3.22.69 \(\int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)} \, dx\) [2169]

Optimal. Leaf size=106 \[ \frac {117649}{1056 (1-2 x)^{3/2}}-\frac {2739541}{3872 \sqrt {1-2 x}}-\frac {5992353 \sqrt {1-2 x}}{10000}+\frac {169209 (1-2 x)^{3/2}}{2000}-\frac {43011 (1-2 x)^{5/2}}{4000}+\frac {729 (1-2 x)^{7/2}}{1120}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{75625 \sqrt {55}} \]

[Out]

117649/1056/(1-2*x)^(3/2)+169209/2000*(1-2*x)^(3/2)-43011/4000*(1-2*x)^(5/2)+729/1120*(1-2*x)^(7/2)-2/4159375*
arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-2739541/3872/(1-2*x)^(1/2)-5992353/10000*(1-2*x)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {89, 45, 65, 212} \begin {gather*} \frac {729 (1-2 x)^{7/2}}{1120}-\frac {43011 (1-2 x)^{5/2}}{4000}+\frac {169209 (1-2 x)^{3/2}}{2000}-\frac {5992353 \sqrt {1-2 x}}{10000}-\frac {2739541}{3872 \sqrt {1-2 x}}+\frac {117649}{1056 (1-2 x)^{3/2}}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{75625 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

117649/(1056*(1 - 2*x)^(3/2)) - 2739541/(3872*Sqrt[1 - 2*x]) - (5992353*Sqrt[1 - 2*x])/10000 + (169209*(1 - 2*
x)^(3/2))/2000 - (43011*(1 - 2*x)^(5/2))/4000 + (729*(1 - 2*x)^(7/2))/1120 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]])/(75625*Sqrt[55])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 65

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 89

Int[(((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegr
and[(e + f*x)^FractionalPart[p], (c + d*x)^n*((e + f*x)^IntegerPart[p]/(a + b*x)), x], x] /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin {align*} \int \frac {(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)} \, dx &=\int \left (\frac {117649}{352 (1-2 x)^{5/2}}-\frac {2739541}{3872 (1-2 x)^{3/2}}+\frac {3946293}{10000 \sqrt {1-2 x}}+\frac {639819 x}{2000 \sqrt {1-2 x}}+\frac {8019 x^2}{50 \sqrt {1-2 x}}+\frac {729 x^3}{20 \sqrt {1-2 x}}+\frac {1}{75625 \sqrt {1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac {117649}{1056 (1-2 x)^{3/2}}-\frac {2739541}{3872 \sqrt {1-2 x}}-\frac {3946293 \sqrt {1-2 x}}{10000}+\frac {\int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{75625}+\frac {729}{20} \int \frac {x^3}{\sqrt {1-2 x}} \, dx+\frac {8019}{50} \int \frac {x^2}{\sqrt {1-2 x}} \, dx+\frac {639819 \int \frac {x}{\sqrt {1-2 x}} \, dx}{2000}\\ &=\frac {117649}{1056 (1-2 x)^{3/2}}-\frac {2739541}{3872 \sqrt {1-2 x}}-\frac {3946293 \sqrt {1-2 x}}{10000}-\frac {\text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{75625}+\frac {729}{20} \int \left (\frac {1}{8 \sqrt {1-2 x}}-\frac {3}{8} \sqrt {1-2 x}+\frac {3}{8} (1-2 x)^{3/2}-\frac {1}{8} (1-2 x)^{5/2}\right ) \, dx+\frac {8019}{50} \int \left (\frac {1}{4 \sqrt {1-2 x}}-\frac {1}{2} \sqrt {1-2 x}+\frac {1}{4} (1-2 x)^{3/2}\right ) \, dx+\frac {639819 \int \left (\frac {1}{2 \sqrt {1-2 x}}-\frac {1}{2} \sqrt {1-2 x}\right ) \, dx}{2000}\\ &=\frac {117649}{1056 (1-2 x)^{3/2}}-\frac {2739541}{3872 \sqrt {1-2 x}}-\frac {5992353 \sqrt {1-2 x}}{10000}+\frac {169209 (1-2 x)^{3/2}}{2000}-\frac {43011 (1-2 x)^{5/2}}{4000}+\frac {729 (1-2 x)^{7/2}}{1120}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{75625 \sqrt {55}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 66, normalized size = 0.62 \begin {gather*} \frac {-\frac {55 \left (1780047848-5374023537 x+2562785082 x^2+611141355 x^3+190531440 x^4+33078375 x^5\right )}{(1-2 x)^{3/2}}-42 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{87346875} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-55*(1780047848 - 5374023537*x + 2562785082*x^2 + 611141355*x^3 + 190531440*x^4 + 33078375*x^5))/(1 - 2*x)^(
3/2) - 42*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/87346875

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Maple [A]
time = 0.12, size = 74, normalized size = 0.70

method	result	size
derivativedivides	\(\frac {117649}{1056 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {169209 \left (1-2 x \right )^{\frac {3}{2}}}{2000}-\frac {43011 \left (1-2 x \right )^{\frac {5}{2}}}{4000}+\frac {729 \left (1-2 x \right )^{\frac {7}{2}}}{1120}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{4159375}-\frac {2739541}{3872 \sqrt {1-2 x}}-\frac {5992353 \sqrt {1-2 x}}{10000}\)	\(74\)
default	\(\frac {117649}{1056 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {169209 \left (1-2 x \right )^{\frac {3}{2}}}{2000}-\frac {43011 \left (1-2 x \right )^{\frac {5}{2}}}{4000}+\frac {729 \left (1-2 x \right )^{\frac {7}{2}}}{1120}-\frac {2 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{4159375}-\frac {2739541}{3872 \sqrt {1-2 x}}-\frac {5992353 \sqrt {1-2 x}}{10000}\)	\(74\)
trager	\(-\frac {\left (33078375 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-5374023537 x [phone]\right ) \sqrt {1-2 x}}{1588125 \left (-1+2 x \right )^{2}}+\frac {\RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{4159375}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

117649/1056/(1-2*x)^(3/2)+169209/2000*(1-2*x)^(3/2)-43011/4000*(1-2*x)^(5/2)+729/1120*(1-2*x)^(7/2)-2/4159375*
arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-2739541/3872/(1-2*x)^(1/2)-5992353/10000*(1-2*x)^(1/2)

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Maxima [A]
time = 0.55, size = 87, normalized size = 0.82 \begin {gather*} \frac {729}{1120} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {43011}{4000} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {169209}{2000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{4159375} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {5992353}{10000} \, \sqrt {-2 \, x + 1} + \frac {16807 \, {\left (489 \, x - 206\right )}}{5808 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

729/1120*(-2*x + 1)^(7/2) - 43011/4000*(-2*x + 1)^(5/2) + 169209/2000*(-2*x + 1)^(3/2) + 1/4159375*sqrt(55)*lo
g(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 5992353/10000*sqrt(-2*x + 1) + 16807/5808*(4
89*x - 206)/(-2*x + 1)^(3/2)

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Fricas [A]
time = 1.52, size = 89, normalized size = 0.84 \begin {gather*} \frac {21 \, \sqrt {55} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (33078375 \, x^{5} + 190531440 \, x^{4} + 611141355 \, x^{3} + 2562785082 \, x^{2} - 5374023537 \, x + 1780047848\right )} \sqrt {-2 \, x + 1}}{87346875 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/87346875*(21*sqrt(55)*(4*x^2 - 4*x + 1)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(33078375*x^
5 + 190531440*x^4 + 611141355*x^3 + 2562785082*x^2 - 5374023537*x + 1780047848)*sqrt(-2*x + 1))/(4*x^2 - 4*x +
1)

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Sympy [A]
time = 71.91, size = 131, normalized size = 1.24 \begin {gather*} \frac {729 \left (1 - 2 x\right )^{\frac {7}{2}}}{1120} - \frac {43011 \left (1 - 2 x\right )^{\frac {5}{2}}}{4000} + \frac {169209 \left (1 - 2 x\right )^{\frac {3}{2}}}{2000} - \frac {5992353 \sqrt {1 - 2 x}}{10000} + \frac {2 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x < - \frac {3}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: x > - \frac {3}{5} \end {cases}\right )}{75625} - \frac {2739541}{3872 \sqrt {1 - 2 x}} + \frac {117649}{1056 \left (1 - 2 x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

729*(1 - 2*x)**(7/2)/1120 - 43011*(1 - 2*x)**(5/2)/4000 + 169209*(1 - 2*x)**(3/2)/2000 - 5992353*sqrt(1 - 2*x)
/10000 + 2*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(1 - 2*x)/11)/55, x < -3/5), (-sqrt(55)*atanh(sqrt(55)*sqrt
(1 - 2*x)/11)/55, x > -3/5))/75625 - 2739541/(3872*sqrt(1 - 2*x)) + 117649/(1056*(1 - 2*x)**(3/2))

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Giac [A]
time = 1.57, size = 111, normalized size = 1.05 \begin {gather*} -\frac {729}{1120} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {43011}{4000} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {169209}{2000} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1}{4159375} \, \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 {5992353}{10000} \, \sqrt {-2 \, x + 1} - \frac {16807 \, {\left (489 \, x - 206\right )}}{5808 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729/1120*(2*x - 1)^3*sqrt(-2*x + 1) - 43011/4000*(2*x - 1)^2*sqrt(-2*x + 1) + 169209/2000*(-2*x + 1)^(3/2) +
1/4159375*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 5992353/10000
*sqrt(-2*x + 1) - 16807/5808*(489*x - 206)/((2*x - 1)*sqrt(-2*x + 1))

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Mupad [B]
time = 0.05, size = 70, normalized size = 0.66 \begin {gather*} \frac {\frac {2739541\,x}{1936}-\frac {1731121}{2904}}{{\left (1-2\,x\right )}^{3/2}}-\frac {5992353\,\sqrt {1-2\,x}}{10000}+\frac {169209\,{\left (1-2\,x\right )}^{3/2}}{2000}-\frac {43011\,{\left (1-2\,x\right )}^{5/2}}{4000}+\frac {729\,{\left (1-2\,x\right )}^{7/2}}{1120}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,2{}\mathrm {i}}{4159375} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2739541*x)/1936 - 1731121/2904)/(1 - 2*x)^(3/2) + (55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*2i)/41593
75 - (5992353*(1 - 2*x)^(1/2))/10000 + (169209*(1 - 2*x)^(3/2))/2000 - (43011*(1 - 2*x)^(5/2))/4000 + (729*(1
- 2*x)^(7/2))/1120

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