3.2168 \(\int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^6} \, dx\)

Optimal. Leaf size=160 \[ \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^5}+\frac {2 (83544 x+55633)}{5145 \sqrt {1-2 x} (3 x+2)^5}-\frac {81737 \sqrt {1-2 x}}{352947 (3 x+2)}-\frac {81737 \sqrt {1-2 x}}{151263 (3 x+2)^2}-\frac {163474 \sqrt {1-2 x}}{108045 (3 x+2)^3}-\frac {163474 \sqrt {1-2 x}}{36015 (3 x+2)^4}-\frac {163474 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{352947 \sqrt {21}} \]

[Out]

11/21*(3+5*x)^2/(1-2*x)^(3/2)/(2+3*x)^5-163474/7411887*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+2/5145*(55
633+83544*x)/(2+3*x)^5/(1-2*x)^(1/2)-163474/36015*(1-2*x)^(1/2)/(2+3*x)^4-163474/108045*(1-2*x)^(1/2)/(2+3*x)^
3-81737/151263*(1-2*x)^(1/2)/(2+3*x)^2-81737/352947*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]  time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^5}+\frac {2 (83544 x+55633)}{5145 \sqrt {1-2 x} (3 x+2)^5}-\frac {81737 \sqrt {1-2 x}}{352947 (3 x+2)}-\frac {81737 \sqrt {1-2 x}}{151263 (3 x+2)^2}-\frac {163474 \sqrt {1-2 x}}{108045 (3 x+2)^3}-\frac {163474 \sqrt {1-2 x}}{36015 (3 x+2)^4}-\frac {163474 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{352947 \sqrt {21}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^3/((1 - 2*x)^(5/2)*(2 + 3*x)^6),x]

[Out]

(-163474*Sqrt[1 - 2*x])/(36015*(2 + 3*x)^4) - (163474*Sqrt[1 - 2*x])/(108045*(2 + 3*x)^3) - (81737*Sqrt[1 - 2*
x])/(151263*(2 + 3*x)^2) - (81737*Sqrt[1 - 2*x])/(352947*(2 + 3*x)) + (11*(3 + 5*x)^2)/(21*(1 - 2*x)^(3/2)*(2
+ 3*x)^5) + (2*(55633 + 83544*x))/(5145*Sqrt[1 - 2*x]*(2 + 3*x)^5) - (163474*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])
/(352947*Sqrt[21])

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 {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^6} \, dx &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^5}-\frac {1}{21} \int \frac {(-266-480 x) (3+5 x)}{(1-2 x)^{3/2} (2+3 x)^6} \, dx\\ &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^5}+\frac {2 (55633+83544 x)}{5145 \sqrt {1-2 x} (2+3 x)^5}+\frac {653896 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^5} \, dx}{5145}\\ &=-\frac {163474 \sqrt {1-2 x}}{36015 (2+3 x)^4}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^5}+\frac {2 (55633+83544 x)}{5145 \sqrt {1-2 x} (2+3 x)^5}+\frac {163474 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx}{5145}\\ &=-\frac {163474 \sqrt {1-2 x}}{36015 (2+3 x)^4}-\frac {163474 \sqrt {1-2 x}}{108045 (2+3 x)^3}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^5}+\frac {2 (55633+83544 x)}{5145 \sqrt {1-2 x} (2+3 x)^5}+\frac {163474 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{21609}\\ &=-\frac {163474 \sqrt {1-2 x}}{36015 (2+3 x)^4}-\frac {163474 \sqrt {1-2 x}}{108045 (2+3 x)^3}-\frac {81737 \sqrt {1-2 x}}{151263 (2+3 x)^2}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^5}+\frac {2 (55633+83544 x)}{5145 \sqrt {1-2 x} (2+3 x)^5}+\frac {81737 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{50421}\\ &=-\frac {163474 \sqrt {1-2 x}}{36015 (2+3 x)^4}-\frac {163474 \sqrt {1-2 x}}{108045 (2+3 x)^3}-\frac {81737 \sqrt {1-2 x}}{151263 (2+3 x)^2}-\frac {81737 \sqrt {1-2 x}}{352947 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^5}+\frac {2 (55633+83544 x)}{5145 \sqrt {1-2 x} (2+3 x)^5}+\frac {81737 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{352947}\\ &=-\frac {163474 \sqrt {1-2 x}}{36015 (2+3 x)^4}-\frac {163474 \sqrt {1-2 x}}{108045 (2+3 x)^3}-\frac {81737 \sqrt {1-2 x}}{151263 (2+3 x)^2}-\frac {81737 \sqrt {1-2 x}}{352947 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^5}+\frac {2 (55633+83544 x)}{5145 \sqrt {1-2 x} (2+3 x)^5}-\frac {81737 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{352947}\\ &=-\frac {163474 \sqrt {1-2 x}}{36015 (2+3 x)^4}-\frac {163474 \sqrt {1-2 x}}{108045 (2+3 x)^3}-\frac {81737 \sqrt {1-2 x}}{151263 (2+3 x)^2}-\frac {81737 \sqrt {1-2 x}}{352947 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^5}+\frac {2 (55633+83544 x)}{5145 \sqrt {1-2 x} (2+3 x)^5}-\frac {163474 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{352947 \sqrt {21}}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 61, normalized size = 0.38 \[ \frac {6125 x^2-68198 x+53531}{1323 (1-2 x)^{3/2} (3 x+2)^5}-\frac {41849344 \sqrt {1-2 x} \, _2F_1\left (\frac {1}{2},6;\frac {3}{2};\frac {3}{7}-\frac {6 x}{7}\right )}{155649627} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^3/((1 - 2*x)^(5/2)*(2 + 3*x)^6),x]

[Out]

(53531 - 68198*x + 6125*x^2)/(1323*(1 - 2*x)^(3/2)*(2 + 3*x)^5) - (41849344*Sqrt[1 - 2*x]*Hypergeometric2F1[1/
2, 6, 3/2, 3/7 - (6*x)/7])/155649627

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fricas [A]  time = 1.03, size = 144, normalized size = 0.90 \[ \frac {408685 \, \sqrt {21} {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (132413940 \, x^{6} + 323678520 \, x^{5} + 232214817 \, x^{4} - 22641149 \, x^{3} - 99751837 \, x^{2} - 42553376 \, x - 5615203\right )} \sqrt {-2 \, x + 1}}{37059435 \, {\left (972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)^(5/2)/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/37059435*(408685*sqrt(21)*(972*x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 112*x^2 + 112*x + 32)*log((3*
x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(132413940*x^6 + 323678520*x^5 + 232214817*x^4 - 22641149*x^3
- 99751837*x^2 - 42553376*x - 5615203)*sqrt(-2*x + 1))/(972*x^7 + 2268*x^6 + 1323*x^5 - 630*x^4 - 840*x^3 - 1
12*x^2 + 112*x + 32)

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giac [A]  time = 1.28, size = 137, normalized size = 0.86 \[ \frac {81737}{7411887} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {1936 \, {\left (279 \, x - 178\right )}}{2470629 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {67655655 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 654366510 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 2361386244 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 3770746490 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 2249364845 \, \sqrt {-2 \, x + 1}}{197650320 \, {\left (3 \, x + 2\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)^(5/2)/(2+3*x)^6,x, algorithm="giac")

[Out]

81737/7411887*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1936/24706
29*(279*x - 178)/((2*x - 1)*sqrt(-2*x + 1)) - 1/197650320*(67655655*(2*x - 1)^4*sqrt(-2*x + 1) + 654366510*(2*
x - 1)^3*sqrt(-2*x + 1) + 2361386244*(2*x - 1)^2*sqrt(-2*x + 1) - 3770746490*(-2*x + 1)^(3/2) + 2249364845*sqr
t(-2*x + 1))/(3*x + 2)^5

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maple [A]  time = 0.02, size = 93, normalized size = 0.58 \[ -\frac {163474 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{7411887}+\frac {10648}{352947 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {90024}{823543 \sqrt {-2 x +1}}+\frac {\frac {9020754 \left (-2 x +1\right )^{\frac {9}{2}}}{823543}-\frac {12464124 \left (-2 x +1\right )^{\frac {7}{2}}}{117649}+\frac {4589672 \left (-2 x +1\right )^{\frac {5}{2}}}{12005}-\frac {628196 \left (-2 x +1\right )^{\frac {3}{2}}}{1029}+\frac {53534 \sqrt {-2 x +1}}{147}}{\left (-6 x -4\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^3/(-2*x+1)^(5/2)/(3*x+2)^6,x)

[Out]

10648/352947/(-2*x+1)^(3/2)+90024/823543/(-2*x+1)^(1/2)+1944/823543*(167051/36*(-2*x+1)^(9/2)-7270739/162*(-2*
x+1)^(7/2)[phone]/1215*(-2*x+1)^(5/2)-377074649/1458*(-2*x+1)^(3/2)[phone]/2916*(-2*x+1)^(1/2))/(-6*x-4)
^5-163474/7411887*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.38, size = 146, normalized size = 0.91 \[ \frac {81737}{7411887} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (33103485 \, {\left (2 \, x - 1\right )}^{6} + 360460170 \, {\left (2 \, x - 1\right )}^{5} + 1537963392 \, {\left (2 \, x - 1\right )}^{4} + 3164039270 \, {\left (2 \, x - 1\right )}^{3} + 2973379535 \, {\left (2 \, x - 1\right )}^{2} + 1324775760 \, x - 1109790220\right )}}{1764735 \, {\left (243 \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - 2835 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + 13230 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 30870 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 36015 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 16807 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)^(5/2)/(2+3*x)^6,x, algorithm="maxima")

[Out]

81737/7411887*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/1764735*(33103485
*(2*x - 1)^6 + 360460170*(2*x - 1)^5 + 1537963392*(2*x - 1)^4 + 3164039270*(2*x - 1)^3 + 2973379535*(2*x - 1)^
2 + 1324775760*x - 1109790220)/(243*(-2*x + 1)^(13/2) - 2835*(-2*x + 1)^(11/2) + 13230*(-2*x + 1)^(9/2) - 3087
0*(-2*x + 1)^(7/2) + 36015*(-2*x + 1)^(5/2) - 16807*(-2*x + 1)^(3/2))

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mupad [B]  time = 1.20, size = 128, normalized size = 0.80 \[ -\frac {163474\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7411887}-\frac {\frac {73568\,x}{11907}+\frac {3467498\,{\left (2\,x-1\right )}^2}{250047}+\frac {25828892\,{\left (2\,x-1\right )}^3}{1750329}+\frac {20924672\,{\left (2\,x-1\right )}^4}{2917215}+\frac {326948\,{\left (2\,x-1\right )}^5}{194481}+\frac {163474\,{\left (2\,x-1\right )}^6}{1058841}-\frac {184888}{35721}}{\frac {16807\,{\left (1-2\,x\right )}^{3/2}}{243}-\frac {12005\,{\left (1-2\,x\right )}^{5/2}}{81}+\frac {3430\,{\left (1-2\,x\right )}^{7/2}}{27}-\frac {490\,{\left (1-2\,x\right )}^{9/2}}{9}+\frac {35\,{\left (1-2\,x\right )}^{11/2}}{3}-{\left (1-2\,x\right )}^{13/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^3/((1 - 2*x)^(5/2)*(3*x + 2)^6),x)

[Out]

- (163474*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/7411887 - ((73568*x)/11907 + (3467498*(2*x - 1)^2)/250
047 + (25828892*(2*x - 1)^3)/1750329 + (20924672*(2*x - 1)^4)/2917215 + (326948*(2*x - 1)^5)/194481 + (163474*
(2*x - 1)^6)/1058841 - 184888/35721)/((16807*(1 - 2*x)^(3/2))/243 - (12005*(1 - 2*x)^(5/2))/81 + (3430*(1 - 2*
x)^(7/2))/27 - (490*(1 - 2*x)^(9/2))/9 + (35*(1 - 2*x)^(11/2))/3 - (1 - 2*x)^(13/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**3/(1-2*x)**(5/2)/(2+3*x)**6,x)

[Out]

Timed out

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