3.21.80 \(\int \frac {3+5 x}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\) [2080]

Optimal. Leaf size=101 \[ \frac {10}{147 \sqrt {1-2 x}}+\frac {1}{63 \sqrt {1-2 x} (2+3 x)^3}-\frac {1}{9 \sqrt {1-2 x} (2+3 x)^2}-\frac {5}{63 \sqrt {1-2 x} (2+3 x)}-\frac {10 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \]

[Out]

-10/1029*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+10/147/(1-2*x)^(1/2)+1/63/(2+3*x)^3/(1-2*x)^(1/2)-1/9/(2
+3*x)^2/(1-2*x)^(1/2)-5/63/(2+3*x)/(1-2*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {79, 44, 53, 65, 212} \begin {gather*} \frac {10}{147 \sqrt {1-2 x}}-\frac {5}{63 \sqrt {1-2 x} (3 x+2)}-\frac {1}{9 \sqrt {1-2 x} (3 x+2)^2}+\frac {1}{63 \sqrt {1-2 x} (3 x+2)^3}-\frac {10 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

10/(147*Sqrt[1 - 2*x]) + 1/(63*Sqrt[1 - 2*x]*(2 + 3*x)^3) - 1/(9*Sqrt[1 - 2*x]*(2 + 3*x)^2) - 5/(63*Sqrt[1 - 2
*x]*(2 + 3*x)) - (10*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(49*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 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}{(1-2 x)^{3/2} (2+3 x)^4} \, dx &=\frac {1}{63 \sqrt {1-2 x} (2+3 x)^3}+\frac {14}{9} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\\ &=\frac {1}{63 \sqrt {1-2 x} (2+3 x)^3}+\frac {4}{9 \sqrt {1-2 x} (2+3 x)^2}+\frac {10}{3} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {1}{63 \sqrt {1-2 x} (2+3 x)^3}+\frac {4}{9 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {1-2 x}}{21 (2+3 x)^2}+\frac {5}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {1}{63 \sqrt {1-2 x} (2+3 x)^3}+\frac {4}{9 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {1-2 x}}{21 (2+3 x)^2}-\frac {5 \sqrt {1-2 x}}{49 (2+3 x)}+\frac {5}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {1}{63 \sqrt {1-2 x} (2+3 x)^3}+\frac {4}{9 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {1-2 x}}{21 (2+3 x)^2}-\frac {5 \sqrt {1-2 x}}{49 (2+3 x)}-\frac {5}{49} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {1}{63 \sqrt {1-2 x} (2+3 x)^3}+\frac {4}{9 \sqrt {1-2 x} (2+3 x)^2}-\frac {5 \sqrt {1-2 x}}{21 (2+3 x)^2}-\frac {5 \sqrt {1-2 x}}{49 (2+3 x)}-\frac {10 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 63, normalized size = 0.62 \begin {gather*} \frac {1+57 x+145 x^2+90 x^3}{49 \sqrt {1-2 x} (2+3 x)^3}-\frac {10 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{49 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)/((1 - 2*x)^(3/2)*(2 + 3*x)^4),x]

[Out]

(1 + 57*x + 145*x^2 + 90*x^3)/(49*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (10*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(49*Sqrt[
21])

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Maple [A]
time = 0.12, size = 66, normalized size = 0.65

method	result	size
risch	\(\frac {90 x^{3}+145 x^{2}+57 x +1}{49 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {10 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\)	\(51\)
derivativedivides	\(\frac {88}{2401 \sqrt {1-2 x}}+\frac {\frac {2034 \left (1-2 x \right )^{\frac {5}{2}}}{2401}-\frac {1544 \left (1-2 x \right )^{\frac {3}{2}}}{343}+\frac {286 \sqrt {1-2 x}}{49}}{\left (-4-6 x \right )^{3}}-\frac {10 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\)	\(66\)
default	\(\frac {88}{2401 \sqrt {1-2 x}}+\frac {\frac {2034 \left (1-2 x \right )^{\frac {5}{2}}}{2401}-\frac {1544 \left (1-2 x \right )^{\frac {3}{2}}}{343}+\frac {286 \sqrt {1-2 x}}{49}}{\left (-4-6 x \right )^{3}}-\frac {10 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{1029}\)	\(66\)
trager	\(-\frac {\left (90 x^{3}+145 x^{2}+57 x +1\right ) \sqrt {1-2 x}}{49 \left (2+3 x \right )^{3} \left (-1+2 x \right )}-\frac {5 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{1029}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)/(1-2*x)^(3/2)/(2+3*x)^4,x,method=_RETURNVERBOSE)

[Out]

88/2401/(1-2*x)^(1/2)+216/2401*(113/12*(1-2*x)^(5/2)-1351/27*(1-2*x)^(3/2)+7007/108*(1-2*x)^(1/2))/(-4-6*x)^3-
10/1029*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.49, size = 101, normalized size = 1.00 \begin {gather*} \frac {5}{1029} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (45 \, {\left (2 \, x - 1\right )}^{3} + 280 \, {\left (2 \, x - 1\right )}^{2} + 1078 \, x - 231\right )}}{49 \, {\left (27 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 189 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 441 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

5/1029*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/49*(45*(2*x - 1)^3 + 280
*(2*x - 1)^2 + 1078*x - 231)/(27*(-2*x + 1)^(7/2) - 189*(-2*x + 1)^(5/2) + 441*(-2*x + 1)^(3/2) - 343*sqrt(-2*
x + 1))

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Fricas [A]
time = 1.39, size = 99, normalized size = 0.98 \begin {gather*} \frac {5 \, \sqrt {21} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (90 \, x^{3} + 145 \, x^{2} + 57 \, x + 1\right )} \sqrt {-2 \, x + 1}}{1029 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/1029*(5*sqrt(21)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) -
21*(90*x^3 + 145*x^2 + 57*x + 1)*sqrt(-2*x + 1))/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)

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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)/(1-2*x)**(3/2)/(2+3*x)**4,x)

[Out]

Timed out

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Giac [A]
time = 1.70, size = 93, normalized size = 0.92 \begin {gather*} \frac {5}{1029} \, \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 {88}{2401 \, \sqrt {-2 \, x + 1}} - \frac {1017 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 5404 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 7007 \, \sqrt {-2 \, x + 1}}{9604 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

5/1029*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 88/2401/sqrt(-2*x
+ 1) - 1/9604*(1017*(2*x - 1)^2*sqrt(-2*x + 1) - 5404*(-2*x + 1)^(3/2) + 7007*sqrt(-2*x + 1))/(3*x + 2)^3

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Mupad [B]
time = 0.07, size = 82, normalized size = 0.81 \begin {gather*} \frac {\frac {44\,x}{27}+\frac {80\,{\left (2\,x-1\right )}^2}{189}+\frac {10\,{\left (2\,x-1\right )}^3}{147}-\frac {22}{63}}{\frac {343\,\sqrt {1-2\,x}}{27}-\frac {49\,{\left (1-2\,x\right )}^{3/2}}{3}+7\,{\left (1-2\,x\right )}^{5/2}-{\left (1-2\,x\right )}^{7/2}}-\frac {10\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1029} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)/((1 - 2*x)^(3/2)*(3*x + 2)^4),x)

[Out]

((44*x)/27 + (80*(2*x - 1)^2)/189 + (10*(2*x - 1)^3)/147 - 22/63)/((343*(1 - 2*x)^(1/2))/27 - (49*(1 - 2*x)^(3
/2))/3 + 7*(1 - 2*x)^(5/2) - (1 - 2*x)^(7/2)) - (10*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/1029

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