3.2022 \(\int \frac{(3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^5} \, dx\)

Optimal. Leaf size=108 \[ -\frac{635 \sqrt{1-2 x}}{8232 (3 x+2)}-\frac{635 \sqrt{1-2 x}}{3528 (3 x+2)^2}+\frac{13 \sqrt{1-2 x}}{252 (3 x+2)^3}-\frac{\sqrt{1-2 x}}{252 (3 x+2)^4}-\frac{635 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{4116 \sqrt{21}} \]

[Out]

-Sqrt[1 - 2*x]/(252*(2 + 3*x)^4) + (13*Sqrt[1 - 2*x])/(252*(2 + 3*x)^3) - (635*Sqrt[1 - 2*x])/(3528*(2 + 3*x)^
2) - (635*Sqrt[1 - 2*x])/(8232*(2 + 3*x)) - (635*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(4116*Sqrt[21])

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Rubi [A]  time = 0.0291196, antiderivative size = 108, normalized size of antiderivative = 1., 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 = {89, 78, 51, 63, 206} \[ -\frac{635 \sqrt{1-2 x}}{8232 (3 x+2)}-\frac{635 \sqrt{1-2 x}}{3528 (3 x+2)^2}+\frac{13 \sqrt{1-2 x}}{252 (3 x+2)^3}-\frac{\sqrt{1-2 x}}{252 (3 x+2)^4}-\frac{635 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{4116 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[1 - 2*x]/(252*(2 + 3*x)^4) + (13*Sqrt[1 - 2*x])/(252*(2 + 3*x)^3) - (635*Sqrt[1 - 2*x])/(3528*(2 + 3*x)^
2) - (635*Sqrt[1 - 2*x])/(8232*(2 + 3*x)) - (635*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(4116*Sqrt[21])

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)^5} \, dx &=-\frac{\sqrt{1-2 x}}{252 (2+3 x)^4}+\frac{1}{252} \int \frac{1127+2100 x}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=-\frac{\sqrt{1-2 x}}{252 (2+3 x)^4}+\frac{13 \sqrt{1-2 x}}{252 (2+3 x)^3}+\frac{635}{252} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{\sqrt{1-2 x}}{252 (2+3 x)^4}+\frac{13 \sqrt{1-2 x}}{252 (2+3 x)^3}-\frac{635 \sqrt{1-2 x}}{3528 (2+3 x)^2}+\frac{635 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{1176}\\ &=-\frac{\sqrt{1-2 x}}{252 (2+3 x)^4}+\frac{13 \sqrt{1-2 x}}{252 (2+3 x)^3}-\frac{635 \sqrt{1-2 x}}{3528 (2+3 x)^2}-\frac{635 \sqrt{1-2 x}}{8232 (2+3 x)}+\frac{635 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{8232}\\ &=-\frac{\sqrt{1-2 x}}{252 (2+3 x)^4}+\frac{13 \sqrt{1-2 x}}{252 (2+3 x)^3}-\frac{635 \sqrt{1-2 x}}{3528 (2+3 x)^2}-\frac{635 \sqrt{1-2 x}}{8232 (2+3 x)}-\frac{635 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{8232}\\ &=-\frac{\sqrt{1-2 x}}{252 (2+3 x)^4}+\frac{13 \sqrt{1-2 x}}{252 (2+3 x)^3}-\frac{635 \sqrt{1-2 x}}{3528 (2+3 x)^2}-\frac{635 \sqrt{1-2 x}}{8232 (2+3 x)}-\frac{635 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{4116 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0158695, size = 47, normalized size = 0.44 \[ \frac{\sqrt{1-2 x} \left (\frac{343 (39 x+25)}{(3 x+2)^4}-5080 \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{86436} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*((343*(25 + 39*x))/(2 + 3*x)^4 - 5080*Hypergeometric2F1[1/2, 3, 3/2, 3/7 - (6*x)/7]))/86436

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Maple [A]  time = 0.011, size = 66, normalized size = 0.6 \begin{align*} 648\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{635\, \left ( 1-2\,x \right ) ^{7/2}}{98784}}-{\frac{6985\, \left ( 1-2\,x \right ) ^{5/2}}{127008}}+{\frac{2717\, \left ( 1-2\,x \right ) ^{3/2}}{18144}}-{\frac{7171\,\sqrt{1-2\,x}}{54432}} \right ) }-{\frac{635\,\sqrt{21}}{86436}{\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)^5/(1-2*x)^(1/2),x)

[Out]

648*(635/98784*(1-2*x)^(7/2)-6985/127008*(1-2*x)^(5/2)+2717/18144*(1-2*x)^(3/2)-7171/54432*(1-2*x)^(1/2))/(-6*
x-4)^4-635/86436*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.77567, size = 149, normalized size = 1.38 \begin{align*} \frac{635}{172872} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{17145 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 146685 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 399399 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 351379 \, \sqrt{-2 \, x + 1}}{4116 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

635/172872*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/4116*(17145*(-2*x +
1)^(7/2) - 146685*(-2*x + 1)^(5/2) + 399399*(-2*x + 1)^(3/2) - 351379*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2
*x - 1)^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 1.64341, size = 297, normalized size = 2.75 \begin{align*} \frac{635 \, \sqrt{21}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (17145 \, x^{3} + 47625 \, x^{2} + 39366 \, x + 10190\right )} \sqrt{-2 \, x + 1}}{172872 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/172872*(635*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x +
2)) - 21*(17145*x^3 + 47625*x^2 + 39366*x + 10190)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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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)**5/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.22074, size = 135, normalized size = 1.25 \begin{align*} \frac{635}{172872} \, \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{17145 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 146685 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 399399 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 351379 \, \sqrt{-2 \, x + 1}}{65856 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

635/172872*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/65856*(1714
5*(2*x - 1)^3*sqrt(-2*x + 1) + 146685*(2*x - 1)^2*sqrt(-2*x + 1) - 399399*(-2*x + 1)^(3/2) + 351379*sqrt(-2*x
+ 1))/(3*x + 2)^4