9/2))/3 - (1 - 2*x)^(11/2)) - (123*21^(1/2)*atanh((21^(1/2)*(1 - 2*
x)^(1/2))/7))/117649

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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)/(1-2*x)**(3/2)/(2+3*x)**6,x)

[Out]

Timed out

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