3.21.23 \(\int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^6} \, dx\) [2023]

Optimal. Leaf size=130 \[ -\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}-\frac {117 \sqrt {1-2 x}}{980 (2+3 x)^3}-\frac {117 \sqrt {1-2 x}}{2744 (2+3 x)^2}-\frac {351 \sqrt {1-2 x}}{19208 (2+3 x)}-\frac {117 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604} \]

[Out]

-117/67228*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1/315*(1-2*x)^(1/2)/(2+3*x)^5+341/8820*(1-2*x)^(1/2)/(
2+3*x)^4-117/980*(1-2*x)^(1/2)/(2+3*x)^3-117/2744*(1-2*x)^(1/2)/(2+3*x)^2-351/19208*(1-2*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.03, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {91, 79, 44, 65, 212} \begin {gather*} -\frac {351 \sqrt {1-2 x}}{19208 (3 x+2)}-\frac {117 \sqrt {1-2 x}}{2744 (3 x+2)^2}-\frac {117 \sqrt {1-2 x}}{980 (3 x+2)^3}+\frac {341 \sqrt {1-2 x}}{8820 (3 x+2)^4}-\frac {\sqrt {1-2 x}}{315 (3 x+2)^5}-\frac {117 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/315*Sqrt[1 - 2*x]/(2 + 3*x)^5 + (341*Sqrt[1 - 2*x])/(8820*(2 + 3*x)^4) - (117*Sqrt[1 - 2*x])/(980*(2 + 3*x)
^3) - (117*Sqrt[1 - 2*x])/(2744*(2 + 3*x)^2) - (351*Sqrt[1 - 2*x])/(19208*(2 + 3*x)) - (117*Sqrt[3/7]*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/9604

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 {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^6} \, dx &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {1}{315} \int \frac {1409+2625 x}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}+\frac {351}{140} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}-\frac {117 \sqrt {1-2 x}}{980 (2+3 x)^3}+\frac {117}{196} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}-\frac {117 \sqrt {1-2 x}}{980 (2+3 x)^3}-\frac {117 \sqrt {1-2 x}}{2744 (2+3 x)^2}+\frac {351 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{2744}\\ &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}-\frac {117 \sqrt {1-2 x}}{980 (2+3 x)^3}-\frac {117 \sqrt {1-2 x}}{2744 (2+3 x)^2}-\frac {351 \sqrt {1-2 x}}{19208 (2+3 x)}+\frac {351 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{19208}\\ &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}-\frac {117 \sqrt {1-2 x}}{980 (2+3 x)^3}-\frac {117 \sqrt {1-2 x}}{2744 (2+3 x)^2}-\frac {351 \sqrt {1-2 x}}{19208 (2+3 x)}-\frac {351 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{19208}\\ &=-\frac {\sqrt {1-2 x}}{315 (2+3 x)^5}+\frac {341 \sqrt {1-2 x}}{8820 (2+3 x)^4}-\frac {117 \sqrt {1-2 x}}{980 (2+3 x)^3}-\frac {117 \sqrt {1-2 x}}{2744 (2+3 x)^2}-\frac {351 \sqrt {1-2 x}}{19208 (2+3 x)}-\frac {117 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9604}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 70, normalized size = 0.54 \begin {gather*} \frac {-\frac {7 \sqrt {1-2 x} \left (298748+1327058 x+2110212 x^2+1468935 x^3+426465 x^4\right )}{2 (2+3 x)^5}-1755 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1008420} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*Sqrt[1 - 2*x]*(298748 + 1327058*x + 2110212*x^2 + 1468935*x^3 + 426465*x^4))/(2*(2 + 3*x)^5) - 1755*Sqrt[
21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/1008420

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Maple [A]
time = 0.12, size = 75, normalized size = 0.58

method	result	size
risch	\(\frac {852930 x^{5}+2511405 x^{4}+2751489 x^{3}+543904 x^{2}-729562 x -298748}{288120 \left (2+3 x \right )^{5} \sqrt {1-2 x}}-\frac {117 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{67228}\)	\(61\)
derivativedivides	\(-\frac {3888 \left (-\frac {117 \left (1-2 x \right )^{\frac {9}{2}}}{153664}+\frac {13 \left (1-2 x \right )^{\frac {7}{2}}}{1568}-\frac {26 \left (1-2 x \right )^{\frac {5}{2}}}{735}+\frac {77587 \left (1-2 x \right )^{\frac {3}{2}}}{1143072}-\frac {5287 \sqrt {1-2 x}}{108864}\right )}{\left (-4-6 x \right )^{5}}-\frac {117 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{67228}\)	\(75\)
default	\(-\frac {3888 \left (-\frac {117 \left (1-2 x \right )^{\frac {9}{2}}}{153664}+\frac {13 \left (1-2 x \right )^{\frac {7}{2}}}{1568}-\frac {26 \left (1-2 x \right )^{\frac {5}{2}}}{735}+\frac {77587 \left (1-2 x \right )^{\frac {3}{2}}}{1143072}-\frac {5287 \sqrt {1-2 x}}{108864}\right )}{\left (-4-6 x \right )^{5}}-\frac {117 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{67228}\)	\(75\)
trager	\(-\frac {\left (426465 x^{4}+1468935 x^{3}+2110212 x^{2}+1327058 x +298748\right ) \sqrt {1-2 x}}{288120 \left (2+3 x \right )^{5}}+\frac {117 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{134456}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3888*(-117/153664*(1-2*x)^(9/2)+13/1568*(1-2*x)^(7/2)-26/735*(1-2*x)^(5/2)+77587/1143072*(1-2*x)^(3/2)-5287/1
08864*(1-2*x)^(1/2))/(-4-6*x)^5-117/67228*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 = 0.98 \begin {gather*} \frac {117}{134456} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {426465 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 4643730 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 19813248 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 38017630 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 27201615 \, \sqrt {-2 \, x + 1}}{144060 \, {\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)^2/(2+3*x)^6/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

117/134456*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/144060*(426465*(-2*x
+ 1)^(9/2) - 4643730*(-2*x + 1)^(7/2) + 19813248*(-2*x + 1)^(5/2) - 38017630*(-2*x + 1)^(3/2) + 27201615*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 = 0.90, size = 120, normalized size = 0.92 \begin {gather*} \frac {1755 \, \sqrt {7} \sqrt {3} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (426465 \, x^{4} + 1468935 \, x^{3} + 2110212 \, x^{2} + 1327058 \, x + 298748\right )} \sqrt {-2 \, x + 1}}{2016840 \, {\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)^2/(2+3*x)^6/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/2016840*(1755*sqrt(7)*sqrt(3)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((sqrt(7)*sqrt(3)*sqr
t(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 7*(426465*x^4 + 1468935*x^3 + 2110212*x^2 + 1327058*x + 298748)*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)**2/(2+3*x)**6/(1-2*x)**(1/2),x)

[Out]

Timed out

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Giac [A]
time = 0.81, size = 116, normalized size = 0.89 \begin {gather*} \frac {117}{134456} \, \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 {426465 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 4643730 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 19813248 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 38017630 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 27201615 \, \sqrt {-2 \, x + 1}}{4609920 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

117/134456*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/4609920*(42
6465*(2*x - 1)^4*sqrt(-2*x + 1) + 4643730*(2*x - 1)^3*sqrt(-2*x + 1) + 19813248*(2*x - 1)^2*sqrt(-2*x + 1) - 3
8017630*(-2*x + 1)^(3/2) + 27201615*sqrt(-2*x + 1))/(3*x + 2)^5

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Mupad [B]
time = 1.20, size = 108, normalized size = 0.83 \begin {gather*} -\frac {117\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{67228}-\frac {\frac {5287\,\sqrt {1-2\,x}}{6804}-\frac {77587\,{\left (1-2\,x\right )}^{3/2}}{71442}+\frac {416\,{\left (1-2\,x\right )}^{5/2}}{735}-\frac {13\,{\left (1-2\,x\right )}^{7/2}}{98}+\frac {117\,{\left (1-2\,x\right )}^{9/2}}{9604}}{\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)^2/((1 - 2*x)^(1/2)*(3*x + 2)^6),x)

[Out]

- (117*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/67228 - ((5287*(1 - 2*x)^(1/2))/6804 - (77587*(1 - 2*x)^(
3/2))/71442 + (416*(1 - 2*x)^(5/2))/735 - (13*(1 - 2*x)^(7/2))/98 + (117*(1 - 2*x)^(9/2))/9604)/((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 - 19208/243)

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