3.2103 \(\int \frac{(3+5 x)^3}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\)

Optimal. Leaf size=120 \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)^4}+\frac{\sqrt{1-2 x} (3789 x+2395)}{1764 (3 x+2)^4}-\frac{39185 \sqrt{1-2 x}}{57624 (3 x+2)}-\frac{39185 \sqrt{1-2 x}}{24696 (3 x+2)^2}-\frac{39185 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28812 \sqrt{21}} \]

[Out]

(-39185*Sqrt[1 - 2*x])/(24696*(2 + 3*x)^2) - (39185*Sqrt[1 - 2*x])/(57624*(2 + 3*x)) + (11*(3 + 5*x)^2)/(7*Sqr
t[1 - 2*x]*(2 + 3*x)^4) + (Sqrt[1 - 2*x]*(2395 + 3789*x))/(1764*(2 + 3*x)^4) - (39185*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/(28812*Sqrt[21])

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Rubi [A]  time = 0.0336001, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 145, 51, 63, 206} \[ \frac{11 (5 x+3)^2}{7 \sqrt{1-2 x} (3 x+2)^4}+\frac{\sqrt{1-2 x} (3789 x+2395)}{1764 (3 x+2)^4}-\frac{39185 \sqrt{1-2 x}}{57624 (3 x+2)}-\frac{39185 \sqrt{1-2 x}}{24696 (3 x+2)^2}-\frac{39185 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28812 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-39185*Sqrt[1 - 2*x])/(24696*(2 + 3*x)^2) - (39185*Sqrt[1 - 2*x])/(57624*(2 + 3*x)) + (11*(3 + 5*x)^2)/(7*Sqr
t[1 - 2*x]*(2 + 3*x)^4) + (Sqrt[1 - 2*x]*(2395 + 3789*x))/(1764*(2 + 3*x)^4) - (39185*ArcTanh[Sqrt[3/7]*Sqrt[1
- 2*x]])/(28812*Sqrt[21])

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 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)^3}{(1-2 x)^{3/2} (2+3 x)^5} \, dx &=\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^4}-\frac{1}{7} \int \frac{(-173-325 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)^5} \, dx\\ &=\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^4}+\frac{\sqrt{1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}+\frac{39185 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{1764}\\ &=-\frac{39185 \sqrt{1-2 x}}{24696 (2+3 x)^2}+\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^4}+\frac{\sqrt{1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}+\frac{39185 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{8232}\\ &=-\frac{39185 \sqrt{1-2 x}}{24696 (2+3 x)^2}-\frac{39185 \sqrt{1-2 x}}{57624 (2+3 x)}+\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^4}+\frac{\sqrt{1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}+\frac{39185 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{57624}\\ &=-\frac{39185 \sqrt{1-2 x}}{24696 (2+3 x)^2}-\frac{39185 \sqrt{1-2 x}}{57624 (2+3 x)}+\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^4}+\frac{\sqrt{1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}-\frac{39185 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{57624}\\ &=-\frac{39185 \sqrt{1-2 x}}{24696 (2+3 x)^2}-\frac{39185 \sqrt{1-2 x}}{57624 (2+3 x)}+\frac{11 (3+5 x)^2}{7 \sqrt{1-2 x} (2+3 x)^4}+\frac{\sqrt{1-2 x} (2395+3789 x)}{1764 (2+3 x)^4}-\frac{39185 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28812 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0174932, size = 59, normalized size = 0.49 \[ \frac{62696 (3 x+2)^4 \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+49 \left (44100 x^2+58389 x+19333\right )}{259308 \sqrt{1-2 x} (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(49*(19333 + 58389*x + 44100*x^2) + 62696*(2 + 3*x)^4*Hypergeometric2F1[-1/2, 3, 1/2, 3/7 - (6*x)/7])/(259308*
Sqrt[1 - 2*x]*(2 + 3*x)^4)

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Maple [A]  time = 0.012, size = 75, normalized size = 0.6 \begin{align*}{\frac{324}{16807\, \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{82631}{144} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{5020939}{1296} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{33905795}{3888} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{25445455}{3888}\sqrt{1-2\,x}} \right ) }-{\frac{39185\,\sqrt{21}}{605052}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{5324}{16807}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

324/16807*(82631/144*(1-2*x)^(7/2)-5020939/1296*(1-2*x)^(5/2)[phone]/3888*(1-2*x)^(3/2)-25445455/3888*(1-2*x
)^(1/2))/(-6*x-4)^4-39185/605052*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+5324/16807/(1-2*x)^(1/2)

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Maxima [A]  time = 2.88735, size = 161, normalized size = 1.34 \begin{align*} \frac{39185}{1210104} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1057995 \,{\left (2 \, x - 1\right )}^{4} + 9051735 \,{\left (2 \, x - 1\right )}^{3} + 28993349 \,{\left (2 \, x - 1\right )}^{2} + 82402418 \, x - 19287625}{28812 \,{\left (81 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 756 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 2646 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 4116 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2401 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

39185/1210104*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/28812*(1057995*(2
*x - 1)^4 + 9051735*(2*x - 1)^3 + 28993349*(2*x - 1)^2 + 82402418*x - 19287625)/(81*(-2*x + 1)^(9/2) - 756*(-2
*x + 1)^(7/2) + 2646*(-2*x + 1)^(5/2) - 4116*(-2*x + 1)^(3/2) + 2401*sqrt(-2*x + 1))

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Fricas [A]  time = 1.58455, size = 356, normalized size = 2.97 \begin{align*} \frac{39185 \, \sqrt{21}{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (2115990 \, x^{4} + 4819755 \, x^{3} + 4093057 \, x^{2} + 1534434 \, x + 213998\right )} \sqrt{-2 \, x + 1}}{1210104 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1210104*(39185*sqrt(21)*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log((3*x + sqrt(21)*sqrt(-2*x + 1
) - 5)/(3*x + 2)) - 21*(2115990*x^4 + 4819755*x^3 + 4093057*x^2 + 1534434*x + 213998)*sqrt(-2*x + 1))/(162*x^5
+ 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.09045, size = 147, normalized size = 1.22 \begin{align*} \frac{39185}{1210104} \, \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{5324}{16807 \, \sqrt{-2 \, x + 1}} - \frac{2231037 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 15062817 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 33905795 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 25445455 \, \sqrt{-2 \, x + 1}}{3226944 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

39185/1210104*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 5324/16807
/sqrt(-2*x + 1) - 1/3226944*(2231037*(2*x - 1)^3*sqrt(-2*x + 1) + 15062817*(2*x - 1)^2*sqrt(-2*x + 1) - 339057
95*(-2*x + 1)^(3/2) + 25445455*sqrt(-2*x + 1))/(3*x + 2)^4