3.2193 \(\int \frac{(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=100 \[ \frac{7 (3 x+2)^2}{33 (1-2 x)^{3/2} (5 x+3)^2}+\frac{17296 x+10217}{39930 \sqrt{1-2 x} (5 x+3)^2}-\frac{7559 \sqrt{1-2 x}}{146410 (5 x+3)}-\frac{7559 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{73205 \sqrt{55}} \]

[Out]

(7*(2 + 3*x)^2)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - (7559*Sqrt[1 - 2*x])/(146410*(3 + 5*x)) + (10217 + 17296*x)
/(39930*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (7559*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(73205*Sqrt[55])

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Rubi [A]  time = 0.0256688, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 144, 51, 63, 206} \[ \frac{7 (3 x+2)^2}{33 (1-2 x)^{3/2} (5 x+3)^2}+\frac{17296 x+10217}{39930 \sqrt{1-2 x} (5 x+3)^2}-\frac{7559 \sqrt{1-2 x}}{146410 (5 x+3)}-\frac{7559 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{73205 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(2 + 3*x)^2)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - (7559*Sqrt[1 - 2*x])/(146410*(3 + 5*x)) + (10217 + 17296*x)
/(39930*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (7559*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(73205*Sqrt[55])

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 144

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :>
Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(
m + n + 2)) + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m +
1) + d^2*e*g*(m + n + 2)))*x)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1))/(b*d*(b*c - a*d)^2*(m + 1)*(n + 1)), x] -
Dist[(a^2*d^2*f*h*(2 + 3*n + n^2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h
*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b
*c - a*d)^2*(m + 1)*(n + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, h
}, x] && LtQ[m, -1] && LtQ[n, -1]

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{(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)^3} \, dx &=\frac{7 (2+3 x)^2}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{1}{33} \int \frac{(-20-9 x) (2+3 x)}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\\ &=\frac{7 (2+3 x)^2}{33 (1-2 x)^{3/2} (3+5 x)^2}+\frac{10217+17296 x}{39930 \sqrt{1-2 x} (3+5 x)^2}+\frac{7559 \int \frac{1}{\sqrt{1-2 x} (3+5 x)^2} \, dx}{13310}\\ &=\frac{7 (2+3 x)^2}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{7559 \sqrt{1-2 x}}{146410 (3+5 x)}+\frac{10217+17296 x}{39930 \sqrt{1-2 x} (3+5 x)^2}+\frac{7559 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{146410}\\ &=\frac{7 (2+3 x)^2}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{7559 \sqrt{1-2 x}}{146410 (3+5 x)}+\frac{10217+17296 x}{39930 \sqrt{1-2 x} (3+5 x)^2}-\frac{7559 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{146410}\\ &=\frac{7 (2+3 x)^2}{33 (1-2 x)^{3/2} (3+5 x)^2}-\frac{7559 \sqrt{1-2 x}}{146410 (3+5 x)}+\frac{10217+17296 x}{39930 \sqrt{1-2 x} (3+5 x)^2}-\frac{7559 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{73205 \sqrt{55}}\\ \end{align*}

Mathematica [C]  time = 0.0187099, size = 62, normalized size = 0.62 \[ -\frac{60472 \left (10 x^2+x-3\right )^2 \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{5}{11} (1-2 x)\right )-1331 \left (3267 x^2+3103 x+2348\right )}{2415765 (1-2 x)^{3/2} (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-1331*(2348 + 3103*x + 3267*x^2) + 60472*(-3 + x + 10*x^2)^2*Hypergeometric2F1[1/2, 3, 3/2, (5*(1 - 2*x))/11
])/(2415765*(1 - 2*x)^(3/2)*(3 + 5*x)^2)

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Maple [A]  time = 0.013, size = 66, normalized size = 0.7 \begin{align*}{\frac{343}{3993} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{294}{14641}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{50}{14641\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{209}{50} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{2321}{250}\sqrt{1-2\,x}} \right ) }-{\frac{7559\,\sqrt{55}}{4026275}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/3993/(1-2*x)^(3/2)+294/14641/(1-2*x)^(1/2)+50/14641*(209/50*(1-2*x)^(3/2)-2321/250*(1-2*x)^(1/2))/(-10*x-6
)^2-7559/4026275*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 2.98886, size = 124, normalized size = 1.24 \begin{align*} \frac{7559}{8052550} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{113385 \,{\left (2 \, x - 1\right )}^{3} + 20438 \,{\left (2 \, x - 1\right )}^{2} - 3083080 \, x - 741125}{219615 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 121 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7559/8052550*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 1/219615*(113385*(2*
x - 1)^3 + 20438*(2*x - 1)^2 - 3083080*x - 741125)/(25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 121*(-2*x + 1
)^(3/2))

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Fricas [A]  time = 1.06081, size = 301, normalized size = 3.01 \begin{align*} \frac{22677 \, \sqrt{55}{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (453540 \, x^{3} - 639434 \, x^{2} - 1242261 \, x - 417036\right )} \sqrt{-2 \, x + 1}}{24157650 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24157650*(22677*sqrt(55)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x
+ 3)) - 55*(453540*x^3 - 639434*x^2 - 1242261*x - 417036)*sqrt(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9
)

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.98475, size = 120, normalized size = 1.2 \begin{align*} \frac{7559}{8052550} \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{49 \,{\left (36 \, x - 95\right )}}{43923 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} + \frac{95 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 211 \, \sqrt{-2 \, x + 1}}{26620 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7559/8052550*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 49/43923*(
36*x - 95)/((2*x - 1)*sqrt(-2*x + 1)) + 1/26620*(95*(-2*x + 1)^(3/2) - 211*sqrt(-2*x + 1))/(5*x + 3)^2