3.21.34 \(\int \frac {(3+5 x)^3}{\sqrt {1-2 x} (2+3 x)^6} \, dx\) [2034]

Optimal. Leaf size=120 \[ -\frac {5293 \sqrt {1-2 x}}{18522 (2+3 x)^2}-\frac {5293 \sqrt {1-2 x}}{43218 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac {\sqrt {1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}-\frac {5293 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609 \sqrt {21}} \]

[Out]

-5293/453789*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-5293/18522*(1-2*x)^(1/2)/(2+3*x)^2-5293/43218*(1-2*x
)^(1/2)/(2+3*x)+1/105*(3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)^5+1/6615*(1255+1971*x)*(1-2*x)^(1/2)/(2+3*x)^4

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Rubi [A]
time = 0.02, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 150, 44, 65, 212} \begin {gather*} \frac {\sqrt {1-2 x} (5 x+3)^2}{105 (3 x+2)^5}+\frac {\sqrt {1-2 x} (1971 x+1255)}{6615 (3 x+2)^4}-\frac {5293 \sqrt {1-2 x}}{43218 (3 x+2)}-\frac {5293 \sqrt {1-2 x}}{18522 (3 x+2)^2}-\frac {5293 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609 \sqrt {21}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5293*Sqrt[1 - 2*x])/(18522*(2 + 3*x)^2) - (5293*Sqrt[1 - 2*x])/(43218*(2 + 3*x)) + (Sqrt[1 - 2*x]*(3 + 5*x)^
2)/(105*(2 + 3*x)^5) + (Sqrt[1 - 2*x]*(1255 + 1971*x))/(6615*(2 + 3*x)^4) - (5293*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/(21609*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 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 100

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

Rule 150

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(
n + 1), x] + Dist[f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] &&  !L
tQ[n, -2]))

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)^3}{\sqrt {1-2 x} (2+3 x)^6} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}-\frac {1}{105} \int \frac {(-488-850 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac {\sqrt {1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}+\frac {5293 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{1323}\\ &=-\frac {5293 \sqrt {1-2 x}}{18522 (2+3 x)^2}+\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac {\sqrt {1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}+\frac {5293 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{6174}\\ &=-\frac {5293 \sqrt {1-2 x}}{18522 (2+3 x)^2}-\frac {5293 \sqrt {1-2 x}}{43218 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac {\sqrt {1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}+\frac {5293 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{43218}\\ &=-\frac {5293 \sqrt {1-2 x}}{18522 (2+3 x)^2}-\frac {5293 \sqrt {1-2 x}}{43218 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac {\sqrt {1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}-\frac {5293 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{43218}\\ &=-\frac {5293 \sqrt {1-2 x}}{18522 (2+3 x)^2}-\frac {5293 \sqrt {1-2 x}}{43218 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^2}{105 (2+3 x)^5}+\frac {\sqrt {1-2 x} (1255+1971 x)}{6615 (2+3 x)^4}-\frac {5293 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21609 \sqrt {21}}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 70, normalized size = 0.58 \begin {gather*} \frac {-\frac {21 \sqrt {1-2 x} \left (816938+4450198 x+8806422 x^2+7383735 x^3+2143665 x^4\right )}{2 (2+3 x)^5}-26465 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2268945} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-21*Sqrt[1 - 2*x]*(816938 + 4450198*x + 8806422*x^2 + 7383735*x^3 + 2143665*x^4))/(2*(2 + 3*x)^5) - 26465*Sq
rt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/2268945

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Maple [A]
time = 0.12, size = 75, normalized size = 0.62

method	result	size
risch	\(\frac {4287330 x^{5}[phone] x^{4}[phone] x^{3}+93974 x^{2}-2816322 x -816938}{216090 \left (2+3 x \right )^{5} \sqrt {1-2 x}}-\frac {5293 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{453789}\)	\(61\)
derivativedivides	\(\frac {\frac {47637 \left (1-2 x \right )^{\frac {9}{2}}}{2401}-\frac {10586 \left (1-2 x \right )^{\frac {7}{2}}}{49}+\frac {628504 \left (1-2 x \right )^{\frac {5}{2}}}{735}-\frac {648694 \left (1-2 x \right )^{\frac {3}{2}}}{441}+\frac {58781 \sqrt {1-2 x}}{63}}{\left (-4-6 x \right )^{5}}-\frac {5293 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{453789}\)	\(75\)
default	\(\frac {\frac {47637 \left (1-2 x \right )^{\frac {9}{2}}}{2401}-\frac {10586 \left (1-2 x \right )^{\frac {7}{2}}}{49}+\frac {628504 \left (1-2 x \right )^{\frac {5}{2}}}{735}-\frac {648694 \left (1-2 x \right )^{\frac {3}{2}}}{441}+\frac {58781 \sqrt {1-2 x}}{63}}{\left (-4-6 x \right )^{5}}-\frac {5293 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{453789}\)	\(75\)
trager	\(-\frac {\left (2143665 x^{4}+7383735 x^{3}+8806422 x^{2}+4450198 x +816938\right ) \sqrt {1-2 x}}{216090 \left (2+3 x \right )^{5}}-\frac {5293 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{907578}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1944*(5293/518616*(1-2*x)^(9/2)-5293/47628*(1-2*x)^(7/2)+78563/178605*(1-2*x)^(5/2)-324347/428652*(1-2*x)^(3/2
)+58781/122472*(1-2*x)^(1/2))/(-4-6*x)^5-5293/453789*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]
time = 0.53, size = 128, normalized size = 1.07 \begin {gather*} \frac {5293}{907578} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2143665 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 23342130 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 92390088 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 158930030 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 100809415 \, \sqrt {-2 \, x + 1}}{108045 \, {\left (243 \, {\left (2 \, x - 1\right )}^{5} + 2835 \, {\left (2 \, x - 1\right )}^{4} + 13230 \, {\left (2 \, x - 1\right )}^{3} + 30870 \, {\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5293/907578*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/108045*(2143665*(-2
*x + 1)^(9/2) - 23342130*(-2*x + 1)^(7/2) + 92390088*(-2*x + 1)^(5/2) - 158930030*(-2*x + 1)^(3/2) + 100809415
*sqrt(-2*x + 1))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208
)

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Fricas [A]
time = 1.29, size = 114, normalized size = 0.95 \begin {gather*} \frac {26465 \, \sqrt {21} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (2143665 \, x^{4} + 7383735 \, x^{3} + 8806422 \, x^{2} + 4450198 \, x + 816938\right )} \sqrt {-2 \, x + 1}}{4537890 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4537890*(26465*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x + sqrt(21)*sqrt(-2*x
+ 1) - 5)/(3*x + 2)) - 21*(2143665*x^4 + 7383735*x^3 + 8806422*x^2 + 4450198*x + 816938)*sqrt(-2*x + 1))/(243*
x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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

[Out]

Timed out

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Giac [A]
time = 0.72, size = 116, normalized size = 0.97 \begin {gather*} \frac {5293}{907578} \, \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 {2143665 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 23342130 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 92390088 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 158930030 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 100809415 \, \sqrt {-2 \, x + 1}}{3457440 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5293/907578*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/3457440*(2
143665*(2*x - 1)^4*sqrt(-2*x + 1) + 23342130*(2*x - 1)^3*sqrt(-2*x + 1) + 92390088*(2*x - 1)^2*sqrt(-2*x + 1)
- 158930030*(-2*x + 1)^(3/2) + 100809415*sqrt(-2*x + 1))/(3*x + 2)^5

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Mupad [B]
time = 0.07, size = 108, normalized size = 0.90 \begin {gather*} -\frac {5293\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{453789}-\frac {\frac {58781\,\sqrt {1-2\,x}}{15309}-\frac {648694\,{\left (1-2\,x\right )}^{3/2}}{107163}+\frac {628504\,{\left (1-2\,x\right )}^{5/2}}{178605}-\frac {10586\,{\left (1-2\,x\right )}^{7/2}}{11907}+\frac {5293\,{\left (1-2\,x\right )}^{9/2}}{64827}}{\frac {24010\,x}{81}+\frac {3430\,{\left (2\,x-1\right )}^2}{27}+\frac {490\,{\left (2\,x-1\right )}^3}{9}+\frac {35\,{\left (2\,x-1\right )}^4}{3}+{\left (2\,x-1\right )}^5-\frac {19208}{243}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (5293*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/453789 - ((58781*(1 - 2*x)^(1/2))/15309 - (648694*(1 - 2
*x)^(3/2))/107163 + (628504*(1 - 2*x)^(5/2))/178605 - (10586*(1 - 2*x)^(7/2))/11907 + (5293*(1 - 2*x)^(9/2))/6
4827)/((24010*x)/81 + (3430*(2*x - 1)^2)/27 + (490*(2*x - 1)^3)/9 + (35*(2*x - 1)^4)/3 + (2*x - 1)^5 - 19208/2
43)

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