3.22.53 \(\int \frac {(3+5 x)^2}{(1-2 x)^{5/2} (2+3 x)^2} \, dx\) [2153]

Optimal. Leaf size=81 \[ -\frac {130}{1029 \sqrt {1-2 x}}+\frac {121}{42 (1-2 x)^{3/2} (2+3 x)}-\frac {365}{294 \sqrt {1-2 x} (2+3 x)}+\frac {130 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{343 \sqrt {21}} \]

[Out]

121/42/(1-2*x)^(3/2)/(2+3*x)+130/7203*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-130/1029/(1-2*x)^(1/2)-365/
294/(2+3*x)/(1-2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 81, 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 = {91, 79, 53, 65, 212} \begin {gather*} -\frac {130}{1029 \sqrt {1-2 x}}-\frac {365}{294 \sqrt {1-2 x} (3 x+2)}+\frac {121}{42 (1-2 x)^{3/2} (3 x+2)}+\frac {130 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{343 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-130/(1029*Sqrt[1 - 2*x]) + 121/(42*(1 - 2*x)^(3/2)*(2 + 3*x)) - 365/(294*Sqrt[1 - 2*x]*(2 + 3*x)) + (130*ArcT
anh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(343*Sqrt[21])

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)^2} \, dx &=\frac {121}{42 (1-2 x)^{3/2} (2+3 x)}-\frac {1}{42} \int \frac {-15+525 x}{(1-2 x)^{3/2} (2+3 x)^2} \, dx\\ &=\frac {121}{42 (1-2 x)^{3/2} (2+3 x)}-\frac {365}{294 \sqrt {1-2 x} (2+3 x)}-\frac {65}{147} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=-\frac {130}{1029 \sqrt {1-2 x}}+\frac {121}{42 (1-2 x)^{3/2} (2+3 x)}-\frac {365}{294 \sqrt {1-2 x} (2+3 x)}-\frac {65}{343} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {130}{1029 \sqrt {1-2 x}}+\frac {121}{42 (1-2 x)^{3/2} (2+3 x)}-\frac {365}{294 \sqrt {1-2 x} (2+3 x)}+\frac {65}{343} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {130}{1029 \sqrt {1-2 x}}+\frac {121}{42 (1-2 x)^{3/2} (2+3 x)}-\frac {365}{294 \sqrt {1-2 x} (2+3 x)}+\frac {130 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{343 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 70, normalized size = 0.86 \begin {gather*} \frac {-5929+3465 (1-2 x)-390 (1-2 x)^2}{1029 (-7+3 (1-2 x)) (1-2 x)^{3/2}}+\frac {130 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{343 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5929 + 3465*(1 - 2*x) - 390*(1 - 2*x)^2)/(1029*(-7 + 3*(1 - 2*x))*(1 - 2*x)^(3/2)) + (130*ArcTanh[Sqrt[3/7]*
Sqrt[1 - 2*x]])/(343*Sqrt[21])

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Maple [A]
time = 0.16, size = 54, normalized size = 0.67

method	result	size
risch	\(-\frac {780 x^{2}+2685 x +1427}{1029 \left (-1+2 x \right ) \sqrt {1-2 x}\, \left (2+3 x \right )}+\frac {130 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7203}\)	\(53\)
derivativedivides	\(\frac {121}{147 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {44}{343 \sqrt {1-2 x}}+\frac {2 \sqrt {1-2 x}}{1029 \left (-\frac {4}{3}-2 x \right )}+\frac {130 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7203}\)	\(54\)
default	\(\frac {121}{147 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {44}{343 \sqrt {1-2 x}}+\frac {2 \sqrt {1-2 x}}{1029 \left (-\frac {4}{3}-2 x \right )}+\frac {130 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7203}\)	\(54\)
trager	\(\frac {\left (780 x^{2}+2685 x +1427\right ) \sqrt {1-2 x}}{1029 \left (-1+2 x \right )^{2} \left (2+3 x \right )}+\frac {65 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{7203}\)	\(79\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

121/147/(1-2*x)^(3/2)-44/343/(1-2*x)^(1/2)+2/1029*(1-2*x)^(1/2)/(-4/3-2*x)+130/7203*arctanh(1/7*21^(1/2)*(1-2*
x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.50, size = 74, normalized size = 0.91 \begin {gather*} -\frac {65}{7203} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (195 \, {\left (2 \, x - 1\right )}^{2} + 3465 \, x + 1232\right )}}{1029 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 7 \, {\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)^2,x, algorithm="maxima")

[Out]

-65/7203*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/1029*(195*(2*x - 1)^2
+ 3465*x + 1232)/(3*(-2*x + 1)^(5/2) - 7*(-2*x + 1)^(3/2))

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Fricas [A]
time = 0.76, size = 85, normalized size = 1.05 \begin {gather*} \frac {65 \, \sqrt {21} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 7 \, {\left (780 \, x^{2} + 2685 \, x + 1427\right )} \sqrt {-2 \, x + 1}}{7203 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 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)^2,x, algorithm="fricas")

[Out]

1/7203*(65*sqrt(21)*(12*x^3 - 4*x^2 - 5*x + 2)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 7*(780*x^2
+ 2685*x + 1427)*sqrt(-2*x + 1))/(12*x^3 - 4*x^2 - 5*x + 2)

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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)**2,x)

[Out]

Timed out

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Giac [A]
time = 2.04, size = 77, normalized size = 0.95 \begin {gather*} -\frac {65}{7203} \, \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 {11 \, {\left (24 \, x + 65\right )}}{1029 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} - \frac {\sqrt {-2 \, x + 1}}{343 \, {\left (3 \, x + 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)^2,x, algorithm="giac")

[Out]

-65/7203*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 11/1029*(24*x +
65)/((2*x - 1)*sqrt(-2*x + 1)) - 1/343*sqrt(-2*x + 1)/(3*x + 2)

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Mupad [B]
time = 0.06, size = 55, normalized size = 0.68 \begin {gather*} \frac {\frac {110\,x}{49}+\frac {130\,{\left (2\,x-1\right )}^2}{1029}+\frac {352}{441}}{\frac {7\,{\left (1-2\,x\right )}^{3/2}}{3}-{\left (1-2\,x\right )}^{5/2}}+\frac {130\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7203} \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)^2),x)

[Out]

((110*x)/49 + (130*(2*x - 1)^2)/1029 + 352/441)/((7*(1 - 2*x)^(3/2))/3 - (1 - 2*x)^(5/2)) + (130*21^(1/2)*atan
h((21^(1/2)*(1 - 2*x)^(1/2))/7))/7203

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