3.22.56 \(\int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^5} \, dx\) [2156]

Optimal. Leaf size=141 \[ \frac {20465}{201684 \sqrt {1-2 x}}+\frac {121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac {727}{588 \sqrt {1-2 x} (2+3 x)^4}-\frac {4093}{12348 \sqrt {1-2 x} (2+3 x)^3}-\frac {4093}{24696 \sqrt {1-2 x} (2+3 x)^2}-\frac {20465}{172872 \sqrt {1-2 x} (2+3 x)}-\frac {20465 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{67228 \sqrt {21}} \]

[Out]

121/42/(1-2*x)^(3/2)/(2+3*x)^4-20465/1411788*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+20465/201684/(1-2*x)
^(1/2)-727/588/(2+3*x)^4/(1-2*x)^(1/2)-4093/12348/(2+3*x)^3/(1-2*x)^(1/2)-4093/24696/(2+3*x)^2/(1-2*x)^(1/2)-2
0465/172872/(2+3*x)/(1-2*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {91, 79, 44, 53, 65, 212} \begin {gather*} \frac {20465}{201684 \sqrt {1-2 x}}-\frac {20465}{172872 \sqrt {1-2 x} (3 x+2)}-\frac {4093}{24696 \sqrt {1-2 x} (3 x+2)^2}-\frac {4093}{12348 \sqrt {1-2 x} (3 x+2)^3}-\frac {727}{588 \sqrt {1-2 x} (3 x+2)^4}+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)^4}-\frac {20465 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{67228 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

20465/(201684*Sqrt[1 - 2*x]) + 121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)^4) - 727/(588*Sqrt[1 - 2*x]*(2 + 3*x)^4) - 40
93/(12348*Sqrt[1 - 2*x]*(2 + 3*x)^3) - 4093/(24696*Sqrt[1 - 2*x]*(2 + 3*x)^2) - 20465/(172872*Sqrt[1 - 2*x]*(2
+ 3*x)) - (20465*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(67228*Sqrt[21])

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 53

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 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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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 {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^5} \, dx &=\frac {121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac {1}{42} \int \frac {-1104+525 x}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\\ &=\frac {121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac {727}{588 \sqrt {1-2 x} (2+3 x)^4}+\frac {4093}{588} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\\ &=\frac {121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac {727}{588 \sqrt {1-2 x} (2+3 x)^4}+\frac {4093}{2058 \sqrt {1-2 x} (2+3 x)^3}+\frac {4093}{196} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=\frac {121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac {727}{588 \sqrt {1-2 x} (2+3 x)^4}+\frac {4093}{2058 \sqrt {1-2 x} (2+3 x)^3}-\frac {4093 \sqrt {1-2 x}}{4116 (2+3 x)^3}+\frac {20465 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{4116}\\ &=\frac {121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac {727}{588 \sqrt {1-2 x} (2+3 x)^4}+\frac {4093}{2058 \sqrt {1-2 x} (2+3 x)^3}-\frac {4093 \sqrt {1-2 x}}{4116 (2+3 x)^3}-\frac {20465 \sqrt {1-2 x}}{57624 (2+3 x)^2}+\frac {20465 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{19208}\\ &=\frac {121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac {727}{588 \sqrt {1-2 x} (2+3 x)^4}+\frac {4093}{2058 \sqrt {1-2 x} (2+3 x)^3}-\frac {4093 \sqrt {1-2 x}}{4116 (2+3 x)^3}-\frac {20465 \sqrt {1-2 x}}{57624 (2+3 x)^2}-\frac {20465 \sqrt {1-2 x}}{134456 (2+3 x)}+\frac {20465 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{134456}\\ &=\frac {121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac {727}{588 \sqrt {1-2 x} (2+3 x)^4}+\frac {4093}{2058 \sqrt {1-2 x} (2+3 x)^3}-\frac {4093 \sqrt {1-2 x}}{4116 (2+3 x)^3}-\frac {20465 \sqrt {1-2 x}}{57624 (2+3 x)^2}-\frac {20465 \sqrt {1-2 x}}{134456 (2+3 x)}-\frac {20465 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{134456}\\ &=\frac {121}{42 (1-2 x)^{3/2} (2+3 x)^4}-\frac {727}{588 \sqrt {1-2 x} (2+3 x)^4}+\frac {4093}{2058 \sqrt {1-2 x} (2+3 x)^3}-\frac {4093 \sqrt {1-2 x}}{4116 (2+3 x)^3}-\frac {20465 \sqrt {1-2 x}}{57624 (2+3 x)^2}-\frac {20465 \sqrt {1-2 x}}{134456 (2+3 x)}-\frac {20465 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{67228 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.27, size = 75, normalized size = 0.53 \begin {gather*} \frac {-\frac {7 \left (-401410-2528226 x-3646863 x^2+3769653 x^3+11787840 x^4+6630660 x^5\right )}{2 (1-2 x)^{3/2} (2+3 x)^4}-20465 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1411788} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*(-401410 - 2528226*x - 3646863*x^2 + 3769653*x^3 + 11787840*x^4 + 6630660*x^5))/(2*(1 - 2*x)^(3/2)*(2 + 3
*x)^4) - 20465*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/1411788

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Maple [A]
time = 0.11, size = 84, normalized size = 0.60

method	result	size
risch	\(\frac {6630660 x^{5}[phone] x^{4}+3769653 x^{3}-3646863 x^{2}-2528226 x -401410}{403368 \left (2+3 x \right )^{4} \sqrt {1-2 x}\, \left (-1+2 x \right )}-\frac {20465 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1411788}\)	\(68\)
derivativedivides	\(\frac {968}{50421 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {8360}{117649 \sqrt {1-2 x}}+\frac {\frac {1159245 \left (1-2 x \right )^{\frac {7}{2}}}{470596}-\frac {1220439 \left (1-2 x \right )^{\frac {5}{2}}}{67228}+\frac {425155 \left (1-2 x \right )^{\frac {3}{2}}}{9604}-\frac {49065 \sqrt {1-2 x}}{1372}}{\left (-4-6 x \right )^{4}}-\frac {20465 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1411788}\)	\(84\)
default	\(\frac {968}{50421 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {8360}{117649 \sqrt {1-2 x}}+\frac {\frac {1159245 \left (1-2 x \right )^{\frac {7}{2}}}{470596}-\frac {1220439 \left (1-2 x \right )^{\frac {5}{2}}}{67228}+\frac {425155 \left (1-2 x \right )^{\frac {3}{2}}}{9604}-\frac {49065 \sqrt {1-2 x}}{1372}}{\left (-4-6 x \right )^{4}}-\frac {20465 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1411788}\)	\(84\)
trager	\(-\frac {\left (6630660 x^{5}[phone] x^{4}+3769653 x^{3}-3646863 x^{2}-2528226 x -401410\right ) \sqrt {1-2 x}}{403368 \left (2+3 x \right )^{4} \left (-1+2 x \right )^{2}}-\frac {20465 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{2823576}\)	\(94\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

968/50421/(1-2*x)^(3/2)+8360/117649/(1-2*x)^(1/2)+648/117649*(42935/96*(1-2*x)^(7/2)-2847691/864*(1-2*x)^(5/2)
[phone]/2592*(1-2*x)^(3/2)-5609765/864*(1-2*x)^(1/2))/(-4-6*x)^4-20465/1411788*arctanh(1/7*21^(1/2)*(1-2*x)^
(1/2))*21^(1/2)

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Maxima [A]
time = 0.52, size = 128, normalized size = 0.91 \begin {gather*} \frac {20465}{2823576} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1657665 \, {\left (2 \, x - 1\right )}^{5} + 14182245 \, {\left (2 \, x - 1\right )}^{4} + 43921983 \, {\left (2 \, x - 1\right )}^{3} + 55955403 \, {\left (2 \, x - 1\right )}^{2} + 36945216 \, x - 27769280}{201684 \, {\left (81 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 756 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 2646 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 4116 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 2401 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20465/2823576*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/201684*(1657665*(
2*x - 1)^5 + 14182245*(2*x - 1)^4 + 43921983*(2*x - 1)^3 + 55955403*(2*x - 1)^2 + 36945216*x - 27769280)/(81*(
-2*x + 1)^(11/2) - 756*(-2*x + 1)^(9/2) + 2646*(-2*x + 1)^(7/2) - 4116*(-2*x + 1)^(5/2) + 2401*(-2*x + 1)^(3/2
))

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Fricas [A]
time = 0.77, size = 129, normalized size = 0.91 \begin {gather*} \frac {20465 \, \sqrt {21} {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 7 \, {\left (6630660 \, x^{5} + 11787840 \, x^{4} + 3769653 \, x^{3} - 3646863 \, x^{2} - 2528226 \, x - 401410\right )} \sqrt {-2 \, x + 1}}{2823576 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2823576*(20465*sqrt(21)*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*log((3*x + sqrt(21)*sqr
t(-2*x + 1) - 5)/(3*x + 2)) - 7*(6630660*x^5 + 11787840*x^4 + 3769653*x^3 - 3646863*x^2 - 2528226*x - 401410)*
sqrt(-2*x + 1))/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)

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Sympy [F(-1)] Timed out
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((3+5*x)**2/(1-2*x)**(5/2)/(2+3*x)**5,x)

[Out]

Timed out

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Giac [A]
time = 1.40, size = 121, normalized size = 0.86 \begin {gather*} \frac {20465}{2823576} \, \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 {176 \, {\left (285 \, x - 181\right )}}{352947 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {1159245 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 8543073 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 20832595 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 16829295 \, \sqrt {-2 \, x + 1}}{7529536 \, {\left (3 \, x + 2\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20465/2823576*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 176/352947
*(285*x - 181)/((2*x - 1)*sqrt(-2*x + 1)) - 1/7529536*(1159245*(2*x - 1)^3*sqrt(-2*x + 1) + 8543073*(2*x - 1)^
2*sqrt(-2*x + 1) - 20832595*(-2*x + 1)^(3/2) + 16829295*sqrt(-2*x + 1))/(3*x + 2)^4

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Mupad [B]
time = 1.20, size = 108, normalized size = 0.77 \begin {gather*} -\frac {20465\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1411788}-\frac {\frac {2992\,x}{1323}+\frac {126883\,{\left (2\,x-1\right )}^2}{37044}+\frac {298789\,{\left (2\,x-1\right )}^3}{111132}+\frac {225115\,{\left (2\,x-1\right )}^4}{259308}+\frac {20465\,{\left (2\,x-1\right )}^5}{201684}-\frac {20240}{11907}}{\frac {2401\,{\left (1-2\,x\right )}^{3/2}}{81}-\frac {1372\,{\left (1-2\,x\right )}^{5/2}}{27}+\frac {98\,{\left (1-2\,x\right )}^{7/2}}{3}-\frac {28\,{\left (1-2\,x\right )}^{9/2}}{3}+{\left (1-2\,x\right )}^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (20465*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/1411788 - ((2992*x)/1323 + (126883*(2*x - 1)^2)/37044 +
(298789*(2*x - 1)^3)/111132 + (225115*(2*x - 1)^4)/259308 + (20465*(2*x - 1)^5)/201684 - 20240/11907)/((2401*
(1 - 2*x)^(3/2))/81 - (1372*(1 - 2*x)^(5/2))/27 + (98*(1 - 2*x)^(7/2))/3 - (28*(1 - 2*x)^(9/2))/3 + (1 - 2*x)^
(11/2))

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