3.2023 \(\int \frac{(3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^6} \, dx\)

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

[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

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Rubi [A]  time = 0.0382197, antiderivative size = 130, normalized size of antiderivative = 1., 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} \[ -\frac{351 \sqrt{1-2 x}}{19208 (3 x+2)}-\frac{117 \sqrt{1-2 x}}{2744 (3 x+2)^2}-\frac{117 \sqrt{1-2 x}}{980 (3 x+2)^3}+\frac{341 \sqrt{1-2 x}}{8820 (3 x+2)^4}-\frac{\sqrt{1-2 x}}{315 (3 x+2)^5}-\frac{117 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9604} \]

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 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 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 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 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*}

Mathematica [C]  time = 0.0178667, size = 47, normalized size = 0.36 \[ \frac{\sqrt{1-2 x} \left (\frac{1029 (341 x+218)}{(3 x+2)^5}-50544 \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{3025260} \]

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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Maple [A]  time = 0.009, size = 75, normalized size = 0.6 \begin{align*} -3888\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ( -{\frac{117\, \left ( 1-2\,x \right ) ^{9/2}}{153664}}+{\frac{13\, \left ( 1-2\,x \right ) ^{7/2}}{1568}}-{\frac{26\, \left ( 1-2\,x \right ) ^{5/2}}{735}}+{\frac{77587\, \left ( 1-2\,x \right ) ^{3/2}}{1143072}}-{\frac{5287\,\sqrt{1-2\,x}}{108864}} \right ) }-{\frac{117\,\sqrt{21}}{67228}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

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)

[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))/(-6*x-4)^5-117/67228*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 3.12548, size = 173, normalized size = 1.33 \begin{align*} \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{align*}

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 = 1.63692, size = 379, normalized size = 2.92 \begin{align*} \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{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Giac [A]  time = 2.19724, size = 157, normalized size = 1.21 \begin{align*} \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{align*}

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