3.19.84 \(\int \frac {(3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^6} \, dx\)

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

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Rubi [A]  time = 0.04, 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 = {89, 78, 51, 63, 206} \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]

-Sqrt[1 - 2*x]/(315*(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 51

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 63

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 78

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] || LtQ
[p, n]))))

Rule 89

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 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 \operatorname {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 [C]  time = 0.03, size = 47, normalized size = 0.36 \begin {gather*} \frac {\sqrt {1-2 x} \left (\frac {1029 (341 x+218)}{(3 x+2)^5}-50544 \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{3025260} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*((1029*(218 + 341*x))/(2 + 3*x)^5 - 50544*Hypergeometric2F1[1/2, 4, 3/2, 3/7 - (6*x)/7]))/30252
60

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IntegrateAlgebraic [A]  time = 0.28, size = 90, normalized size = 0.69 \begin {gather*} \frac {\left (426465 (1-2 x)^4-4643730 (1-2 x)^3+19813248 (1-2 x)^2-38017630 (1-2 x)[phone]\right ) \sqrt {1-2 x}}{144060 (3 (1-2 x)-7)^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]

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

[Out]

((27201615 - 38017630*(1 - 2*x) + 19813248*(1 - 2*x)^2 - 4643730*(1 - 2*x)^3 + 426465*(1 - 2*x)^4)*Sqrt[1 - 2*
x])/(144060*(-7 + 3*(1 - 2*x))^5) - (117*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/9604

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fricas [A]  time = 1.58, 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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giac [A]  time = 1.02, 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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maple [A]  time = 0.01, size = 75, normalized size = 0.58 \begin {gather*} -\frac {117 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{67228}-\frac {3888 \left (-\frac {117 \left (-2 x +1\right )^{\frac {9}{2}}}{153664}+\frac {13 \left (-2 x +1\right )^{\frac {7}{2}}}{1568}-\frac {26 \left (-2 x +1\right )^{\frac {5}{2}}}{735}+\frac {77587 \left (-2 x +1\right )^{\frac {3}{2}}}{1143072}-\frac {5287 \sqrt {-2 x +1}}{108864}\right )}{\left (-6 x -4\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3888*(-117/153664*(-2*x+1)^(9/2)+13/1568*(-2*x+1)^(7/2)-26/735*(-2*x+1)^(5/2)+77587/1143072*(-2*x+1)^(3/2)-52
87/108864*(-2*x+1)^(1/2))/(-6*x-4)^5-117/67228*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.32, 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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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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sympy [F(-1)]  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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