3.2082 \(\int \frac{3+5 x}{(1-2 x)^{3/2} (2+3 x)^6} \, dx\)

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

[Out]

123/(16807*Sqrt[1 - 2*x]) + 1/(105*Sqrt[1 - 2*x]*(2 + 3*x)^5) - 41/(735*Sqrt[1 - 2*x]*(2 + 3*x)^4) - 41/(1715*
Sqrt[1 - 2*x]*(2 + 3*x)^3) - 41/(3430*Sqrt[1 - 2*x]*(2 + 3*x)^2) - 41/(4802*Sqrt[1 - 2*x]*(2 + 3*x)) - (123*Sq
rt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/16807

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Rubi [A]  time = 0.0477408, 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} \[ -\frac{369 \sqrt{1-2 x}}{33614 (3 x+2)}-\frac{123 \sqrt{1-2 x}}{4802 (3 x+2)^2}-\frac{123 \sqrt{1-2 x}}{1715 (3 x+2)^3}-\frac{369 \sqrt{1-2 x}}{1715 (3 x+2)^4}+\frac{328}{735 \sqrt{1-2 x} (3 x+2)^4}+\frac{1}{105 \sqrt{1-2 x} (3 x+2)^5}-\frac{123 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{16807} \]

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 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 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 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*}

Mathematica [C]  time = 0.0177121, size = 42, normalized size = 0.29 \[ \frac{5248 \, _2F_1\left (-\frac{1}{2},5;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+\frac{16807}{(3 x+2)^5}}{1764735 \sqrt{1-2 x}} \]

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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Maple [A]  time = 0.013, size = 84, normalized size = 0.6 \begin{align*}{\frac{7776}{117649\, \left ( -6\,x-4 \right ) ^{5}} \left ({\frac{509}{32} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{7903}{48} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{29302}{45} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{169099}{144} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{6321833}{7776}\sqrt{1-2\,x}} \right ) }-{\frac{123\,\sqrt{21}}{117649}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{352}{117649}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7776/117649*(509/32*(1-2*x)^(9/2)-7903/48*(1-2*x)^(7/2)+29302/45*(1-2*x)^(5/2)-169099/144*(1-2*x)^(3/2)+632183
3/7776*(1-2*x)^(1/2))/(-6*x-4)^5-123/117649*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+352/117649/(1-2*x)^(1
/2)

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Maxima [A]  time = 4.43122, size = 185, normalized size = 1.29 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.65621, size = 417, normalized size = 2.92 \begin{align*} \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{align*}

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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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Giac [A]  time = 2.22629, size = 169, normalized size = 1.18 \begin{align*} \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{align*}

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