3.2155 \(\int \frac{(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=121 \[ \frac{1415}{7203 \sqrt{1-2 x}}-\frac{1415}{6174 \sqrt{1-2 x} (3 x+2)}-\frac{283}{882 \sqrt{1-2 x} (3 x+2)^2}-\frac{1091}{882 \sqrt{1-2 x} (3 x+2)^3}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)^3}-\frac{1415 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401 \sqrt{21}} \]

[Out]

1415/(7203*Sqrt[1 - 2*x]) + 121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - 1091/(882*Sqrt[1 - 2*x]*(2 + 3*x)^3) - 283/
(882*Sqrt[1 - 2*x]*(2 + 3*x)^2) - 1415/(6174*Sqrt[1 - 2*x]*(2 + 3*x)) - (1415*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
)/(2401*Sqrt[21])

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Rubi [A]  time = 0.0396978, antiderivative size = 128, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {89, 78, 51, 63, 206} \[ -\frac{1415 \sqrt{1-2 x}}{4802 (3 x+2)}-\frac{1415 \sqrt{1-2 x}}{2058 (3 x+2)^2}+\frac{566}{441 \sqrt{1-2 x} (3 x+2)^2}-\frac{1091}{882 \sqrt{1-2 x} (3 x+2)^3}+\frac{121}{42 (1-2 x)^{3/2} (3 x+2)^3}-\frac{1415 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - 1091/(882*Sqrt[1 - 2*x]*(2 + 3*x)^3) + 566/(441*Sqrt[1 - 2*x]*(2 + 3*x)
^2) - (1415*Sqrt[1 - 2*x])/(2058*(2 + 3*x)^2) - (1415*Sqrt[1 - 2*x])/(4802*(2 + 3*x)) - (1415*ArcTanh[Sqrt[3/7
]*Sqrt[1 - 2*x]])/(2401*Sqrt[21])

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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)^2}{(1-2 x)^{5/2} (2+3 x)^4} \, dx &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^3}-\frac{1}{42} \int \frac{-741+525 x}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^3}-\frac{1091}{882 \sqrt{1-2 x} (2+3 x)^3}+\frac{283}{63} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^3}-\frac{1091}{882 \sqrt{1-2 x} (2+3 x)^3}+\frac{566}{441 \sqrt{1-2 x} (2+3 x)^2}+\frac{1415}{147} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^3}-\frac{1091}{882 \sqrt{1-2 x} (2+3 x)^3}+\frac{566}{441 \sqrt{1-2 x} (2+3 x)^2}-\frac{1415 \sqrt{1-2 x}}{2058 (2+3 x)^2}+\frac{1415}{686} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^3}-\frac{1091}{882 \sqrt{1-2 x} (2+3 x)^3}+\frac{566}{441 \sqrt{1-2 x} (2+3 x)^2}-\frac{1415 \sqrt{1-2 x}}{2058 (2+3 x)^2}-\frac{1415 \sqrt{1-2 x}}{4802 (2+3 x)}+\frac{1415 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{4802}\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^3}-\frac{1091}{882 \sqrt{1-2 x} (2+3 x)^3}+\frac{566}{441 \sqrt{1-2 x} (2+3 x)^2}-\frac{1415 \sqrt{1-2 x}}{2058 (2+3 x)^2}-\frac{1415 \sqrt{1-2 x}}{4802 (2+3 x)}-\frac{1415 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{4802}\\ &=\frac{121}{42 (1-2 x)^{3/2} (2+3 x)^3}-\frac{1091}{882 \sqrt{1-2 x} (2+3 x)^3}+\frac{566}{441 \sqrt{1-2 x} (2+3 x)^2}-\frac{1415 \sqrt{1-2 x}}{2058 (2+3 x)^2}-\frac{1415 \sqrt{1-2 x}}{4802 (2+3 x)}-\frac{1415 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{2401 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0217526, size = 59, normalized size = 0.49 \[ -\frac{2264 (2 x-1) (3 x+2)^3 \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )-49 (1091 x+725)}{21609 (1-2 x)^{3/2} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-49*(725 + 1091*x) + 2264*(-1 + 2*x)*(2 + 3*x)^3*Hypergeometric2F1[-1/2, 3, 1/2, 3/7 - (6*x)/7])/(21609*(1 -
2*x)^(3/2)*(2 + 3*x)^3)

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Maple [A]  time = 0.014, size = 75, normalized size = 0.6 \begin{align*}{\frac{108}{16807\, \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{1721}{12} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{17395}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{78155}{108}\sqrt{1-2\,x}} \right ) }-{\frac{1415\,\sqrt{21}}{50421}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{484}{7203} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2728}{16807}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

108/16807*(1721/12*(1-2*x)^(5/2)-17395/27*(1-2*x)^(3/2)+78155/108*(1-2*x)^(1/2))/(-6*x-4)^3-1415/50421*arctanh
(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+484/7203/(1-2*x)^(3/2)+2728/16807/(1-2*x)^(1/2)

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Maxima [A]  time = 2.92978, size = 149, normalized size = 1.23 \begin{align*} \frac{1415}{100842} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{38205 \,{\left (2 \, x - 1\right )}^{4} + 237720 \,{\left (2 \, x - 1\right )}^{3} + 457611 \,{\left (2 \, x - 1\right )}^{2} + 375144 \, x - 353584}{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)^2/(1-2*x)^(5/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

1415/100842*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/7203*(38205*(2*x -
1)^4 + 237720*(2*x - 1)^3 + 457611*(2*x - 1)^2 + 375144*x - 353584)/(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 = 1.87037, size = 335, normalized size = 2.77 \begin{align*} \frac{1415 \, \sqrt{21}{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 7 \,{\left (152820 \, x^{4} + 169800 \, x^{3} - 26319 \, x^{2} - 83655 \, x - 23872\right )} \sqrt{-2 \, x + 1}}{100842 \,{\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)^2/(1-2*x)^(5/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/100842*(1415*sqrt(21)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5
)/(3*x + 2)) - 7*(152820*x^4 + 169800*x^3 - 26319*x^2 - 83655*x - 23872)*sqrt(-2*x + 1))/(108*x^5 + 108*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)**2/(1-2*x)**(5/2)/(2+3*x)**4,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.43393, size = 128, normalized size = 1.06 \begin{align*} \frac{1415}{100842} \, \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{38205 \,{\left (2 \, x - 1\right )}^{4} + 237720 \,{\left (2 \, x - 1\right )}^{3} + 457611 \,{\left (2 \, x - 1\right )}^{2} + 375144 \, x - 353584}{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)^2/(1-2*x)^(5/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

1415/100842*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/7203*(3820
5*(2*x - 1)^4 + 237720*(2*x - 1)^3 + 457611*(2*x - 1)^2 + 375144*x - 353584)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2*x
+ 1))^3