3.2144 \(\int \frac{3+5 x}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=116 \[ \frac{160}{2401 \sqrt{1-2 x}}-\frac{16}{147 (1-2 x)^{3/2} (3 x+2)}+\frac{160}{3087 (1-2 x)^{3/2}}-\frac{16}{147 (1-2 x)^{3/2} (3 x+2)^2}+\frac{1}{63 (1-2 x)^{3/2} (3 x+2)^3}-\frac{160 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401} \]

[Out]

160/(3087*(1 - 2*x)^(3/2)) + 160/(2401*Sqrt[1 - 2*x]) + 1/(63*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - 16/(147*(1 - 2*x)
^(3/2)*(2 + 3*x)^2) - 16/(147*(1 - 2*x)^(3/2)*(2 + 3*x)) - (160*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/24
01

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Rubi [A]  time = 0.031155, antiderivative size = 130, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ -\frac{240 \sqrt{1-2 x}}{2401 (3 x+2)}-\frac{80 \sqrt{1-2 x}}{343 (3 x+2)^2}+\frac{64}{147 \sqrt{1-2 x} (3 x+2)^2}+\frac{64}{441 (1-2 x)^{3/2} (3 x+2)^2}+\frac{1}{63 (1-2 x)^{3/2} (3 x+2)^3}-\frac{160 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(63*(1 - 2*x)^(3/2)*(2 + 3*x)^3) + 64/(441*(1 - 2*x)^(3/2)*(2 + 3*x)^2) + 64/(147*Sqrt[1 - 2*x]*(2 + 3*x)^2)
- (80*Sqrt[1 - 2*x])/(343*(2 + 3*x)^2) - (240*Sqrt[1 - 2*x])/(2401*(2 + 3*x)) - (160*Sqrt[3/7]*ArcTanh[Sqrt[3
/7]*Sqrt[1 - 2*x]])/2401

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)^{5/2} (2+3 x)^4} \, dx &=\frac{1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac{32}{21} \int \frac{1}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\\ &=\frac{1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac{64}{441 (1-2 x)^{3/2} (2+3 x)^2}+\frac{32}{21} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\\ &=\frac{1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac{64}{441 (1-2 x)^{3/2} (2+3 x)^2}+\frac{64}{147 \sqrt{1-2 x} (2+3 x)^2}+\frac{160}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac{64}{441 (1-2 x)^{3/2} (2+3 x)^2}+\frac{64}{147 \sqrt{1-2 x} (2+3 x)^2}-\frac{80 \sqrt{1-2 x}}{343 (2+3 x)^2}+\frac{240}{343} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac{64}{441 (1-2 x)^{3/2} (2+3 x)^2}+\frac{64}{147 \sqrt{1-2 x} (2+3 x)^2}-\frac{80 \sqrt{1-2 x}}{343 (2+3 x)^2}-\frac{240 \sqrt{1-2 x}}{2401 (2+3 x)}+\frac{240 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{2401}\\ &=\frac{1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac{64}{441 (1-2 x)^{3/2} (2+3 x)^2}+\frac{64}{147 \sqrt{1-2 x} (2+3 x)^2}-\frac{80 \sqrt{1-2 x}}{343 (2+3 x)^2}-\frac{240 \sqrt{1-2 x}}{2401 (2+3 x)}-\frac{240 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2401}\\ &=\frac{1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac{64}{441 (1-2 x)^{3/2} (2+3 x)^2}+\frac{64}{147 \sqrt{1-2 x} (2+3 x)^2}-\frac{80 \sqrt{1-2 x}}{343 (2+3 x)^2}-\frac{240 \sqrt{1-2 x}}{2401 (2+3 x)}-\frac{160 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401}\\ \end{align*}

Mathematica [C]  time = 0.0161928, size = 42, normalized size = 0.36 \[ \frac{256 \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+\frac{343}{(3 x+2)^3}}{21609 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(343/(2 + 3*x)^3 + 256*Hypergeometric2F1[-3/2, 3, -1/2, 3/7 - (6*x)/7])/(21609*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.012, size = 75, normalized size = 0.7 \begin{align*}{\frac{648}{16807\, \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{43}{3} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{1960}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2450}{27}\sqrt{1-2\,x}} \right ) }-{\frac{160\,\sqrt{21}}{16807}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{88}{7203} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{776}{16807}{\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)^(5/2)/(2+3*x)^4,x)

[Out]

648/16807*(43/3*(1-2*x)^(5/2)-1960/27*(1-2*x)^(3/2)+2450/27*(1-2*x)^(1/2))/(-6*x-4)^3-160/16807*arctanh(1/7*21
^(1/2)*(1-2*x)^(1/2))*21^(1/2)+88/7203/(1-2*x)^(3/2)+776/16807/(1-2*x)^(1/2)

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Maxima [A]  time = 2.76228, size = 149, normalized size = 1.28 \begin{align*} \frac{80}{16807} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{8 \,{\left (1620 \,{\left (2 \, x - 1\right )}^{4} + 10080 \,{\left (2 \, x - 1\right )}^{3} + 19404 \,{\left (2 \, x - 1\right )}^{2} + 18816 \, x - 13181\right )}}{7203 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 343 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

80/16807*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 8/7203*(1620*(2*x - 1)^4
+ 10080*(2*x - 1)^3 + 19404*(2*x - 1)^2 + 18816*x - 13181)/(27*(-2*x + 1)^(9/2) - 189*(-2*x + 1)^(7/2) + 441*
(-2*x + 1)^(5/2) - 343*(-2*x + 1)^(3/2))

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Fricas [A]  time = 1.65516, size = 346, normalized size = 2.98 \begin{align*} \frac{240 \, \sqrt{7} \sqrt{3}{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \,{\left (25920 \, x^{4} + 28800 \, x^{3} - 4464 \, x^{2} - 11280 \, x - 2237\right )} \sqrt{-2 \, x + 1}}{50421 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/50421*(240*sqrt(7)*sqrt(3)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log((sqrt(7)*sqrt(3)*sqrt(-2*x +
1) + 3*x - 5)/(3*x + 2)) - 7*(25920*x^4 + 28800*x^3 - 4464*x^2 - 11280*x - 2237)*sqrt(-2*x + 1))/(108*x^5 + 10
8*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)

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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)**(5/2)/(2+3*x)**4,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.12175, size = 128, normalized size = 1.1 \begin{align*} \frac{80}{16807} \, \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{8 \,{\left (1620 \,{\left (2 \, x - 1\right )}^{4} + 10080 \,{\left (2 \, x - 1\right )}^{3} + 19404 \,{\left (2 \, x - 1\right )}^{2} + 18816 \, x - 13181\right )}}{7203 \,{\left (3 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 7 \, \sqrt{-2 \, x + 1}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

80/16807*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 8/7203*(1620*(2
*x - 1)^4 + 10080*(2*x - 1)^3 + 19404*(2*x - 1)^2 + 18816*x - 13181)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x + 1))^3