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

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

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Rubi [A]  time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 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 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 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*}

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Mathematica [C]  time = 0.03, size = 62, normalized size = 0.62 \begin {gather*} -\frac {60472 \left (10 x^2+x-3\right )^2 \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};-\frac {5}{11} (2 x-1)\right )-1331 \left (3267 x^2+3103 x+2348\right )}{2415765 (1-2 x)^{3/2} (5 x+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2415765*(-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])/((1 - 2*x)^(3/2)*(3 + 5*x)^2)

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IntegrateAlgebraic [A]  time = 0.19, size = 79, normalized size = 0.79 \begin {gather*} \frac {113385 (1-2 x)^3-20438 (1-2 x)^2-1541540 (1-2 x)+2282665}{219615 (5 (1-2 x)-11)^2 (1-2 x)^{3/2}}-\frac {7559 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{73205 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2282665 - 1541540*(1 - 2*x) - 20438*(1 - 2*x)^2 + 113385*(1 - 2*x)^3)/(219615*(-11 + 5*(1 - 2*x))^2*(1 - 2*x)
^(3/2)) - (7559*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(73205*Sqrt[55])

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fricas [A]  time = 1.19, size = 99, normalized size = 0.99 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.25, size = 89, normalized size = 0.89 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.02, size = 66, normalized size = 0.66 \begin {gather*} -\frac {7559 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{4026275}+\frac {343}{3993 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {294}{14641 \sqrt {-2 x +1}}+\frac {\frac {19 \left (-2 x +1\right )^{\frac {3}{2}}}{1331}-\frac {211 \sqrt {-2 x +1}}{6655}}{\left (-10 x -6\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 92, normalized size = 0.92 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 1.22, size = 71, normalized size = 0.71 \begin {gather*} \frac {\frac {5096\,x}{9075}-\frac {1858\,{\left (2\,x-1\right )}^2}{499125}-\frac {7559\,{\left (2\,x-1\right )}^3}{366025}+\frac {49}{363}}{\frac {121\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {22\,{\left (1-2\,x\right )}^{5/2}}{5}+{\left (1-2\,x\right )}^{7/2}}-\frac {7559\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{4026275} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5096*x)/9075 - (1858*(2*x - 1)^2)/499125 - (7559*(2*x - 1)^3)/366025 + 49/363)/((121*(1 - 2*x)^(3/2))/25 - (
22*(1 - 2*x)^(5/2))/5 + (1 - 2*x)^(7/2)) - (7559*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/4026275

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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