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]

(-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])

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Rubi [A]  time = 0.0533644, antiderivative size = 160, normalized size of antiderivative = 1., 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 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{(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*}

Mathematica [C]  time = 0.0534539, 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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Maple [A]  time = 0.015, size = 93, normalized size = 0.6 \begin{align*}{\frac{1944}{823543\, \left ( -6\,x-4 \right ) ^{5}} \left ({\frac{167051}{36} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{7270739}{162} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}+{\frac{196782187}{1215} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{377074649}{1458} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{449872969}{2916}\sqrt{1-2\,x}} \right ) }-{\frac{163474\,\sqrt{21}}{7411887}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{10648}{352947} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{90024}{823543}{\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)^(5/2)/(2+3*x)^6,x)

[Out]

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

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Maxima [A]  time = 3.75382, size = 197, normalized size = 1.23 \begin{align*} \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 )}} \end{align*}

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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Fricas [A]  time = 1.56203, size = 475, normalized size = 2.97 \begin{align*} \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 )}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 2.00963, size = 185, normalized size = 1.16 \begin{align*} \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}} \end{align*}

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