3.19.14 \(\int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx\) [1814]

Optimal. Leaf size=148 \[ -\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}+\frac {1613 \sqrt {1-2 x}}{158760 (2+3 x)^3}+\frac {1613 \sqrt {1-2 x}}{444528 (2+3 x)^2}+\frac {1613 \sqrt {1-2 x}}{1037232 (2+3 x)}+\frac {1613 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{518616 \sqrt {21}} \]

[Out]

-1/378*(1-2*x)^(3/2)/(2+3*x)^6+137/4410*(1-2*x)^(3/2)/(2+3*x)^5+1613/10890936*arctanh(1/7*21^(1/2)*(1-2*x)^(1/
2))*21^(1/2)-1613/7560*(1-2*x)^(1/2)/(2+3*x)^4+1613/158760*(1-2*x)^(1/2)/(2+3*x)^3+1613/444528*(1-2*x)^(1/2)/(
2+3*x)^2+1613/1037232*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.03, antiderivative size = 148, 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, 43, 44, 65, 212} \begin {gather*} \frac {137 (1-2 x)^{3/2}}{4410 (3 x+2)^5}-\frac {(1-2 x)^{3/2}}{378 (3 x+2)^6}+\frac {1613 \sqrt {1-2 x}}{1037232 (3 x+2)}+\frac {1613 \sqrt {1-2 x}}{444528 (3 x+2)^2}+\frac {1613 \sqrt {1-2 x}}{158760 (3 x+2)^3}-\frac {1613 \sqrt {1-2 x}}{7560 (3 x+2)^4}+\frac {1613 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{518616 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

-1/378*(1 - 2*x)^(3/2)/(2 + 3*x)^6 + (137*(1 - 2*x)^(3/2))/(4410*(2 + 3*x)^5) - (1613*Sqrt[1 - 2*x])/(7560*(2
+ 3*x)^4) + (1613*Sqrt[1 - 2*x])/(158760*(2 + 3*x)^3) + (1613*Sqrt[1 - 2*x])/(444528*(2 + 3*x)^2) + (1613*Sqrt
[1 - 2*x])/(1037232*(2 + 3*x)) + (1613*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(518616*Sqrt[21])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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 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 {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^7} \, dx &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {1}{378} \int \frac {\sqrt {1-2 x} (1689+3150 x)}{(2+3 x)^6} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}+\frac {1613}{630} \int \frac {\sqrt {1-2 x}}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}-\frac {1613 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx}{7560}\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}+\frac {1613 \sqrt {1-2 x}}{158760 (2+3 x)^3}-\frac {1613 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{31752}\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}+\frac {1613 \sqrt {1-2 x}}{158760 (2+3 x)^3}+\frac {1613 \sqrt {1-2 x}}{444528 (2+3 x)^2}-\frac {1613 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{148176}\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}+\frac {1613 \sqrt {1-2 x}}{158760 (2+3 x)^3}+\frac {1613 \sqrt {1-2 x}}{444528 (2+3 x)^2}+\frac {1613 \sqrt {1-2 x}}{1037232 (2+3 x)}-\frac {1613 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1037232}\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}+\frac {1613 \sqrt {1-2 x}}{158760 (2+3 x)^3}+\frac {1613 \sqrt {1-2 x}}{444528 (2+3 x)^2}+\frac {1613 \sqrt {1-2 x}}{1037232 (2+3 x)}+\frac {1613 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1037232}\\ &=-\frac {(1-2 x)^{3/2}}{378 (2+3 x)^6}+\frac {137 (1-2 x)^{3/2}}{4410 (2+3 x)^5}-\frac {1613 \sqrt {1-2 x}}{7560 (2+3 x)^4}+\frac {1613 \sqrt {1-2 x}}{158760 (2+3 x)^3}+\frac {1613 \sqrt {1-2 x}}{444528 (2+3 x)^2}+\frac {1613 \sqrt {1-2 x}}{1037232 (2+3 x)}+\frac {1613 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{518616 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.31, size = 75, normalized size = 0.51 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \left (-3136864-7772840 x+1791558 x^2+14197626 x^3+8056935 x^4+1959795 x^5\right )}{2 (2+3 x)^6}+8065 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{54454680} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^2)/(2 + 3*x)^7,x]

[Out]

((21*Sqrt[1 - 2*x]*(-3136864 - 7772840*x + 1791558*x^2 + 14197626*x^3 + 8056935*x^4 + 1959795*x^5))/(2*(2 + 3*
x)^6) + 8065*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/54454680

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Maple [A]
time = 0.11, size = 84, normalized size = 0.57

method	result	size
risch	\(-\frac {3919590 x^{6}[phone] x^{5}[phone] x^{4}-10614510 x^{3}-17337238 x^{2}+1499112 x +3136864}{5186160 \left (2+3 x \right )^{6} \sqrt {1-2 x}}+\frac {1613 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{10890936}\)	\(66\)
derivativedivides	\(\frac {-\frac {14517 \left (1-2 x \right )^{\frac {11}{2}}}{19208}+\frac {27421 \left (1-2 x \right )^{\frac {9}{2}}}{2744}-\frac {53229 \left (1-2 x \right )^{\frac {7}{2}}}{980}+\frac {113751 \left (1-2 x \right )^{\frac {5}{2}}}{980}-\frac {86837 \left (1-2 x \right )^{\frac {3}{2}}}{1512}-\frac {11291 \sqrt {1-2 x}}{216}}{\left (-4-6 x \right )^{6}}+\frac {1613 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{10890936}\)	\(84\)
default	\(\frac {-\frac {14517 \left (1-2 x \right )^{\frac {11}{2}}}{19208}+\frac {27421 \left (1-2 x \right )^{\frac {9}{2}}}{2744}-\frac {53229 \left (1-2 x \right )^{\frac {7}{2}}}{980}+\frac {113751 \left (1-2 x \right )^{\frac {5}{2}}}{980}-\frac {86837 \left (1-2 x \right )^{\frac {3}{2}}}{1512}-\frac {11291 \sqrt {1-2 x}}{216}}{\left (-4-6 x \right )^{6}}+\frac {1613 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{10890936}\)	\(84\)
trager	\(\frac {\left (1959795 x^{5}+8056935 x^{4}[phone] x^{3}+1791558 x^{2}-7772840 x -3136864\right ) \sqrt {1-2 x}}{5186160 \left (2+3 x \right )^{6}}-\frac {1613 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{21781872}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^7,x,method=_RETURNVERBOSE)

[Out]

23328*(-1613/49787136*(1-2*x)^(11/2)+27421/64012032*(1-2*x)^(9/2)-17743/7620480*(1-2*x)^(7/2)+4213/846720*(1-2
*x)^(5/2)-86837/35271936*(1-2*x)^(3/2)-11291/5038848*(1-2*x)^(1/2))/(-4-6*x)^6+1613/10890936*arctanh(1/7*21^(1
/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.53, size = 146, normalized size = 0.99 \begin {gather*} -\frac {1613}{21781872} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {1959795 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 25912845 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 140843934 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 300985146 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 148925455 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 135548455 \, \sqrt {-2 \, x + 1}}{2593080 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1613/21781872*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/2593080*(1959795
*(-2*x + 1)^(11/2) - 25912845*(-2*x + 1)^(9/2) + 140843934*(-2*x + 1)^(7/2) - 300985146*(-2*x + 1)^(5/2) + 148
925455*(-2*x + 1)^(3/2) + 135548455*sqrt(-2*x + 1))/(729*(2*x - 1)^6 + 10206*(2*x - 1)^5 + 59535*(2*x - 1)^4 +
185220*(2*x - 1)^3 + 324135*(2*x - 1)^2 + 605052*x - 184877)

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Fricas [A]
time = 0.81, size = 130, normalized size = 0.88 \begin {gather*} \frac {8065 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (1959795 \, x^{5} + 8056935 \, x^{4} + 14197626 \, x^{3} + 1791558 \, x^{2} - 7772840 \, x - 3136864\right )} \sqrt {-2 \, x + 1}}{108909360 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^7,x, algorithm="fricas")

[Out]

1/108909360*(8065*sqrt(21)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log((3*x - sqrt(
21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(1959795*x^5 + 8056935*x^4 + 14197626*x^3 + 1791558*x^2 - 7772840*x -
3136864)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

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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)**(1/2)/(2+3*x)**7,x)

[Out]

Timed out

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Giac [A]
time = 1.22, size = 132, normalized size = 0.89 \begin {gather*} -\frac {1613}{21781872} \, \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 {1959795 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 25912845 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 140843934 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 300985146 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 148925455 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 135548455 \, \sqrt {-2 \, x + 1}}{165957120 \, {\left (3 \, x + 2\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^7,x, algorithm="giac")

[Out]

-1613/21781872*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1659571
20*(1959795*(2*x - 1)^5*sqrt(-2*x + 1) + 25912845*(2*x - 1)^4*sqrt(-2*x + 1) + 140843934*(2*x - 1)^3*sqrt(-2*x
+ 1) + 300985146*(2*x - 1)^2*sqrt(-2*x + 1) - 148925455*(-2*x + 1)^(3/2) - 135548455*sqrt(-2*x + 1))/(3*x + 2
)^6

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Mupad [B]
time = 0.08, size = 126, normalized size = 0.85 \begin {gather*} \frac {1613\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{10890936}-\frac {\frac {11291\,\sqrt {1-2\,x}}{157464}+\frac {86837\,{\left (1-2\,x\right )}^{3/2}}{1102248}-\frac {4213\,{\left (1-2\,x\right )}^{5/2}}{26460}+\frac {17743\,{\left (1-2\,x\right )}^{7/2}}{238140}-\frac {27421\,{\left (1-2\,x\right )}^{9/2}}{2000376}+\frac {1613\,{\left (1-2\,x\right )}^{11/2}}{1555848}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(1/2)*(5*x + 3)^2)/(3*x + 2)^7,x)

[Out]

(1613*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/10890936 - ((11291*(1 - 2*x)^(1/2))/157464 + (86837*(1 - 2
*x)^(3/2))/1102248 - (4213*(1 - 2*x)^(5/2))/26460 + (17743*(1 - 2*x)^(7/2))/238140 - (27421*(1 - 2*x)^(9/2))/2
000376 + (1613*(1 - 2*x)^(11/2))/1555848)/((67228*x)/81 + (12005*(2*x - 1)^2)/27 + (6860*(2*x - 1)^3)/27 + (24
5*(2*x - 1)^4)/3 + 14*(2*x - 1)^5 + (2*x - 1)^6 - 184877/729)

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